Assignment 11 Due: 11:59pm on Wednesday, April 30, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 13.2 The gravitational force of a star on orbiting planet 1 is F1 . Planet 2, which is twice as massive as planet 1 and orbits at twice the distance from the star, experiences gravitational force F2 . Part A What is the ratio F1 ? F2 ANSWER: F1 = 2 F2 Correct Conceptual Question 13.3 A 1500kg satellite and a 2200kg satellite follow exactly the same orbit around the earth. Part A What is the ratio ANSWER: F1 = 0.682 F2 Correct F1 of the force on the first satellite to that on the second satellite? F2 Part B a1 What is the ratio a of the acceleration of the first satellite to that of the second satellite? 2 ANSWER: a1 a2 = 1 Correct Problem 13.2 The centers of a 15.0kg lead ball and a 90.0g lead ball are separated by 9.00cm . Part A What gravitational force does each exert on the other? Express your answer with the appropriate units. ANSWER: 1.11×10 −8 N Correct Part B What is the ratio of this gravitational force to the weight of the 90.0g ball? ANSWER: 1.26×10−8 Typesetting math: 100% Correct Problem 13.6 The space shuttle orbits 310km above the surface of the earth. Part A What is the gravitational force on a 7.5kg sphere inside the space shuttle? Express your answer with the appropriate units. ANSWER: Fe on s = 67.0 N Correct ± A Satellite in Orbit A satellite used in a cellular telephone network has a mass of 2310kg and is in a circular orbit at a height of 650km above the surface of the earth. Part A What is the gravitational force Fgrav on the satellite? Take the gravitational constant to be G = 6.67×10−11N ⋅ m2 /kg2 , the mass of the earth to be me = 5.97×1024kg , and the radius of the Earth to be re = 6.38×106m . Express your answer in newtons. Hint 1. How to approach the problem Use the equation for the law of gravitation to calculate the force on the satellite. Be careful about the units when performing the calculations. Typesetting math: 100% Hint 2. Law of gravitation According to Newton's law of gravitation, F = Gm1 m2 /r2 , where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of mass of the two objects. Hint 3. Calculate the distance between the centers of mass What is the distance r from the center of mass of the satellite to the center of mass of the earth? Express your answer in meters. ANSWER: r = 7.03×106 m ANSWER: Fgrav = 1.86×104 N Correct Part B What fraction is this of the satellite's weight at the surface of the earth? Take the free-fall acceleration at the surface of the earth to be g = 9.80m/s2 . Hint 1. How to approach the problem All you need to do is to take the ratio of the gravitational force on the satellite to the weight of the satellite at ground level. There are two ways to do this, depending on how you define the force of gravity at the surface of the earth. ANSWER: 0.824 Typesetting math: 100% Correct Although it is easy to find the weight of the satellite using the constant acceleration due to gravity, it is instructional to consider the weight calculated using the law of gravitation: w = Gme m/r2e. Dividing the gravitational force on the satellite Fgrav = Gme m/(re + h)2 by w, we find that the ratio of the forces due to the earth's gravity is simply the square of the ratio of the earth's radius to the sum of the earth's radius and the height of the orbit of the satellite above the earth, [re /(re + h)]2. This will also be the fraction of the weight of, say, an astronaut in an orbit at the same altitude. Notice that an astronaut's weight is never zero. When people speak of "weightlessness" in space, what they really mean is "free fall." Problem 13.8 Part A What is the free-fall acceleration at the surface of the moon? Express your answer with the appropriate units. ANSWER: m gmoon = 1.62 s 2 Correct Part B What is the free-fall acceleration at the surface of the Jupiter? Express your answer with the appropriate units. ANSWER: gJupiter = 25.9 m2 s Correct Typesetting math: 100% Enhanced EOC: Problem 13.14 A rocket is launched straight up from the earth's surface at a speed of 1.90×104m/s . You may want to review ( pages 362 - 365) . For help with math skills, you may want to review: Mathematical Expressions Involving Squares Part A What is its speed when it is very far away from the earth? Express your answer with the appropriate units. Hint 1. How to approach the problem What is conserved in this problem? What is the rocket's initial kinetic energy in terms of its unknown mass, m? What is the rocket's initial gravitational potential energy in terms of its unknown mass, m? When the rocket is very far away from the Earth, what is its gravitational potential energy? Using conservation of energy, what is the rocket's kinetic energy when it is very far away from the Earth? Therefore, what is the rocket's velocity when it is very far away from the Earth? ANSWER: m 1.54×104 s Correct Problem 13.13 Part A Typesetting math: 100% What is the escape speed from Venus? Express your answer with the appropriate units. ANSWER: vescape = 10.4 km s Correct Problem 13.17 The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 4.2 earth years. Part A What is the asteroid's orbital radius? Express your answer with the appropriate units. ANSWER: R = 3.89×1011 m Correct Part B What is the asteroid's orbital speed? Express your answer with the appropriate units. ANSWER: v = 1.85×104 m s Typesetting math: 100% Correct Problem 13.32 Part A At what height above the earth is the acceleration due to gravity 15.0% of its value at the surface? Express your answer with the appropriate units. ANSWER: 7 1.01×10 m Correct Part B What is the speed of a satellite orbiting at that height? Express your answer with the appropriate units. ANSWER: m 4920 s Correct Problem 13.36 Two meteoroids are heading for earth. Their speeds as they cross the moon's orbit are 2km/s . Typesetting math: 100% Part A The first meteoroid is heading straight for earth. What is its speed of impact? Express your answer with the appropriate units. ANSWER: v1 = 11.3 km s Correct Part B The second misses the earth by 5500km . What is its speed at its closest point? Express your answer with the appropriate units. ANSWER: v2 = Incorrect; Try Again Problem 14.2 An air-track glider attached to a spring oscillates between the 11.0cm mark and the 67.0cm mark on the track. The glider completes 11.0 oscillations in 32.0s . Part A What is the period of the oscillations? Express your answer with the appropriate units. Typesetting math: 100% ANSWER: 2.91 s Correct Part B What is the frequency of the oscillations? Express your answer with the appropriate units. ANSWER: 0.344 Hz Correct Part C What is the angular frequency of the oscillations? Express your answer with the appropriate units. ANSWER: 2.16 rad s Correct Part D What is the amplitude? Express your answer with the appropriate units. Typesetting math: 100% ANSWER: 28.0 cm Correct Part E What is the maximum speed of the glider? Express your answer with the appropriate units. ANSWER: cm 60.5 s Correct Good Vibes: Introduction to Oscillations Learning Goal: To learn the basic terminology and relationships among the main characteristics of simple harmonic motion. Motion that repeats itself over and over is called periodic motion. There are many examples of periodic motion: the earth revolving around the sun, an elastic ball bouncing up and down, or a block attached to a spring oscillating back and forth. The last example differs from the first two, in that it represents a special kind of periodic motion called simple harmonic motion. The conditions that lead to simple harmonic motion are as follows: There must be a position of stable equilibrium. There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F ⃗ is given by constant that depends on the properties of the oscillating system. The resistive forces in the system must be reasonably small. F ⃗ = −kx,⃗ where x⃗ is the displacement from equilibrium and k is a In this problem, we will introduce some of the basic quantities that describe oscillations and the relationships among them. Consider a block of mass m attached to a spring with force constant k, as shown in the figure. The spring can be either stretched or compressed. The100% block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x = 0. If the Typesetting math: block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from x = A, it will ANSWER: remain at rest. move to the left until it reaches equilibrium and stop there. x = −A and stop there. move to the left until it reaches x = −A and then begin to move to the right. move to the left until it reaches Correct As the block begins its motion to the left, it accelerates. Although the restoring force decreases as the block approaches equilibrium, it still pulls the block to the left, so by the time the equilibrium position is reached, the block has gained some speed. It will, therefore, pass the equilibrium position and keep moving, compressing the spring. The spring will now be pushing the block to the right, and the block will slow down, temporarily coming to rest at x = −A. After x = −A is reached, the block will begin its motion to the right, pushed by the spring. The block will pass the equilibrium position and continue until it reaches cycle of motion. The motion will then repeat; if, as we've assumed, there is no friction, the motion will repeat indefinitely. The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds. The frequency, denoted f , is the number of cycles that are completed per unit of time: Typesetting math: 100% f = 1/T . In SI units, f is measured in inverse seconds, or hertz (Hz). x = A, completing one Part B If the period is doubled, the frequency is ANSWER: unchanged. doubled. halved. Correct Part C An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f ? Express your answer in hertz. ANSWER: f = 10 Hz Correct Typesetting math: 100% Part D If the frequency is 40 Hz, what is the period T ? Express your answer in seconds. ANSWER: T = 0.025 s Correct The following questions refer to the figure that graphically depicts the oscillations of the block on the spring. Note that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents time. Part E Which points on the x axis are located a distance A from the equilibrium position? ANSWER: Typesetting math: 100% R only Q only both R and Q Correct Part F Suppose that the period is T . Which of the following points on the t axis are separated by the time interval T ? ANSWER: K and L K and M K and P L and N M and P Correct Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s. Part G What is the period T ? Express your answer in seconds. Hint 1. How to approach the problem In moving from the point Typesetting math: 100% t = 0 to the point K, what fraction of a full wavelength is covered? Call that fraction a. Then you can set aT = 0.005 s. Dividing by the fraction a will give the period T . ANSWER: T = 0.02 s Correct Part H How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement? Express your answer in seconds. ANSWER: t = 0.01 s Correct Part I What distance d does the object cover during one period of oscillation? Express your answer in meters. ANSWER: d = 0.48 m Correct Part J What distance d does the object cover between the moments labeled K and N on the graph? Typesetting math: 100% Express your answer in meters. ANSWER: d = 0.36 m Correct Problem 14.4 Part A What is the amplitude of the oscillation shown in the figure? Express your answer to three significant figures and include the appropriate units. ANSWER: A = 20.0 cm Correct Typesetting math: 100% Part B What is the frequency of this oscillation? Express your answer to two significant figures and include the appropriate units. ANSWER: f = 0.25 Hz Correct Part C What is the phase constant? Express your answer to two significant figures and include the appropriate units. ANSWER: ϕ0 = Incorrect; Try Again Problem 14.10 An air-track glider attached to a spring oscillates with a period of 1.50s . At t = 0 s the glider is 4.60cm left of the equilibrium position and moving to the right at 33.4cm/s . Part A What is the phase constant? Express your answer to three significant figures and include the appropriate units. ANSWER: Typesetting math: 100% ϕ0 = Incorrect; Try Again Part B This question will be shown after you complete previous question(s). Part C This question will be shown after you complete previous question(s). Part D This question will be shown after you complete previous question(s). Problem 14.12 A 140g air-track glider is attached to a spring. The glider is pushed in 12.2cm and released. A student with a stopwatch finds that 14.0 oscillations take 19.0s . Part A What is the spring constant? Express your answer with the appropriate units. ANSWER: Typesetting math: 100% N 3.00 m Correct Problem 14.14 The position of a 50 g oscillating mass is given by x(t) = (2.0 cm)cos(10t − π/4), where t is in s. If necessary, round your answers to three significant figures. Determine: Part A The amplitude. Express your answer to three significant figures and include the appropriate units. ANSWER: 2.00 cm Correct Part B The period. Express your answer to three significant figures and include the appropriate units. ANSWER: 0.628 s Correct Part C Typesetting math: 100% The spring constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Part D The phase constant. Express your answer to three significant figures and include the appropriate units. ANSWER: Incorrect; Try Again Part E This question will be shown after you complete previous question(s). Part F This question will be shown after you complete previous question(s). Part G Typesetting math: 100% This question will be shown after you complete previous question(s). Part H This question will be shown after you complete previous question(s). Part I This question will be shown after you complete previous question(s). Enhanced EOC: Problem 14.17 A spring with spring constant 16N/m hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 4.0cm and released. The ball makes 35 oscillations in 18s seconds. You may want to review ( pages 389 - 391) . For help with math skills, you may want to review: Differentiation of Trigonometric Functions Part A What is its the mass of the ball? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the period of oscillation? What is the angular frequency of the oscillations? How is the angular frequency related to the mass and spring constant? What is the mass? Typesetting math: 100% ANSWER: m = 110 g Correct Part B What is its maximum speed? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the amplitude of the oscillations? How is the maximum speed related to the amplitude of the oscillations and the angular frequency? ANSWER: vmax = 49 cm s Correct Changing the Period of a Pendulum A simple pendulum consisting of a bob of mass m attached to a string of length L swings with a period T . Part A If the bob's mass is doubled, approximately what will the pendulum's new period be? Hint 1. Period of a simple pendulum Typesetting The math: period 100% T of a simple pendulum of length L is given by −− T = 2π√ Lg , where g is the acceleration due to gravity. ANSWER: T /2 T √2T 2T Correct Part B If the pendulum is brought on the moon where the gravitational acceleration is about g/6 , approximately what will its period now be? Hint 1. How to approach the problem Recall the formula of the period of a simple pendulum. Since the gravitational acceleration appears in the denominator, the period must increase when the gravitational acceleration decreases. ANSWER: T /6 T /√6 √6T 6T Typesetting math: 100% Correct Part C If the pendulum is taken into the orbiting space station what will happen to the bob? Hint 1. How to approach the problem Recall that the oscillations of a simple pendulum occur when a pendulum bob is raised above its equilibrium position and let go, causing the pendulum bob to fall. The gravitational force acts to bring the bob back to its equilibrium position. In the space station, the earth's gravity acts on both the station and everything inside it, giving them the same acceleration. These objects are said to be in free fall. ANSWER: It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero. Correct In the space station, where all objects undergo the same acceleration due to the earth's gravity, the tension in the string is zero and the bob does not fall relative to the point to which the string is attached. Problem 14.20 A 175g ball is tied to a string. It is pulled to an angle of 8.0� and released to swing as a pendulum. A student with a stopwatch finds that 15 oscillations take 13s . Part A How long is the string? Express your answer to two significant figures and include the appropriate units. Typesetting math: 100% ANSWER: L = 19 cm Correct Problem 14.22 Part A What is the length of a pendulum whose period on the moon matches the period of a 2.1-m-long pendulum on the earth? Express your answer to two significant figures and include the appropriate units. ANSWER: lmoon = 0.35 m Correct Problem 14.42 An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk ( m = 0.17g ) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil. Part A 4 The maximum restoring force that can be applied to the disk without breaking it is 4.4×10 Express your answer to two significant figures and include the appropriate units. ANSWER: amax = 6.6 µm Typesetting math: 100% N . What is the maximum oscillation amplitude that won't rupture the disk? Correct Part B What is the disk's maximum speed at this amplitude? Express your answer to two significant figures and include the appropriate units. ANSWER: vmax = 41 m s Correct Score Summary: Your score on this assignment is 81.4%. You received 117.25 out of a possible total of 144 points. Typesetting math: 100% Assignment 6 Due: 11:59pm on Friday, March 7, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 7.7 A small car is pushing a large truck. They are speeding up. Part A Is the force of the truck on the car larger than, smaller than, or equal to the force of the car on the truck? ANSWER: The force of the truck on the car is larger than the force of the car on the truck. The force of the truck on the car is equal to the force of the car on the truck. The force of the truck on the car is smaller than the force of the car on the truck. Correct Conceptual Question 7.12 The figure shows two masses at rest. The string is massless and the pulley is frictionless. The spring scale reads in kg. Assume that . m = 4kg Part A What is the reading of the scale? Express your answer to one significant figure and include the appropriate units. ANSWER: m = 4 kg Correct Problem 7.1 A weightlifter stands up at constant speed from a squatting position while holding a heavy barbell across his shoulders. Part A Draw a free-body diagram for the barbells. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Correct Part B Draw a free-body diagram for the weight lifter. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Correct Problem 7.6 Block A in the figure is sliding down the incline. The rope is massless, and the massless pulley turns on frictionless bearings, but the surface is not frictionless. The rope and the pulley are among the interacting objects, but you'll have to decide if they're part of the system. Part A Draw a free-body diagram for the block A. The orientation of your vectors will be graded. The exact length of your vectors will not be graded. ANSWER: Correct Part B Draw a free-body diagram for the block B. The orientation of your vectors will be graded. The exact length of your vectors will not be graded. ANSWER: Correct A Space Walk Part A An astronaut is taking a space walk near the shuttle when her safety tether breaks. What should the astronaut do to get back to the shuttle? Hint 1. How to approach the problem Newton's 3rd law tells us that forces occur in pairs. Within each pair, the forces, often called action and reaction, have equal magnitude and opposite direction. Which of the actions suggested in the problem will result in the force pushing the astronaut back to the shuttle? ANSWER: Attempt to "swim" toward the shuttle. Take slow steps toward the shuttle. Take a tool from her tool belt and throw it away from the shuttle. Take the portion of the safety tether still attached to her belt and throw it toward the shuttle. Correct As the astronaut throws the tool away from the shuttle, she exerts a force in the direction away from the shuttle. Then, by Newton's 3rd law, the tool will exert an opposite force on her. Thus, as she throws the tool, a force directed toward the shuttle will act on the astronaut. Newton's 2nd law stipulates that the astronaut would acquire an acceleration toward the shuttle. Part B Assuming that the astronaut can throw any tool with the same force, what tool should be thrown to get back to the shuttle as quickly as possible? You should consider how much mass is left behind as the object is thrown as well as the mass of the object itself. Hint 1. How to approach the problem Recall that the force acting on the astronaut is equal in magnitude and opposite in direction to the force that she exerts on the tool. Hint 2. Newton's 2nd law Newton's 2nd law states that F = ma. If force is held constant, then acceleration is inversely proportional to mass. For example, when the same force is applied to objects of different mass, the object with the largest mass will experience the smallest acceleration. ANSWER: The tool with the smallest mass. The tool with the largest mass. Any tool, since the mass of the tool would make no difference. Correct The force that acts on the astronaut must equal in magnitude the force that she exerts on the tool. Therefore, if she exerts the same force on any tool, the force acting on her will be independent of the mass of the tool. However, the acceleration that the astronaut would acquire is inversely proportional to her mass since she is acted upon by a constant force. If she throws the tool with the largest mass, the remaining mass (the astronaut plus her remaining tools) would be smallest—and the acceleration the greatest! Part C If the astronaut throws the tool with a force of 16.0N , what is the magnitude of the acceleration a of the astronaut during the throw? Assume that the total mass of the astronaut after she throws the tool is 80.0kg . Express your answer in meters per second squared. Hint 1. Find the force acting on the astronaut What is the magnitude of the force F acting on the astronaut as she throws the tool? Express your answer in newtons. ANSWER: F = 16.0 N Hint 2. Newton's 2nd law An object of mass ANSWER: a = 0.200 m/s2 m acted upon by a net force F has an acceleration a given by F = ma. Correct Problem 7.10 Blocks with masses of 2kg , 4kg , and 6kg are lined up in a row on a frictionless table. All three are pushed forward by a 60N force applied to the 2kg block. Part A How much force does the 4kg block exert on the 6kg block? Express your answer to one significant figure and include the appropriate units. ANSWER: F = 30 N Correct Part B How much force does the 4kg block exert on the 2kg block? Express your answer to two significant figures and include the appropriate units. ANSWER: F = 50 N Correct Problem 7.9 A 1000kg car pushes a 2100kg truck that has a dead battery. When the driver steps on the accelerator, the drive wheels of the car push against the ground with a force of 4500N . Rolling friction can be neglected. Part A What is the magnitude of the force of the car on the truck? Express your answer to two significant figures and include the appropriate units. ANSWER: F = 3000 N Correct Part B What is the magnitude of the force of the truck on the car? Express your answer to two significant figures and include the appropriate units. ANSWER: F = 3000 N Correct Atwood Machine Special Cases An Atwood machine consists of two blocks (of masses m1 and m2 ) tied together with a massless rope that passes over a fixed, perfect (massless and frictionless) pulley. In this problem you'll investigate some special cases where physical variables describing the Atwood machine take on limiting values. Often, examining special cases will simplify a problem, so that the solution may be found from inspection or from the results of a problem you've already seen. For all parts of this problem, take upward to be the positive direction and take the gravitational constant, g, to be positive. Part A Consider the case where m1 and m2 are both nonzero, and m2 tension in the rope connected to the block of mass > m1. Let T1 be the magnitude of the tension in the rope connected to the block of mass m1 , and let T2 be the magnitude of the m2 . Which of the following statements is true? ANSWER: T1 is always equal to T2 . T2 is greater than T1 by an amount independent of velocity. T2 is greater than T1 but the difference decreases as the blocks increase in velocity. There is not enough information to determine the relationship between T1 and T2 . Correct Part B Now, consider the special case where the block of mass m1 is not present. Find the magnitude, T , of the tension in the rope. Try to do this without equations; instead, think about the physical consequences. Hint 1. How to approach the problem If the block of mass m1 is not present, and the rope connecting the two blocks is massless, will the motion of the block of mass m2 be any different from free fall? Hint 2. Which physical law to use Use Newton's 2nd law on the block of mass m2 . ANSWER: T= 0 Correct Part C For the same special case (the block of mass m1 not present), what is the acceleration of the block of mass m2 ? Express your answer in terms of g, and remember that an upward acceleration should be positive. ANSWER: a2 = -9.80 Correct Part D Next, consider the special case where only the block of mass m1 is present. Find the magnitude, T , of the tension in the rope. ANSWER: T= 0 Correct Part E For the same special case (the block of mass m2 not present) what is the acceleration of the end of the rope where the block of mass m2 would have been attached? Express your answer in terms of g, and remember that an upward acceleration should be positive. ANSWER: a2 = 9.80 Correct Part F Next, consider the special case m1 = m2 = m. What is the magnitude of the tension in the rope connecting the two blocks? Use the variable m in your answer instead of m1 or m2 . ANSWER: T = mg Correct Part G For the same special case (m1 = m2 = m), what is the acceleration of the block of mass m2 ? ANSWER: a2 = 0 Correct Part H Finally, suppose m1 →∞, while m2 remains finite. What value does the the magnitude of the tension approach? Hint 1. Acceleration of block of mass m1 m1 becomes large, the finite tension T will have a neglible effect on the acceleration, a1 . If you ignore T , you can pretend the rope is gone without changing your results for a1 . As m1 →∞, what value does a1 approach? As ANSWER: a1 = -9.80 Hint 2. Acceleration of block of mass As m2 m1 →∞, what value will the acceleration of the block of mass m2 approach? ANSWER: a2 = 9.80 Hint 3. Net force on block of mass m2 What is the magnitude Fnet of the net force on the block of mass Express your answer in terms of m2 . T , m2 , g, and any other given quantities. Take the upward direction to be positive. ANSWER: Fnet = T − m2 g ANSWER: T = 2m2 g Correct Imagining what would happen if one or more of the variables approached infinity is often a good way to investigate the behavior of a system. Problem 7.17 A 5.9kg rope hangs from the ceiling. Part A What is the tension at the midpoint of the rope? Express your answer to two significant figures and include the appropriate units. ANSWER: T = 29 N Correct Problem 7.23 The sled dog in figure drags sleds A and B across the snow. The coefficient of friction between the sleds and the snow is 0.10. Part A If the tension in rope 1 is 100N , what is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: T2 = 180 N Correct Enhanced EOC: Problem 7.31 Two packages at UPS start sliding down the 20� ramp shown in the figure. Package A has a mass of 4.50kg and a coefficient of kinetic friction of 0.200. Package B has a mass of 11.0kg and a coefficient of kinetic friction of 0.150. You may want to review ( pages 177 - 181) . For help with math skills, you may want to review: Vector Components Part A How long does it take package A to reach the bottom? Express your answer with the appropriate units. Hint 1. How to approach the problem Start by drawing force identification diagrams for package A and package B separately. What are the four forces acting on each block? Which of the forces are related by Newton's third law? Draw separate free-body diagrams for block A and for block B. What is a good coordinate system to use to describe the motion of the blocks down the ramp? Label your coordinate system on the free-body diagram. In your coordinate system, compute the x and y components of each force on block A. What are the x and y components of the net force on block A? What are the x and y components of the net force on block B? Given that the coefficient of friction of block A is greater than the coefficient of friction of block B, do you think the blocks will stay together as they slide down the ramp? Assuming that they do stay together, how is the acceleration of the two blocks related? (We can check this assumption later.) Using the components of the forces and Newton's second law, what is the acceleration of the blocks? What is the initial velocity of the blocks? Given the initial velocity and the acceleration, how long does it take block A to go the given distance? To check that the blocks do indeed stay together, solve for the force of block B on block A. If the force is directed toward the bottom of the ramp, then the blocks stay together. ANSWER: 1.48 s Correct Problem 7.33 The 1.0 kg block in the figure is tied to the wall with a rope. It sits on top of the 2.0 kg block. The lower block is pulled to the right with a tension force of 20 N. The coefficient of kinetic friction at both the lower and upper surfaces of the 2.0 kg block is µk = 0.420. Part A What is the tension in the rope holding the 1.0 kg block to the wall? Express your answer with the appropriate units. ANSWER: 4.12 N Correct Part B What is the acceleration of the 2.0 kg block? Express your answer with the appropriate units. ANSWER: 1.77 m s2 Correct Problem 7.38 The 100 kg block in figure takes 5.60s to reach the floor after being released from rest. Part A What is the mass of the block on the left? Express your answer with the appropriate units. ANSWER: 98.7 kg Correct Problem 7.41 Figure shows a block of mass m resting on a 20� slope. The block has coefficients of friction 0.82 and 0.51 with the surface. It is connected via a massless string over a massless, frictionless pulley to a hanging block of mass 2.0 kg. Part A What is the minimum mass m that will stick and not slip? Express your answer to three significant figures and include the appropriate units. ANSWER: m = 1.80 kg Correct If you need to use the rounded answer you submitted here in a subsequent part, instead use the full precision answer and only round as a final step before submitting an answer. Part B If this minimum mass is nudged ever so slightly, it will start being pulled up the incline. What acceleration will it have? Express your answer to three significant figures and include the appropriate units. ANSWER: m a = 1.35 s 2 Correct Problem 7.46 A house painter uses the chair and pulley arrangement of the figure to lift himself up the side of a house. The painter's mass is 75kg and the chair's mass is 12kg . Part A With what force must he pull down on the rope in order to accelerate upward at 0.22m/s2 ? Express your answer to two significant figures and include the appropriate units. ANSWER: F = 440 N Correct Score Summary: Your score on this assignment is 98.6%. You received 104.5 out of a possible total of 106 points. Assignment 7 Due: 11:59pm on Friday, March 21, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 8.5 The figure shows two balls of equal mass moving in vertical circles. Part A Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal speed? ANSWER: The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Correct Part B Is the tension in string A greater than, less than, or equal to the tension in string B if the balls travel over the top of the circle with equal angular velocity? ANSWER: The tension in string A is less than the tension in string B. The tension in string A is equal to the tension in string B. The tension in string A is greater than the tension in string B. Correct A Mass on a Turntable: Conceptual A small metal cylinder rests on a circular turntable that is rotating at a constant rate, as illustrated in the diagram. Part A Which of the following sets of vectors best describes the velocity, acceleration, and net force acting on the cylinder at the point indicated in the diagram? Typesetting math: 100% Hint 1. The direction of acceleration can be determined from Newton's second law According to Newton's second law, the acceleration of an object has the same direction as the net force acting on that object. ANSWER: a b c d e Correct Part B Let R be the distance between the cylinder and the center of the turntable. Now assume that the cylinder is moved to a new location R/2 from the center of the turntable. Which of the following statements accurately describe the motion of the cylinder at the new location? Check all that apply. Typesetting math: 100% Hint 1. Find the speed of the cylinder Find the speed v of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time T . Express your answer in terms of R and T . ANSWER: v = πR T Hint 2. Find the acceleration of the cylinder Find the magnitude of the acceleration a of the cylinder at the new location. Assume that the cylinder makes one complete turn in a period of time T . Express your answer in terms of R and T . Hint 1. Centripetal acceleration Recall that the acceleration of an object that moves in a circular path of radius r with constant speed v has magnitude given by a= Note that both the velocity and radius of the trajectory change when the cylinder is moved. ANSWER: a= 2π 2 R T2 ANSWER: Typesetting math: 100% v2 r . The speed of the cylinder has decreased. The speed of the cylinder has increased. The magnitude of the acceleration of the cylinder has decreased. The magnitude of the acceleration of the cylinder has increased. The speed and the acceleration of the cylinder have not changed. Correct Accelerating along a Racetrack A road race is taking place along the track shown in the figure . All of the cars are moving at constant speeds. The car at point F is traveling along a straight section of the track, whereas all the other cars are moving along curved segments of the track. Part A ⃗ be the velocity of the car at point A. What can you say about the acceleration of the car at that point? Let v A Hint 1. Acceleration along a curved path Typesetting math: 100% v ⃗ is changing, even though the v ⃗ at each point along the curved Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the velocity vector path and is directed toward the center of curvature of the path. ANSWER: ⃗ . The acceleration is parallel to v A ⃗ and directed toward the inside of the track. The acceleration is perpendicular to vA ⃗ and directed toward the outside of the track. The acceleration is perpendicular to vA ⃗ . The acceleration is neither parallel nor perpendicular to vA The acceleration is zero. Correct Part B ⃗ be the velocity of the car at point C. What can you say about the acceleration of the car at that point? Let v C Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the velocity vector path and is directed toward the center of curvature of the path. ANSWER: Typesetting math: 100% v ⃗ is changing, even though the v ⃗ at each point along the curved ⃗ . The acceleration is parallel to v C ⃗ and pointed toward the inside of the track. The acceleration is perpendicular to vC ⃗ and pointed toward the outside of the track. The acceleration is perpendicular to vC ⃗ . The acceleration is neither parallel nor perpendicular to vC The acceleration is zero. Correct Part C ⃗ be the velocity of the car at point D. What can you say about the acceleration of the car at that point? Let v D Hint 1. Acceleration along a curved path Since acceleration is a vector quantity, an object moving at constant speed along a curved path has nonzero acceleration because the direction of its velocity magnitude of its velocity (the speed) is constant. Moreover, if the speed is constant, the object's acceleration is always perpendicular to the velocity vector path and is directed toward the center of curvature of the path. ANSWER: ⃗ . The acceleration is parallel to v D ⃗ and pointed toward the inside of the track. The acceleration is perpendicular to vD ⃗ and pointed toward the outside of the track. The acceleration is perpendicular to vD ⃗ . The acceleration is neither parallel nor perpendicular to vD The acceleration is zero. Correct Typesetting math: 100% v ⃗ is changing, even though the v ⃗ at each point along the curved Part D ⃗ be the velocity of the car at point F. What can you say about the acceleration of the car at that point? Let v F Hint 1. Acceleration along a straight path The velocity of an object that moves along a straight path is always parallel to the direction of the path, and the object has a nonzero acceleration only if the magnitude of its velocity changes in time. ANSWER: ⃗ . The acceleration is parallel to v F ⃗ and pointed toward the inside of the track. The acceleration is perpendicular to vF ⃗ and pointed toward the outside of the track. The acceleration is perpendicular to vF ⃗ . The acceleration is neither parallel nor perpendicular to vF The acceleration is zero. Correct Part E Assuming that all cars have equal speeds, which car has the acceleration of the greatest magnitude, and which one has the acceleration of the least magnitude? Use A for the car at point A, B for the car at point B, and so on. Express your answer as the name the car that has the greatest magnitude of acceleration followed by the car with the least magnitude of accelation, and separate your answers with a comma. Hint 1. How to approach the problem Recall that the magnitude of the acceleration of an object that moves at constant speed along a curved path is inversely proportional to the radius of curvature of the path. ANSWER: Typesetting math: 100% Correct Part F Assume that the car at point A and the one at point E are traveling along circular paths that have the same radius. If the car at point A now moves twice as fast as the car at point E, how is the magnitude of its acceleration related to that of car E. Hint 1. Find the acceleration of the car at point E Let r be the radius of the two curves along which the cars at points A and E are traveling. What is the magnitude aE of the acceleration of the car at point E? Express your answer in terms of the radius of curvature r and the speed vE of car E. Hint 1. Uniform circular motion The magnitude a of the acceleration of an object that moves with constant speed v along a circular path of radius a= r is given by v2 r . ANSWER: 2 aE = vE r Hint 2. Find the acceleration of the car at point A If vA = 2vE , what is the acceleration aA of the car at point A? Let r be the radius of the two curves along which the cars at points A and E are traveling. Express your answer in terms of the speed vE of the car at E and the radius r. Typesetting math: 100% Hint 1. Uniform circular motion The magnitude of the acceleration of an object that moves with constant speed v along a circular path of radius a= v2 r . ANSWER: 2 aA = 4vE r ANSWER: The magnitude of the acceleration of the car at point A is twice that of the car at point E. The magnitude of the acceleration of the car at point A is the same as that of the car at point E. The magnitude of the acceleration of the car at point A is half that of the car at point E. The magnitude of the acceleration of the car at point A is four times that of the car at point E. Correct Problem 8.5 A 1300kg car takes a 50-m-radius unbanked curve at 13m/s . Part A What is the size of the friction force on the car? Express your answer to two significant figures and include the appropriate units. ANSWER: Typesetting math: 100% r is given by fs = 4400 N Correct Problem 8.10 It is proposed that future space stations create an artificial gravity by rotating. Suppose a space station is constructed as a 1600-m-diameter cylinder that rotates about its axis. The inside surface is the deck of the space station. Part A What rotation period will provide "normal" gravity? Express your answer with the appropriate units. ANSWER: T = 56.8 s Correct Problem 8.7 In the Bohr model of the hydrogen atom, an electron (mass 8.2 × 10 −8 m = 9.1 × 10−31 kg) orbits a proton at a distance of 5.3 × 10−11 m. The proton pulls on the electron with an electric force of N. Part A How many revolutions per second does the electron make? Express your answer with the appropriate units. ANSWER: Typesetting math: 100% rev 6.56×1015 s Correct Problem 8.14 The weight of passengers on a roller coaster increases by 56% as the car goes through a dip with a 38m radius of curvature. Part A What is the car's speed at the bottom of the dip? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 14 m s Correct Problem 8.18 While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 14m , and you use your watch to find that each loop around takes 24s . Part A What is your speed? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 3.7 m s Typesetting math: 100% Correct Part B What is the magnitude of your acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: m a = 0.96 s 2 Correct Part C What is the ratio of your weight at the top of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: wtop = 0.90 FG Correct Part D What is the ratio of your weight at the bottom of the ride to your weight while standing on the ground? Express your answer using two significant figures. ANSWER: Typesetting math: 100% wbottom = 1.1 FG Correct Enhanced EOC: Problem 8.46 A heavy ball with a weight of 120N is hung from the ceiling of a lecture hall on a 4.4-m-long rope. The ball is pulled to one side and released to swing as a pendulum, reaching a speed of 5.6m/s as it passes through the lowest point. You may want to review ( pages 201 - 204) . For help with math skills, you may want to review: Solutions of Systems of Equations Part A What is the tension in the rope at that point? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Start by drawing a free-body diagram indicating the forces acting on the ball when it is at its lowest point. Choose a coordinate system. What is the direction of the acceleration in your chosen coordinate system? What is the magnitude of the acceleration for the mass, which is moving in a circular path? What is Newton's second law applied to the mass at the bottom of its swing? Make sure to use your coordinate system when determining the signs of all the forces and the acceleration. What is the tension in the rope at this point? ANSWER: T = 210 N Typesetting math: 100% Correct Problem 8.43 In an amusement park ride called The Roundup, passengers stand inside a 16.0m -diameter rotating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane, as shown in the figure . Part A Suppose the ring rotates once every 4.80s . If a rider's mass is 54.0kg , with how much force does the ring push on her at the top of the ride? Express your answer with the appropriate units. ANSWER: 211 N Correct Part B Typesetting math: 100% Suppose the ring rotates once every 4.80s . If a rider's mass is 54.0kg , with how much force does the ring push on her at the bottom of the ride? Express your answer with the appropriate units. ANSWER: 1270 N Correct Part C What is the longest rotation period of the wheel that will prevent the riders from falling off at the top? Express your answer with the appropriate units. ANSWER: 5.68 s Correct Conceptual Question 9.9 A 2kg object is moving to the right with a speed of 1 ^ i m/s when it experiences an impulse of 6 ^i N s. Part A What is the object's speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: v= 4 m s Typesetting math: 100% Correct Part B What is the object's direction after the impulse? ANSWER: to the right to the left Correct Conceptual Question 9.10 A 2kg object is moving to the right with a speed of 2 ^ i m/s when it experiences an impulse of -6 ^i N s. Part A What is the object's speed after the impulse? Express your answer as an integer and include the appropriate units. ANSWER: v= 1 m s Correct Part B What is the object's direction after the impulse? Typesetting math: 100% ANSWER: to the right to the left Correct Problem 9.5 Part A In the figure , what value of Fmax gives an impulse of 6.4N s ? Express your answer to two significant figures and include the appropriate units. ANSWER: Fmax = 1.6×103 N Correct Typesetting math: 100% Impulse on a Baseball Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force ΣF ⃗ acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse J ⃗ is a measure of both these effects. For a constant net force, the impulse is given by ⃗ . J ⃗ = F ∆t The impulse is a vector pointing in the same direction as the force vector. The units of J ⃗ are N ⋅ s or kg ⋅ m/s. Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object's momentum (p ⃗ describes the effect that an impulse has on an object's motion: = mv)⃗ will also change. The impulse-momentum theorem ⃗ . ∆p ⃗ = J ⃗ = F ∆t So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time. In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 N to the ball for a time of 0.70 × 10−3 s. Part A Assuming that this force is constant, what is the magnitude J of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER: J = 8.4 N ⋅ s Correct We often visualize the impulse by drawing a graph of force versus time. For a constant net force such as that used in the previous part, the graph will look like the one shown in the figure. Typesetting math: 100% Part B The net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER: length height For this graph, the area of the rectangle corresponds to the impulse. slope Correct The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant. A more realistic graph of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the force decaying to zero. It will look like the graph in the figure. Typesetting math: 100% Part C If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common? ANSWER: maximum force area slope Typesetting math: 100% Correct ⃗ acting on the baseball during When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force F avg time ∆t. This average net force is treated as a constant force that acts on the ball for time ∆t. The impulse on the ball can then be found as ⃗ ∆t. J ⃗ = F avg Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively. The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that one-dimensional problem. J ⃗ = ∆p ⃗ = m(vf⃗ − vi⃗ ). Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a Part D Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction. ANSWER: Typesetting math: 100% positive The impulse on the ball caused by the bat will be in the negative x direction. Correct Part E Now assume that the pitcher in Part D throws a 0.145-kg baseball parallel to the ground with a speed of 32 m/s in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball's velocity just after leaving the bat if the bat applies an impulse of −8.4 N ⋅ s to the baseball? Enter your answer numerically in meters per second using two significant figures. ANSWER: v ⃗ = -26 m/s Correct The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive. Problem 9.9 A 2.6kg object is moving to the right with a speed of 1.0m/s when it experiences the force shown in the figure. Typesetting math: 100% Part A What is the object's speed after the force ends? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 0.62 m s Correct Part B What is the object's direction after the force ends? ANSWER: to the right to the left Correct Enhanced EOC: Problem 9.27 A tennis player swings her 1000 g racket with a speed of 11.0m/s . She hits a 60 g tennis ball that was approaching her at a speed of 19.0m/s . The ball rebounds at 41.0m/s . You may want to review ( pages 226 - 232) . For help with math skills, you may want to review: Typesetting math: 100% Solving Algebraic Equations Part A How fast is her racket moving immediately after the impact? You can ignore the interaction of the racket with her hand for the brief duration of the collision. Express your answer with the appropriate units. Hint 1. How to approach the problem Given that you can ignore the interaction of the racket with her hand during the collision, what is conserved during the collision? Draw a picture indicating the direction of the racket and ball before the collision and a separate picture for after the collision. Place a coordinate system on your pictures, indicating the positive x direction. Keeping in mind that velocity can be either positive or negative in your coordinate system, what is the initial momentum of the ball–racket system? What is the final momentum of the ball–racket system in terms of the velocity of the racket after the collision? Using conservation of momentum, what are the velocity and speed of the racket after the collision? ANSWER: m 7.40 s Correct Part B If the tennis ball and racket are in contact for 8.00ms , what is the average force that the racket exerts on the ball? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the impulse on the ball related to the change in momentum of the ball? What is the change in momentum of the ball? How are the impulse on the ball and the collision time related to the average force on the ball? Typesetting math: 100% ANSWER: 450 N Correct Problem 9.14 4 A 2.00×10 kg railroad car is rolling at 6.00m/s when a 6000kg load of gravel is suddenly dropped in. Part A What is the car's speed just after the gravel is loaded? Express your answer with the appropriate units. ANSWER: m 4.62 s Correct Problem 9.17 A 330g bird flying along at 5.0m/s sees a 9.0g insect heading straight toward it with a speed of 34m/s (as measured by an observer on the ground, not by the bird). The bird opens its mouth wide and enjoys a nice lunch. Part A What is the bird's speed immediately after swallowing? Express your answer to two significant figures and include the appropriate units. ANSWER: Typesetting math: 100% v = 4.0 m s Correct Problem 9.20 A 50.0kg archer, standing on frictionless ice, shoots a 200g arrow at a speed of 200m/s . Part A What is the recoil speed of the archer? Express your answer with the appropriate units. ANSWER: m 0.800 s Correct Problem 9.25 A 40.0g ball of clay traveling east at 4.50m/s collides and sticks together with a 50.0g ball of clay traveling north at 4.50m/s . Part A What is the speed of the resulting ball of clay? Express your answer with the appropriate units. ANSWER: m 3.20 s Typesetting math: 100% Correct Problem 9.32 A particle of mass m is at rest at t = 0. Its momentum for t > 0 is given by px = 6t2 kg m/s, where t is in s. Part A Find an expression for Fx (t), the force exerted on the particle as a function of time. Express your answer in terms of the given quantities. ANSWER: Fx = 12t N Correct Problem 9.37 Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0km and a mass of 1.2×1013kg hits the earth with an impact speed of 4.5×104m/s . Part A What is the earth's recoil speed after such a collision? (Use a reference frame in which the earth was initially at rest.) Assume that Express your answer to two significant figures and include the appropriate units. ANSWER: v = 9.0×10−8 m s Typesetting math: 100% MEarth = 5.98 × 1024 kg. Correct Part B What percentage is this of the earth's speed around the sun? (Use the astronomical data in the textbook.) Express your answer using two significant figures. ANSWER: v = 3.0×10−10 % of the earth's speed Correct Problem 9.42 One billiard ball is shot east at 1.8m/s . A second, identical billiard ball is shot west at 1.2m/s . The balls have a glancing collision, not a head-on collision, deflecting the second ball by 90� and sending it north at 1.50m/s . Part A What is the speed of the first ball after the collision? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 1.6 m s Correct Part B What is the direction of the first ball after the collision? Give the direction as an angle south of east. Typesetting math: 100% Express your answer to two significant figures and include the appropriate units. ANSWER: θ = 68 � Correct Problem 9.49 Two 490g blocks of wood are 2.0 m apart on a frictionless table. A 12g bullet is fired at 420m/s toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterward is 5.6m/s . Part A What is the speed of the second block after the bullet stops? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 4.6 m s Correct Score Summary: Your score on this assignment is 99.5%. You received 156.21 out of a possible total of 157 points. Typesetting math: 100% Assignment 8 Due: 11:59pm on Friday, April 4, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 10.3 Part A If a particle's speed increases by a factor of 5, by what factor does its kinetic energy change? ANSWER: K2 = 25 K1 Correct Conceptual Question 10.11 A spring is compressed 1.5cm . Part A How far must you compress a spring with twice the spring constant to store the same amount of energy? Express your answer to two significant figures and include the appropriate units. ANSWER: ∆x = 1.1 cm Correct Problem 10.2 The lowest point in Death Valley is 85m below sea level. The summit of nearby Mt. Whitney has an elevation of 4420 m. Part A What is the change in potential energy of an energetic 80kg hiker who makes it from the floor of Death Valley to the top of Mt.Whitney? Express your answer to two significant figures and include the appropriate units. ANSWER: ∆U = 3.5×106 J Correct Problem 10.3 Part A 4 At what speed does a 1800kg compact car have the same kinetic energy as a 1.80×10 kg truck going 21.0km/hr ? Express your answer with the appropriate units. ANSWER: vc = 66.4 km hr Correct Problem 10.5 A boy reaches out of a window and tosses a ball straight up with a speed of 13m/s . The ball is 21m above the ground as he releases it. Part A Use energy to find the ball's maximum height above the ground. Express your answer to two significant figures and include the appropriate units. ANSWER: Hmax = 30 m Correct Part B Use energy to find the ball's speed as it passes the window on its way down. Express your answer to two significant figures and include the appropriate units. ANSWER: v = 13 m s Correct Part C Use energy to find the speed of impact on the ground. Express your answer to two significant figures and include the appropriate units. ANSWER: v = 24 m s Correct Problem 10.8 A 59.0kg skateboarder wants to just make it to the upper edge of a "quarter pipe," a track that is one-quarter of a circle with a radius of 2.30m . Part A What speed does he need at the bottom? Express your answer with the appropriate units. ANSWER: m 6.71 s Correct Problem 10.12 A 1500 kg car traveling at 12m/s suddenly runs out of gas while approaching the valley shown in the figure. The alert driver immediately puts the car in neutral so that it will roll. Part A What will be the car’s speed as it coasts into the gas station on the other side of the valley? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 6.8 m s Correct Ups and Downs Learning Goal: To apply the law of conservation of energy to an object launched upward in the gravitational field of the earth. In the absence of nonconservative forces such as friction and air resistance, the total mechanical energy in a closed system is conserved. This is one particular case of the law of conservation of energy. In this problem, you will apply the law of conservation of energy to different objects launched from the earth. The energy transformations that take place involve the object's kinetic energy K = (1/2)mv2 and its gravitational potential energy U = mgh. The law of conservation of energy for such cases implies that the sum of the object's kinetic energy and potential energy does not change with time. This idea can be expressed by the equation K i + Ui = K f + Uf , where "i" denotes the "initial" moment and "f" denotes the "final" moment. Since any two moments will work, the choice of the moments to consider is, technically, up to you. That choice, though, is usually suggested by the question posed in the problem. First, let us consider an object launched vertically upward with an initial speed v. Neglect air resistance. Part A As the projectile goes upward, what energy changes take place? ANSWER: Both kinetic and potential energy decrease. Both kinetic and potential energy increase. Kinetic energy decreases; potential energy increases. Kinetic energy increases; potential energy decreases. Correct Part B At the top point of the flight, what can be said about the projectile's kinetic and potential energy? ANSWER: Both kinetic and potential energy are at their maximum values. Both kinetic and potential energy are at their minimum values. Kinetic energy is at a maximum; potential energy is at a minimum. Kinetic energy is at a minimum; potential energy is at a maximum. Correct Strictly speaking, it is not the ball that possesses potential energy; rather, it is the system "Earth-ball." Although we will often talk about "the gravitational potential energy of an elevated object," it is useful to keep in mind that the energy, in fact, is associated with the interactions between the earth and the elevated object. Part C The potential energy of the object at the moment of launch __________. ANSWER: is negative is positive is zero depends on the choice of the "zero level" of potential energy Correct Usually, the zero level is chosen so as to make the relevant calculations simpler. In this case, it makes good sense to assume that only choice! Part D Using conservation of energy, find the maximum height Express your answer in terms of hmax to which the object will rise. v and the magnitude of the acceleration of gravity g. ANSWER: v2 hmax = 2g Correct You may remember this result from kinematics. It is comforting to know that our new approach yields the same answer. Part E At what height h above the ground does the projectile have a speed of 0.5v? Express your answer in terms of ANSWER: h= 2 v 3 8g v and the magnitude of the acceleration of gravity g. U = 0 at the ground level--but this is not, by any means, the Correct Part F What is the speed u of the object at the height of Express your answer in terms of (1/2)hmax ? v and g. Use three significant figures in the numeric coefficient. Hint 1. How to approach the problem = 0), the speed is v. All of the energy is kinetic energy, and so, the total energy is (1/2)mv . At the maximum height, all of the energy is potential energy. Since the gravitational potential energy is proportional to h, half of the initial kinetic energy must have been You are being asked for the speed at half of the maximum height. You know that at the initial height (h 2 converted to potential energy when the projectile is at (1/2)hmax . Thus, the kinetic energy must be half of its original value (i.e., (1/4)mv2 when h = (1/2)hmax). You need to determine the speed, as a multiple of v, that corresponds to such a kinetic energy. ANSWER: u = 0.707v Correct Let us now consider objects launched at an angle. For such situations, using conservation of energy leads to a quicker solution than can be produced by kinematics. Part G A ball is launched as a projectile with initial speed v at an angle θ above the horizontal. Using conservation of energy, find the maximum height hmax of the ball's flight. Express your answer in terms of v, g, and θ . Hint 1. Find the final kinetic energy Find the final kinetic energy K f of the ball. Here, the best choice of "final" moment is the point at which the ball reaches its maximum height, since this is the point we are interested in. Express your answer in terms of v, m, and θ . Hint 1. Find the speed at the maximum height The speed of the ball at the maximum height is __________. ANSWER: 0 v v cos θ v sin θ v tan θ ANSWER: K f = 0.5m(vcos(θ))2 ANSWER: 2 hmax = (vsin(θ)) 2g Correct Part H A ball is launched with initial speed v from ground level up a frictionless slope. The slope makes an angle θ with the horizontal. Using conservation of energy, find the maximum vertical height to which the ball will climb. Express your answer in terms of v, g, and θ . You may or may not use all of these quantities. hmax ANSWER: v2 hmax = 2g Correct Interestingly, the answer does not depend on θ . The difference between this situation and the projectile case is that the ball moving up a slope has no kinetic energy at the top of its trajectory whereas the projectile launched at an angle does. Part I A ball is launched with initial speed v from the ground level up a frictionless hill. The hill becomes steeper as the ball slides up; however, the ball remains in contact with the hill at all times. Using conservation of energy, find the maximum vertical height hmax to which the ball will climb. Express your answer in terms of v and g. ANSWER: v2 hmax = 2g Correct The profile of the hill does not matter; the equation K i + Ui = K f + Uf would have the same terms regardless of the steepness of the hill. Problem 10.14 A 12-cm-long spring is attached to the ceiling. When a 2.2kg mass is hung from it, the spring stretches to a length of 17cm . Part A What is the spring constant k? Express your answer to two significant figures and include the appropriate units. ANSWER: N k = 430 m Correct Part B How long is the spring when a 3.0 kg mass is suspended from it? Express your answer to two significant figures and include the appropriate units. ANSWER: y ′ = 19 cm Correct Enhanced EOC: Problem 10.17 A 6.2kg mass hanging from a spring scale is slowly lowered onto a vertical spring, as shown in . You may want to review ( pages 255 - 257) . For help with math skills, you may want to review: Solving Algebraic Equations Part A What does the spring scale read just before the mass touches the lower spring? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture showing the forces acting on the mass before it touches the scale. What is the net force on the mass? What is the force on the mass due to gravity? What is the force on the mass due to the scale? ANSWER: F = 61 N Correct Part B The scale reads 22N when the lower spring has been compressed by 2.7cm . What is the value of the spring constant for the lower spring? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture showing the forces acting on the mass. What is the net force on the mass? What is the force on the mass due to gravity? What is the force on the mass due to the scale? Use these to determine the force on the mass by the spring, taking note of the directions from your picture. How is the spring constant ANSWER: N k = 1400 m k related to the force by the spring and the compression of the spring? Check your units. Correct Part C At what compression length will the scale read zero? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Draw a picture showing the forces on the mass. When the scale reads zero, what is the force on the mass due to the scale? What is the gravitational force on the mass? What is the force on the mass by the spring? How is the compression length related to the force by the spring and the spring constant? Check your units. ANSWER: ∆y = 4.2 cm Correct Problem 10.18 Part A How far must you stretch a spring with k = 800N/m to store 180J of energy? Express your answer to two significant figures and include the appropriate units. ANSWER: ∆s = 0.67 m Correct Problem 10.22 A 15kg runaway grocery cart runs into a spring with spring constant 230N/m and compresses it by 57cm . Part A What was the speed of the cart just before it hit the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 2.2 m s Correct Spring Gun A spring-loaded toy gun is used to shoot a ball straight up in the air. The ball reaches a maximum height position of the spring. H , measured from the equilibrium Part A The same ball is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the ball go this time? Neglect friction. Assume that the spring is ideal and that the distance by which the spring is compressed is negligible compared to H . Hint 1. Potential energy of the spring The potential energy of a spring is proportional to the square of the distance the spring is compressed. The spring was compressed half the distance, so the mass, when launched, has one quarter of the energy as in the first trial. Hint 2. Potential energy of the ball At the highest point in the ball's trajectory, all of the spring's potential energy has been converted into gravitational potential energy of the ball. ANSWER: height = H 4 Correct A Bullet Is Fired into a Wooden Block A bullet of mass mb is fired horizontally with speed vi at a wooden block of mass mw resting on a frictionless table. The bullet hits the block and becomes completely embedded within it. After the bullet has come to rest within the block, the block, with the bullet in it, is traveling at speed vf . Part A Which of the following best describes this collision? Hint 1. Types of collisions An inelastic collision is a collision in which kinetic energy is not conserved. In a partially inelastic collision, kinetic energy is lost, but the objects colliding do not stick together. From this information, you can infer what completely inelastic and elastic collisions are. ANSWER: perfectly elastic partially inelastic perfectly inelastic Correct Part B Which of the following quantities, if any, are conserved during this collision? Hint 1. When is kinetic energy conserved? Kinetic energy is conserved only in perfectly elastic collisions. ANSWER: kinetic energy only momentum only kinetic energy and momentum neither momentum nor kinetic energy Correct Part C What is the speed of the block/bullet system after the collision? Express your answer in terms of vi , mw , and mb . Hint 1. Find the momentum after the collision What is the total momentum ptotal of the block/bullet system after the collision? Express your answer in terms of vf and other given quantities. ANSWER: ptotal = (mw + mb )vf Hint 2. Use conservation of momentum The momentum of the block/bullet system is conserved. Therefore, the momentum before the collision is the same as the momentum after the collision. Find a second expression for ptotal, this time expressed as the total momentum of the system before the collision. Express your answer in terms of vi and other given quantities. ANSWER: ptotal = mb vi ANSWER: vf = m b m +vimw b Correct Problem 10.31 Ball 1, with a mass of 150g and traveling at 15.0m/s , collides head on with ball 2, which has a mass of 340g and is initially at rest. Part A What are the final velocities of each ball if the collision is perfectly elastic? Express your answer with the appropriate units. ANSWER: (vfx )1 = -5.82 m s Correct Part B Express your answer with the appropriate units. ANSWER: (vfx )2 = 9.18 m s Correct Part C What are the final velocities of each ball if the collision is perfectly inelastic? Express your answer with the appropriate units. ANSWER: (vfx )1 = 4.59 m s Correct Part D Express your answer with the appropriate units. ANSWER: (vfx )2 = 4.59 m s Correct Enhanced EOC: Problem 10.43 A package of mass m is released from rest at a warehouse loading dock and slides down the h = 2.2m - high, frictionless chute to a waiting truck. Unfortunately, the truck driver went on a break without having removed the previous package, of mass 2m, from the bottom of the chute. You may want to review ( pages 265 - 269) . For help with math skills, you may want to review: Solving Algebraic Equations Part A Suppose the packages stick together. What is their common speed after the collision? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem There are two parts to this problem: the block sliding down the frictionless incline and the collision. What conservation laws are valid in each part? In terms of m, what are the kinetic and potential energies of the block at the top of the incline? What is the potential energy of the same block at the bottom just before the collision? What are the kinetic energy and velocity of block m just before the collision? What is conserved during the collision? What is the total momentum of the two blocks before the collision? What is the momentum of the two blocks stuck together after the collision? What is the velocity of the two blocks after the collision? ANSWER: v = 2.2 m s Correct Part B Suppose the collision between the packages is perfectly elastic. To what height does the package of mass m rebound? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem There are three parts to this problem: the block sliding down the incline, the collision, and mass m going back up the incline. What conservation laws are valid in each part? What is an elastic collision? For an elastic collision, how are the initial and final velocities related when one of the masses is initially at rest? Using the velocity of m just before the collision from Part A, what is the velocity of m just after the collision in this case? What are the kinetic and potential energies of mass What is the kinetic energy of mass height? m just after the collision? m at its maximum rebound height? Using conservation of energy, what is the potential energy of mass m at its maximum height? What is the maximum ANSWER: h = 24 cm Correct Problem 10.35 A cannon tilted up at a 35.0� angle fires a cannon ball at 79.0m/s from atop a 21.0m -high fortress wall. Part A What is the ball's impact speed on the ground below? Express your answer with the appropriate units. ANSWER: vf = 81.6 m s Correct Problem 10.45 A 1000kg safe is 2.5m above a heavy-duty spring when the rope holding the safe breaks. The safe hits the spring and compresses it 48cm . Part A What is the spring constant of the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: N k = 2.5×105 m Correct Problem 10.49 A 100g block on a frictionless table is firmly attached to one end of a spring with k = 21N/m . The other end of the spring is anchored to the wall. A 30g ball is thrown horizontally toward the block with a speed of 6.0m/s . Part A If the collision is perfectly elastic, what is the ball's speed immediately after the collision? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 3.2 m s Correct Part B What is the maximum compression of the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: ∆x = 0.19 m Correct Part C Repeat part A for the case of a perfectly inelastic collision. Express your answer to two significant figures and include the appropriate units. ANSWER: v = 1.4 m s Correct Part D Repeat part B for the case of a perfectly inelastic collision. Express your answer to two significant figures and include the appropriate units. ANSWER: ∆x = 0.11 m Correct Score Summary: Your score on this assignment is 99.4%. You received 120.28 out of a possible total of 121 points. Assignment 9 Due: 11:59pm on Friday, April 11, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 11.2 Part A Evaluate the dot product A⃗ ⋅ B⃗ if A⃗ = 5^i − 6^j and B⃗ = −9^i − 5^j. Express your answer using two significant figures. ANSWER: A⃗ ⋅ B⃗ = -15 Correct Part B Evaluate the dot product A⃗ ⋅ B⃗ if A⃗ = −5^i + 9^j and B⃗ = 5^i + 6^j. Express your answer using two significant figures. ANSWER: A⃗ ⋅ B⃗ = 29 Correct Problem 11.4 Part A What is the angle θ between vectors A⃗ and B⃗ if A⃗ = 2 ^ı + 5 ^ȷ and B⃗ = −2 ^ı − 4 ^ȷ? Express your answer as an integer and include the appropriate units. ANSWER: θ = 175 � Correct Part B What is the angle θ between vectors A⃗ and B⃗ if A⃗ = −6 ^ı + 2 ^ȷ and B⃗ = − ^ı − 3 ^ȷ? Express your answer as an integer and include the appropriate units. ANSWER: θ = 90 � Correct ± All Work and No Play Learning Goal: To be able to calculate work done by a constant force directed at different angles relative to displacement If an object undergoes displacement while being acted upon by a force (or several forces), it is said that work is being done on the object. If the object is moving in a straight line and the displacement and the force are known, the work done by the force can be calculated as W = F ⃗ ⋅ s ⃗ = ∣∣F ∣∣⃗ ∣s ∣⃗ cos θ, where W is the work done by force F ⃗ on the object that undergoes displacement s ⃗ directed at angle θ relative to F .⃗ Note that depending on the value of cos θ, the work done can be positive, negative, or zero. In this problem, you will practice calculating work done on an object moving in a straight line. The first series of questions is related to the accompanying figure. Part A What can be said about the sign of the work done by the force F 1⃗ ? ANSWER: It is positive. It is negative. It is zero. There is not enough information to answer the question. Correct When θ Part B = 90� , the cosine of θ is zero, and therefore the work done is zero. What can be said about the work done by force F 2⃗ ? ANSWER: It is positive. It is negative. It is zero. Correct When 0� < θ < 90� , cos θ is positive, and so the work done is positive. Part C The work done by force F 3⃗ is ANSWER: positive negative zero Correct When 90� < θ < 180� , cos θ is negative, and so the work done is negative. Part D The work done by force F 4⃗ is ANSWER: positive negative zero Correct Part E The work done by force F 5⃗ is ANSWER: positive negative zero Correct Part F The work done by force F 6⃗ is ANSWER: positive negative zero Correct Part G The work done by force F 7⃗ is ANSWER: positive negative zero Correct In the next series of questions, you will use the formula W = F ⃗ ⋅ s ⃗ = ∣∣F ∣∣⃗ ∣s ∣⃗ cos θ to calculate the work done by various forces on an object that moves 160 meters to the right. Part H Find the work W done by the 18-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: W = 2900 J Correct Part I Find the work W done by the 30-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: W = 4200 J Correct Part J Find the work W done by the 12-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: W = -1900 J Correct Part K Find the work W done by the 15-newton force. Use two significant figures in your answer. Express your answer in joules. ANSWER: W = -1800 J Correct Introduction to Potential Energy Learning Goal: Understand that conservative forces can be removed from the work integral by incorporating them into a new form of energy called potential energy that must be added to the kinetic energy to get the total mechanical energy. The first part of this problem contains short-answer questions that review the work-energy theorem. In the second part we introduce the concept of potential energy. But for now, please answer in terms of the work-energy theorem. Work-Energy Theorem The work-energy theorem states K f = K i + Wall, where Wall is the work done by all forces that act on the object, and K i and K f are the initial and final kinetic energies, respectively. Part A The work-energy theorem states that a force acting on a particle as it moves over a ______ changes the ______ energy of the particle if the force has a component parallel to the motion. Choose the best answer to fill in the blanks above: ANSWER: distance / potential distance / kinetic vertical displacement / potential none of the above Correct It is important that the force have a component acting in the direction of motion. For example, if a ball is attached to a string and whirled in uniform circular motion, the string does apply a force to the ball, but since the string's force is always perpendicular to the motion it does no work and cannot change the kinetic energy of the ball. Part B To calculate the change in energy, you must know the force as a function of _______. The work done by the force causes the energy change. Choose the best answer to fill in the blank above: ANSWER: acceleration work distance potential energy Correct Part C To illustrate the work-energy concept, consider the case of a stone falling from xi to xf under the influence of gravity. Using the work-energy concept, we say that work is done by the gravitational _____, resulting in an increase of the ______ energy of the stone. Choose the best answer to fill in the blanks above: ANSWER: force / kinetic potential energy / potential force / potential potential energy / kinetic Correct Potential Energy You should read about potential energy in your text before answering the following questions. Potential energy is a concept that builds on the work-energy theorem, enlarging the concept of energy in the most physically useful way. The key aspect that allows for potential energy is the existence of conservative forces, forces for which the work done on an object does not depend on the path of the object, only the initial and final positions of the object. The gravitational force is conservative; the frictional force is not. The change in potential energy is the negative of the work done by conservative forces. Hence considering the initial and final potential energies is equivalent to calculating the work done by the conservative forces. When potential energy is used, it replaces the work done by the associated conservative force. Then only the work due to nonconservative forces needs to be calculated. In summary, when using the concept of potential energy, only nonconservative forces contribute to the work, which now changes the total energy: K f + Uf = Ef = Wnc + Ei = Wnc + K i + U,i where Uf and Ui are the final and initial potential energies, and Wnc is the work due only to nonconservative forces. Now, we will revisit the falling stone example using the concept of potential energy. Part D Rather than ascribing the increased kinetic energy of the stone to the work of gravity, we now (when using potential energy rather than work-energy) say that the increased kinetic energy comes from the ______ of the _______ energy. Choose the best answer to fill in the blanks above: ANSWER: work / potential force / kinetic change / potential Correct Part E This process happens in such a way that total mechanical energy, equal to the ______ of the kinetic and potential energies, is _______. Choose the best answer to fill in the blanks above: ANSWER: sum / conserved sum / zero sum / not conserved difference / conserved Correct Problem 11.7 Part A How much work is done by the force F ⃗ = (− 2.2 ^i + 6.6 ^j ) N on a particle that moves through displacement ∆r ⃗ = 3.9 ^i m Express your answer to two significant figures and include the appropriate units. ANSWER: W = -8.6 J Correct Part B How much work is done by the force F ⃗ = (− 2.2 ^i + 6.6 ^j ) N on a particle that moves through displacement ∆r ⃗ = 3.9 ^j m? Express your answer to two significant figures and include the appropriate units. ANSWER: W = 26 J Correct Problem 11.10 A 1.8kg book is lying on a 0.80-m-high table. You pick it up and place it on a bookshelf 2.27m above the floor. Part A How much work does gravity do on the book? Express your answer to two significant figures and include the appropriate units. ANSWER: Wg = -26 J Correct Part B How much work does your hand do on the book? Express your answer to two significant figures and include the appropriate units. ANSWER: WH = 26 J Correct Problem 11.12 The three ropes shown in the bird's-eye view of the figure are used to drag a crate 3.3m across the floor. Part A How much work is done by each of the three forces? Express your answers using two significant figures. Enter your answers numerically separated by commas. ANSWER: W1 , W2 , W3 = 1.9,1.2,-2.1 kJ Correct Enhanced EOC: Problem 11.16 A 1.2kg particle moving along the x-axis experiences the force shown in the figure. The particle's velocity is 4.6m/s at You may want to review ( x = 0 m. pages 286 - 287) . For help with math skills, you may want to review: The Definite Integral Part A What is its velocity at x = 2 m? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the work–kinetic energy theorem? What is the kinetic energy at x = 0 m? How is the work done in going from x = 0 m to x = 2 m related to force shown in the graph? Using the work–kinetic energy theorem, what is the kinetic energy at ANSWER: x = 2 m? What is the velocity at x = 2 m? v = 6.2 m s Correct Part B What is its velocity at x = 4 m? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the work–kinetic energy theorem? What is the kinetic energy at x = 0 m? How is the work done in going from x = 0 m to x = 4 m related to force shown in the graph? Can the work be negative? Using the work–kinetic energy theorem, what is the kinetic energy at x = 4 m? What is the velocity at x = 4 m? ANSWER: v = 4.6 m s Correct Work on a Sliding Block A block of weight w sits on a frictionless inclined plane, which makes an angle θ with respect to the horizontal, as shown. A force of magnitude F , applied parallel to the incline, pulls the block up the plane at constant speed. Part A The block moves a distance L up the incline. The block does not stop after moving this distance but continues to move with constant speed. What is the total work Wtot done on the block by all forces? (Include only the work done after the block has started moving, not the work needed to start the block moving from rest.) Express your answer in terms of given quantities. Hint 1. What physical principle to use To find the total work done on the block, use the work-energy theorem: Wtot = K f − K i. Hint 2. Find the change in kinetic energy What is the change in the kinetic energy of the block, from the moment it starts moving until it has been pulled a distance L? Remember that the block is pulled at constant speed. Hint 1. Consider kinetic energy If the block's speed does not change, its kinetic energy cannot change. ANSWER: Kf − Ki = 0 ANSWER: Wtot = 0 Correct Part B What is Wg , the work done on the block by the force of gravity as the block moves a distance L up the incline? Express the work done by gravity in terms of the weight w and any other quantities given in the problem introduction. Hint 1. Force diagram Hint 2. Force of gravity component What is the component of the force of gravity in the direction of the block's displacement (along the inclined plane)? Express your answer in terms of w and θ . Hint 1. Relative direction of the force and the motion Remember that the force of gravity acts down the plane, whereas the block's displacement is directed up the plane. ANSWER: Fg|| = −wsin(θ) ANSWER: Wg = −wLsin(θ) Correct Part C What is WF , the work done on the block by the applied force F as the block moves a distance L up the incline? Express your answer in terms of F and other given quantities. Hint 1. How to find the work done by a constant force Remember that the work done on an object by a particular force is the integral of the dot product of the force and the instantaneous displacement of the object, over the path followed by the object. In this case, since the force is constant and the path is a straight segment of length L up the inclined plane, the dot product becomes simple multiplication. ANSWER: WF = FL Correct Part D What is Wnormal , the work done on the block by the normal force as the block moves a distance L up the inclined plane? Express your answer in terms of given quantities. Hint 1. First step in computing the work The work done by the normal force is equal to the dot product of the force vector and the block's displacement vector. The normal force and the block's displacement vector are perpendicular. Therefore, what is their dot product? ANSWER: N ⃗ ⋅ L⃗ = 0 ANSWER: Wnormal = 0 Correct Problem 11.20 A particle moving along the y -axis has the potential energy U = 3.2y 3 J, where y is in m. Part A What is the y-component of the force on the particle at y = 0 m? Express your answer to two significant figures and include the appropriate units. ANSWER: Fy = 0 N Correct Part B What is the y-component of the force on the particle at y = 1 m? Express your answer to two significant figures and include the appropriate units. ANSWER: Fy = -9.6 N Correct Part C What is the y-component of the force on the particle at y = 2 m? Express your answer to two significant figures and include the appropriate units. ANSWER: Fy = -38 N Correct Problem 11.28 A cable with 25.0N of tension pulls straight up on a 1.08kg block that is initially at rest. Part A What is the block's speed after being lifted 2.40m ? Solve this problem using work and energy. Express your answer with the appropriate units. ANSWER: vf = 8.00 m s Correct Problem 11.29 Part A How much work does an elevator motor do to lift a 1500kg elevator a height of 110m ? Express your answer with the appropriate units. ANSWER: Wext = 1.62×106 J Correct Part B How much power must the motor supply to do this in 50s at constant speed? Express your answer with the appropriate units. ANSWER: P = 3.23×104 W Correct Problem 11.32 How many energy is consumed by a 1.20kW hair dryer used for 10.0min and a 11.0W night light left on for 16.0hr ? Part A Hair dryer: Express your answer with the appropriate units. ANSWER: W = 7.20×105 J Correct Part B Night light: Express your answer with the appropriate units. ANSWER: W = 6.34×105 J Correct Problem 11.42 A 2500kg elevator accelerates upward at 1.20m/s2 for 10.0m , starting from rest. Part A How much work does gravity do on the elevator? Express your answer with the appropriate units. ANSWER: −2.45×105 J Correct Part B How much work does the tension in the elevator cable do on the elevator? Express your answer with the appropriate units. ANSWER: 2.75×105 J Correct Part C Use the work-kinetic energy theorem to find the kinetic energy of the elevator as it reaches 10.0m . Express your answer with the appropriate units. ANSWER: 3.00×104 J Correct Part D What is the speed of the elevator as it reaches 10.0m ? Express your answer with the appropriate units. ANSWER: m 4.90 s Correct Problem 11.47 A horizontal spring with spring constant 130N/m is compressed 17cm and used to launch a 2.4kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Part A Use work and energy to find how far the box slides across the rough surface before stopping. Express your answer to two significant figures and include the appropriate units. ANSWER: l = 53 cm Correct Problem 11.49 Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling friction as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at 6.0� and the coefficient of rolling friction is 0.45. Part A Use work and energy to find the length of a ramp that will stop a 15,000 kg truck that enters the ramp at 30m/s . Express your answer to two significant figures and include the appropriate units. ANSWER: l = 83 m Correct Problem 11.51 Use work and energy to find an expression for the speed of the block in the following figure just before it hits the floor. Part A Find an expression for the speed of the block if the coefficient of kinetic friction for the block on the table is Express your answer in terms of the variables M , m, h, µk , and free fall acceleration g. ANSWER: v= Part B Find an expression for the speed of the block if the table is frictionless. Express your answer in terms of the variables M , ANSWER: m, h, and free fall acceleration g. µk . v= Problem 11.57 The spring shown in the figure is compressed 60cm and used to launch a 100 kg physics student. The track is frictionless until it starts up the incline. The student's coefficient of kinetic friction on the 30� incline is 0.12 . Part A What is the student's speed just after losing contact with the spring? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 17 m s Correct Part B How far up the incline does the student go? Express your answer to two significant figures and include the appropriate units. ANSWER: ∆s = 41 m Correct Score Summary: Your score on this assignment is 93.6%. You received 112.37 out of a possible total of 120 points. Assignment 10 1 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Assignment 10 Due: 11:59pm on Friday, April 18, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 12.3 Part A The figure shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies to . Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER: 4/11/2014 1:13 PM Assignment 10 2 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Incorrect; Try Again Conceptual Question 12.6 You have two steel solid spheres. Sphere 2 has twice the radius of sphere 1. Part A By what factor does the moment of inertia of sphere 2 exceed the moment of inertia of sphere 1? ANSWER: = 32 Correct Problem 12.2 A high-speed drill reaches 2500 in 0.59 . Part A What is the drill's angular acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: = Part B Through how many revolutions does it turn during this first 0.59 ? Express your answer to two significant figures and include the appropriate units. ANSWER: = Constant Angular Acceleration in the Kitchen 4/11/2014 1:13 PM Assignment 10 3 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Dario, a prep cook at an Italian restaurant, spins a salad spinner and observes that it rotates 20.0 times in 5.00 seconds and then stops spinning it. The salad spinner rotates 6.00 more times before it comes to rest. Assume that the spinner slows down with constant angular acceleration. Part A What is the angular acceleration of the salad spinner as it slows down? Express your answer numerically in degrees per second per second. You did not open hints for this part. ANSWER: = Part B This question will be shown after you complete previous question(s). ± A Spinning Electric Fan An electric fan is turned off, and its angular velocity decreases uniformly from 540 interval of length 4.40 . to 250 in a time Part A Find the angular acceleration in revolutions per second per second. You did not open hints for this part. ANSWER: = Part B Find the number of revolutions made by the fan blades during the time that they are slowing down in Part A. 4/11/2014 1:13 PM Assignment 10 4 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... You did not open hints for this part. ANSWER: Part C How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in Part A? You did not open hints for this part. ANSWER: Problem 12.8 A 100 ball and a 230 ball are connected by a 34mass at 130 . -long, massless, rigid rod. The balls rotate about their center of Part A What is the speed of the 100 ball? Express your answer to two significant figures and include the appropriate units. ANSWER: = Problem 12.10 A thin, 60.0 disk with a diameter of 9.00 rotates about an axis through its center with 0.200 of kinetic energy. Part A What is the speed of a point on the rim? 4/11/2014 1:13 PM Assignment 10 5 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Express your answer with the appropriate units. ANSWER: Problem 12.12 A drum major twirls a 95- -long, 470 baton about its center of mass at 150 . Part A What is the baton's rotational kinetic energy? Express your answer to two significant figures and include the appropriate units. ANSWER: = Net Torque on a Pulley The figure below shows two blocks suspended by a cord over a pulley. The mass of block B is twice the mass of block A, while the mass of the pulley is equal to the mass of block A. The blocks are let free to move and the cord moves on the pulley without slipping or stretching. There is no friction in the pulley axle, and the cord's weight can be ignored. Part A Which of the following statements correctly describes the system shown in the figure? Check all that apply. 4/11/2014 1:13 PM Assignment 10 6 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... You did not open hints for this part. ANSWER: The acceleration of the blocks is zero. The net torque on the pulley is zero. The angular acceleration of the pulley is nonzero. Part B This question will be shown after you complete previous question(s). Problem 12.18 Part A In the figure , what is the magnitude of net torque about the axle? Express your answer to two significant figures and include the appropriate units. ANSWER: = Part B What is the direction of net torque about the axle? ANSWER: 4/11/2014 1:13 PM Assignment 10 7 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Clockwise Counterclockwise Problem 12.22 An athlete at the gym holds a 3.5 steel ball in his hand. His arm is 78 center of mass of the arm is at the geometrical center of the arm. long and has a mass of 3.6 . Assume the Part A What is the magnitude of the torque about his shoulder if he holds his arm straight out to his side, parallel to the floor? Express your answer to two significant figures and include the appropriate units. ANSWER: = Part B What is the magnitude of the torque about his shoulder if he holds his arm straight, but below horizontal? Express your answer to two significant figures and include the appropriate units. ANSWER: = Parallel Axis Theorem The parallel axis theorem relates , the moment of inertia of an object about an axis passing through its center of mass, to , the moment of inertia of the same object about a parallel axis passing through point p. The mathematical statement of the theorem is through point p, and , where is the perpendicular distance from the center of mass to the axis that passes is the mass of the object. Part A Suppose a uniform slender rod has length and mass . The moment of inertia of the rod about about an axis that is perpendicular to the rod and that passes through its center of mass is given by . Find , the moment of inertia of the rod with respect to a parallel axis through one end of the rod. Express in terms of and . Use fractions rather than decimal numbers in your answer. 4/11/2014 1:13 PM Assignment 10 8 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... You did not open hints for this part. ANSWER: = Part B Now consider a cube of mass with edges of length . The moment of inertia its center of mass and perpendicular to one of its faces is given by of the cube about an axis through . Find , the moment of inertia about an axis p through one of the edges of the cube Express in terms of and . Use fractions rather than decimal numbers in your answer. You did not open hints for this part. ANSWER: = Problem 12.26 Starting from rest, a 12-diameter compact disk takes 2.9 to reach its operating angular velocity of 2000 Assume that the angular acceleration is constant. The disk's moment of inertia is . . Part A How much torque is applied to the disk? Express your answer to two significant figures and include the appropriate units. 4/11/2014 1:13 PM Assignment 10 9 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... ANSWER: = Part B How many revolutions does it make before reaching full speed? Express your answer using two significant figures. ANSWER: = rev Problem 12.23 An object's moment of inertia is 2.20 . Its angular velocity is increasing at the rate of 3.70 . Part A What is the total torque on the object? ANSWER: Problem 12.31 A 5.1 cat and a 2.5 bowl of tuna fish are at opposite ends of the 4.0- -long seesaw. 4/11/2014 1:13 PM Assignment 10 10 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Part A How far to the left of the pivot must a 3.8 cat stand to keep the seesaw balanced? Express your answer to two significant figures and include the appropriate units. ANSWER: = Static Equilibrium of the Arm You are able to hold out your arm in an outstretched horizontal position because of the action of the deltoid muscle. Assume the humerus bone has a mass , length and its center of mass is a distance from the scapula. (For this problem ignore the rest of the arm.) The deltoid muscle attaches to the humerus a distance from the scapula. The deltoid muscle makes an angle of with the horizontal, as shown. Use throughout the problem. Part A Find the tension in the deltoid muscle. Express the tension in newtons, to the nearest integer. You did not open hints for this part. ANSWER: 4/11/2014 1:13 PM Assignment 10 11 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... = N Part B Using the conditions for static equilibrium, find the magnitude of the vertical component of the force exerted by the scapula on the humerus (where the humerus attaches to the rest of the body). Express your answer in newtons, to the nearest integer. You did not open hints for this part. ANSWER: = Part C Now find the magnitude of the horizontal component of the force exerted by the scapula on the humerus. Express your answer in newtons, to the nearest integer. ANSWER: = ± Moments around a Rod A rod is bent into an L shape and attached at one point to a pivot. The rod sits on a frictionless table and the diagram is a view from above. This means that gravity can be ignored for this problem. There are three forces that are applied to the rod at different points and angles: , , and . Note that the dimensions of the bent rod are in centimeters in the figure, although the answers are requested in SI units (kilograms, meters, seconds). 4/11/2014 1:13 PM Assignment 10 12 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Part A If and , what does the magnitude of have to be for there to be rotational equilibrium? Answer numerically in newtons to two significant figures. You did not open hints for this part. ANSWER: = N Part B If the L-shaped rod has a moment of inertia time , would it take for the object to move through ( , , and again , how long a /4 radians)? Assume that as the object starts to move, each force moves with the object so as to retain its initial angle relative to the object. Express the time in seconds to two significant figures. You did not open hints for this part. ANSWER: = s Part C Now consider the situation in which the rod. What does and , but now a force with nonzero magnitude is acting on have to be to obtain equilibrium? Give a numerical answer, without trigonometric functions, in newtons, to two significant figures. You did not open hints for this part. ANSWER: = N 4/11/2014 1:13 PM Assignment 10 13 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Problem 12.32 A car tire is 55.0 in diameter. The car is traveling at a speed of 24.0 . Part A What is the tire's rotation frequency, in rpm? Express your answer to three significant figures and include the appropriate units. ANSWER: Part B What is the speed of a point at the top edge of the tire? Express your answer to three significant figures and include the appropriate units. ANSWER: Part C What is the speed of a point at the bottom edge of the tire? Express your answer as an integer and include the appropriate units. ANSWER: Problem 12.33 A 460 , 8.00-cm-diameter solid cylinder rolls across the floor at 1.30 . Part A What is the can's kinetic energy? Express your answer with the appropriate units. 4/11/2014 1:13 PM Assignment 10 14 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... ANSWER: Problem 12.45 Part A What is the magnitude of the angular momentum of the 780 rotating bar in the figure ? ANSWER: Part B What is the direction of the angular momentum of the bar ? ANSWER: into the page out of the page Problem 12.46 4/11/2014 1:13 PM Assignment 10 15 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Part A What is the magnitude of the angular momentum of the 2.20 , 4.60-cm-diameter rotating disk in the figure ? ANSWER: Part B What is its direction? ANSWER: x direction -x direction y direction -y direction z direction -z direction Problem 12.60 A 3.0- -long ladder, as shown in the following figure, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.46. 4/11/2014 1:13 PM Assignment 10 16 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Part A What is the minimum angle the ladder can make with the floor without slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: = Problem 12.61 The 3.0- -long, 90 support 1. rigid beam in the following figure is supported at each end. An 70 student stands 2.0 from Part A How much upward force does the support 1 exert on the beam? Express your answer to two significant figures and include the appropriate units. ANSWER: 4/11/2014 1:13 PM Assignment 10 17 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... = Part B How much upward force does the support 2 exert on the beam? Express your answer to two significant figures and include the appropriate units. ANSWER: = Enhanced EOC: Problem 12.63 A 44 , 5.5the beam. -long beam is supported, but not attached to, the two posts in the figure . A 22 You may want to review ( boy starts walking along pages 330 - 334) . For help with math skills, you may want to review: The Vector Cross Product Part A How close can he get to the right end of the beam without it falling over? Express your answer to two significant figures and include the appropriate units. You did not open hints for this part. ANSWER: 4/11/2014 1:13 PM Assignment 10 18 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... = Problem 12.68 Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.6 diameter and a mass of 270 . Its maximum angular velocity is 1500 . Part A A motor spins up the flywheel with a constant torque of 54 speed? . How long does it take the flywheel to reach top Express your answer to two significant figures and include the appropriate units. ANSWER: = Part B How much energy is stored in the flywheel? Express your answer to two significant figures and include the appropriate units. ANSWER: = Part C The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.2 . What is the average power delivered to the machine? Express your answer to two significant figures and include the appropriate units. ANSWER: = 4/11/2014 1:13 PM Assignment 10 19 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Part D How much torque does the flywheel exert on the machine? Express your answer to two significant figures and include the appropriate units. ANSWER: = Problem 12.71 The 3.30 350 . , 40.0-cm-diameter disk in the figure is spinning at Part A How much friction force must the brake apply to the rim to bring the disk to a halt in 2.10 ? Express your answer with the appropriate units. ANSWER: Problem 12.74 A 5.0 , 60-diameter cylinder rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. 4/11/2014 1:13 PM Assignment 10 20 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... Part A What is the magnitude of the cylinder's initial angular acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: = Part B What is the magnitude of the cylinder's angular velocity when it is directly below the axle? Express your answer to two significant figures and include the appropriate units. ANSWER: = Problem 12.82 A 45 figure skater is spinning on the toes of her skates at 0.90 . Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 , 20 average diameter, 160 tall) plus two rod-like arms (2.5 each, 67 long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 , 20-diameter, 200-tall cylinder. Part A What is her new rotation frequency, in revolutions per second? Express your answer to two significant figures and include the appropriate units. ANSWER: 4/11/2014 1:13 PM Assignment 10 21 of 21 http://session.masteringphysics.com/myct/assignmentPrintView?displayM... = Score Summary: Your score on this assignment is 4.0%. You received 7.84 out of a possible total of 198 points. 4/11/2014 1:13 PM Physics 220 - HW #1 1 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... aalromia::app-6@purdue Summer-2013-PHYS-22000-01-XLST, Summer 1 2013 Instructor: Shawn Slavin WebAssign Physics 220 - HW #1 (Homework) Current Score : – / 20 Due : Wednesday, May 22 2013 11:59 PM EDT 1. –/2 points SerCP9 1.P.006. 2 2 Kinetic energy KE has dimensions kg · m /s . It can be written in terms of the momentum p and mass m as 2 KE = P . 2m (a) Determine the proper units for momentum using dimensional analysis. 2 kg · m/s kg · m/s 2 kg · m /s 2 kg · m/s 2 (b) Force has the SI units kg · m/s . Given the units of force, write a simple equation relating a constant force F exerted on an object, an interval of time t during which the force is applied, and the resulting momentum of the object, p. (Do this on paper. Your instructor may ask you to turn in this work.) Show My Work (Optional) 2. –/2 points SerCP9 1.P.502.XP. You can obtain a rough estimate of the size of a molecule by the following simple experiment. Let a droplet of oil spread out on a smooth surface of water. The resulting oil slick will be approximately one molecule thick. Given an oil droplet of mass 6.0 10 -7 3 kg and density 1054 kg/m that spreads out into a circle of radius 41.8 cm on the water surface, what is the order of magnitude of the diameter of an oil molecule? 10 −5 10 −7 10 −9 10 −11 10 −14 Show My Work (Optional) 3. –/2 points SerCP9 1.P.016. A small turtle moves at a speed of 459 furlongs per fortnight. Find the speed of the turtle in centimeters per second. Note that 1 furlong = 220 yards and 1 fortnight = 14 days. cm/s Show My Work (Optional) 4. –/2 points SerCP9 1.P.035.MI.FB. A point is located in a polar coordinate system by the coordinates r = 3.0 m and θ = 30°. Find the x- and y-coordinates of this point, assuming that the two coordinate systems have the same origin. x= m y= m Show My Work (Optional) 19-05-2013 13:42 Physics 220 - HW #1 2 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... 5. –/2 points SerCP9 1.P.045. In the figure below, find each of the following. (a) the side opposite θ (b) the side adjacent to (c) cos θ (d) sin (e) tan Show My Work (Optional) 6. –/2 points SerCP9 2.P.028.WI. In 1865, Jules Verne proposed sending men to the Moon by firing a space capsule from a 220-m-long cannon with final speed of 10.97 km/s. What would have been the unrealistically large acceleration experienced by the space travelers during their launch? (A human can stand an acceleration of 15g for a short time.) 2 m/s 2 Compare your answer with the free-fall acceleration, 9.80 m/s (i.e. how many times stronger than gravity is this force?). g Show My Work (Optional) 7. –/2 points SerCP9 2.P.045. A ball is thrown vertically upward with a speed of 18.0 m/s. (a) How high does it rise? m (b) How long does it take to reach its highest point? s (c) How long does the ball take to hit the ground after it reaches its highest point? s (d) What is its velocity when it returns to the level from which it started? m/s Show My Work (Optional) 19-05-2013 13:42 Physics 220 - HW #1 3 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... 8. –/2 points SerCP9 3.P.001. Vector A has a magnitude of 25 units and points in the positive y-direction. When vector B is added to A, the resultant vector A + B points in the negative y-direction with a magnitude of 14 units. Find the magnitude and direction of B? magnitude unit(s) direction Show My Work (Optional) 9. –/2 points SerCP9 3.P.010. A person walks 17.0° north of east for 2.70 km. How far due north and how far due east would she have to walk to arrive at the same location? north km east km Show My Work (Optional) 10.–/2 points SerCP9 3.P.025.WI. The best leaper in the animal kingdom is the puma, which can jump to a height of 3.7 m when leaving the ground at an angle of 45°. With what speed must the animal leave the ground to reach that height? m/s Show My Work (Optional) 19-05-2013 13:42 Physics 220 - HW #1 1 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... aalsadah::app-6@purdue Summer-2013-PHYS-22000-01-XLST, Summer 1 2013 Instructor: Shawn Slavin WebAssign Physics 220 - HW #1 (Homework) Current Score : – / 20 Due : Wednesday, May 22 2013 11:59 PM EDT 1. –/2 points SerCP9 1.P.006. 2 2 Kinetic energy KE has dimensions kg · m /s . It can be written in terms of the momentum p and mass m as 2 KE = P . 2m (a) Determine the proper units for momentum using dimensional analysis. 2 kg · m/s kg · m/s 2 kg · m /s 2 kg · m/s 2 (b) Force has the SI units kg · m/s . Given the units of force, write a simple equation relating a constant force F exerted on an object, an interval of time t during which the force is applied, and the resulting momentum of the object, p. (Do this on paper. Your instructor may ask you to turn in this work.) Show My Work (Optional) 2. –/2 points SerCP9 1.P.502.XP. You can obtain a rough estimate of the size of a molecule by the following simple experiment. Let a droplet of oil spread out on a smooth surface of water. The resulting oil slick will be approximately one molecule thick. Given an oil droplet of mass 1.0 10 -6 3 kg and density 882 kg/m that spreads out into a circle of radius 41.8 cm on the water surface, what is the order of magnitude of the diameter of an oil molecule? 10 −5 10 −7 10 −9 10 −11 10 −14 Show My Work (Optional) 3. –/2 points SerCP9 1.P.016. A small turtle moves at a speed of 219 furlongs per fortnight. Find the speed of the turtle in centimeters per second. Note that 1 furlong = 220 yards and 1 fortnight = 14 days. cm/s Show My Work (Optional) 4. –/2 points SerCP9 1.P.035.MI.FB. A point is located in a polar coordinate system by the coordinates r = 5.6 m and θ = 22°. Find the x- and y-coordinates of this point, assuming that the two coordinate systems have the same origin. x= m y= m Show My Work (Optional) 19-05-2013 20:37 Physics 220 - HW #1 2 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... 5. –/2 points SerCP9 1.P.045. In the figure below, find each of the following. (a) the side opposite θ (b) the side adjacent to (c) cos θ (d) sin (e) tan Show My Work (Optional) 6. –/2 points SerCP9 2.P.028.WI. In 1865, Jules Verne proposed sending men to the Moon by firing a space capsule from a 220-m-long cannon with final speed of 10.97 km/s. What would have been the unrealistically large acceleration experienced by the space travelers during their launch? (A human can stand an acceleration of 15g for a short time.) 2 m/s 2 Compare your answer with the free-fall acceleration, 9.80 m/s (i.e. how many times stronger than gravity is this force?). g Show My Work (Optional) 7. –/2 points SerCP9 2.P.045. A ball is thrown vertically upward with a speed of 11.0 m/s. (a) How high does it rise? m (b) How long does it take to reach its highest point? s (c) How long does the ball take to hit the ground after it reaches its highest point? s (d) What is its velocity when it returns to the level from which it started? m/s Show My Work (Optional) 19-05-2013 20:37 Physics 220 - HW #1 3 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... 8. –/2 points SerCP9 3.P.001. Vector A has a magnitude of 29 units and points in the positive y-direction. When vector B is added to A, the resultant vector A + B points in the negative y-direction with a magnitude of 12 units. Find the magnitude and direction of B? magnitude unit(s) direction Show My Work (Optional) 9. –/2 points SerCP9 3.P.010. A person walks 27.0° north of east for 2.20 km. How far due north and how far due east would she have to walk to arrive at the same location? north km east km Show My Work (Optional) 10.–/2 points SerCP9 3.P.025.WI. The best leaper in the animal kingdom is the puma, which can jump to a height of 3.7 m when leaving the ground at an angle of 45°. With what speed must the animal leave the ground to reach that height? m/s Show My Work (Optional) 19-05-2013 20:37 Physics 220 - HW #1 1 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... akalajmi::app-6@purdue Summer-2013-PHYS-22000-01-XLST, Summer 1 2013 Instructor: Shawn Slavin WebAssign Physics 220 - HW #1 (Homework) Current Score : – / 20 Due : Wednesday, May 22 2013 11:59 PM EDT 1. –/2 points SerCP9 1.P.006. 2 2 Kinetic energy KE has dimensions kg · m /s . It can be written in terms of the momentum p and mass m as 2 KE = P . 2m (a) Determine the proper units for momentum using dimensional analysis. 2 kg · m/s kg · m/s 2 kg · m /s 2 kg · m/s 2 (b) Force has the SI units kg · m/s . Given the units of force, write a simple equation relating a constant force F exerted on an object, an interval of time t during which the force is applied, and the resulting momentum of the object, p. (Do this on paper. Your instructor may ask you to turn in this work.) Show My Work (Optional) 2. –/2 points SerCP9 1.P.502.XP. You can obtain a rough estimate of the size of a molecule by the following simple experiment. Let a droplet of oil spread out on a smooth surface of water. The resulting oil slick will be approximately one molecule thick. Given an oil droplet of mass 8.0 10 -7 3 kg and density 1122 kg/m that spreads out into a circle of radius 41.8 cm on the water surface, what is the order of magnitude of the diameter of an oil molecule? 10 −5 10 −7 10 −9 10 −11 10 −14 Show My Work (Optional) 3. –/2 points SerCP9 1.P.016. A small turtle moves at a speed of 213 furlongs per fortnight. Find the speed of the turtle in centimeters per second. Note that 1 furlong = 220 yards and 1 fortnight = 14 days. cm/s Show My Work (Optional) 4. –/2 points SerCP9 1.P.035.MI.FB. A point is located in a polar coordinate system by the coordinates r = 5.6 m and θ = 34°. Find the x- and y-coordinates of this point, assuming that the two coordinate systems have the same origin. x= m y= m Show My Work (Optional) 19-05-2013 13:21 Physics 220 - HW #1 2 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... 5. –/2 points SerCP9 1.P.045. In the figure below, find each of the following. (a) the side opposite θ (b) the side adjacent to (c) cos θ (d) sin (e) tan Show My Work (Optional) 6. –/2 points SerCP9 2.P.028.WI. In 1865, Jules Verne proposed sending men to the Moon by firing a space capsule from a 220-m-long cannon with final speed of 10.97 km/s. What would have been the unrealistically large acceleration experienced by the space travelers during their launch? (A human can stand an acceleration of 15g for a short time.) 2 m/s 2 Compare your answer with the free-fall acceleration, 9.80 m/s (i.e. how many times stronger than gravity is this force?). g Show My Work (Optional) 7. –/2 points SerCP9 2.P.045. A ball is thrown vertically upward with a speed of 11.0 m/s. (a) How high does it rise? m (b) How long does it take to reach its highest point? s (c) How long does the ball take to hit the ground after it reaches its highest point? s (d) What is its velocity when it returns to the level from which it started? m/s Show My Work (Optional) 19-05-2013 13:21 Physics 220 - HW #1 3 of 3 http://www.webassign.net/web/Student/Assignment-Responses/last?d... 8. –/2 points SerCP9 3.P.001. Vector A has a magnitude of 29 units and points in the positive y-direction. When vector B is added to A, the resultant vector A + B points in the negative y-direction with a magnitude of 15 units. Find the magnitude and direction of B? magnitude unit(s) direction Show My Work (Optional) 9. –/2 points SerCP9 3.P.010. A person walks 27.0° north of east for 2.90 km. How far due north and how far due east would she have to walk to arrive at the same location? north km east km Show My Work (Optional) 10.–/2 points SerCP9 3.P.025.WI. The best leaper in the animal kingdom is the puma, which can jump to a height of 3.7 m when leaving the ground at an angle of 45°. With what speed must the animal leave the ground to reach that height? m/s Show My Work (Optional) 19-05-2013 13:21 Assignment 1 Due: 11:59pm on Wednesday, February 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 1.6 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Positive Negative Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Negative Positive Correct Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Positive Negative Correct Conceptual Question 1.7 Part A Determine the sign (positive or negative) of the position for the particle in the figure. ANSWER: Positive Negative Correct Part B Determine the sign (positive or negative) of the velocity for the particle in the figure. ANSWER: Positive Negative Correct Part C Determine the sign (positive or negative) of the acceleration for the particle in the figure. ANSWER: Negative Positive Correct Enhanced EOC: Problem 1.18 The figure shows the motion diagram of a drag racer. The camera took one frame every 2 s. You may want to review ( pages 16 - 19) . For help with math skills, you may want to review: Plotting Points on a Graph Part A Make a position-versus-time graph for the drag racer. Hint 1. How to approach the problem Based on Table 1.1 in the book/e-text, what two observables are associated with each point? Which position or point of the drag racer occurs first? Which position occurs last? If you label the first point as happening at t = 0 s, at what time does the next point occur? At what time does the last position point occur? What is the position of a point halfway in between x ANSWER: = 0 m and x = 200 m? Can you think of a way to estimate the positions of the points using a ruler? Correct Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. Two toy rockets are traveling in the same direction (taken to be the x axis). A diagram is shown of a time-exposure image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. Part A At what time(s) do the rockets have the same velocity? Hint 1. How to determine the velocity The diagram shows position, not velocity. You can't find instantaneous velocity from this diagram, but you can determine the average velocity between two times t1 and t2 : vavg[t1 , t2 ] = x(t2 )−x(t1 ) . t2 −t1 Note that no position values are given in the diagram; you will need to estimate these based on the distance between successive positions of the rockets. ANSWER: = 1 only at time t = 4 only at times t = 1 and t = 4 at time t at some instant in time between t at no time shown in the figure Correct = 1 and t = 4 Part B At what time(s) do the rockets have the same x position? ANSWER: = 1 only at time t = 4 only at times t = 1 and t = 4 at time t at some instant in time between t = 1 and t = 4 at no time shown in the figure Correct Part C At what time(s) do the two rockets have the same acceleration? Hint 1. How to determine the acceleration The velocity is related to the spacing between images in a stroboscopic diagram. Since acceleration is the rate at which velocity changes, the acceleration is related to the how much this spacing changes from one interval to the next. ANSWER: = 1 only at time t = 4 only at times t = 1 and t = 4 at time t at some instant in time between t at no time shown in the figure = 1 and t = 4 Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. ANSWER: and nonzero acceleration velocity displacement time Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. ANSWER: and nonzero acceleration velocity displacement time Correct Part F At what time(s) is rocket A ahead of rocket B? Hint 1. Use the diagram You can answer this question by looking at the diagram and identifying the time(s) when rocket A is to the right of rocket B. ANSWER: = 1 only after t = 4 only before t = 1 and after t = 4 between t = 1 and t = 4 before t at no time(s) shown in the figure Correct Dimensions of Physical Quantities Learning Goal: To introduce the idea of physical dimensions and to learn how to find them. Physical quantities are generally not purely numerical: They have a particular dimension or combination of dimensions associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and inches are different units they have the same dimension--length. Part A In classical mechanics there are three base dimensions. Length is one of them. What are the other two? Hint 1. MKS system The current system of units is called the International System (abbreviated SI from the French Système International). In the past this system was called the mks system for its base units: meter, kilogram, and second. What are the dimensions of these quantities? ANSWER: acceleration and mass acceleration and time acceleration and charge mass and time mass and charge time and charge Correct There are three dimensions used in mechanics: length ( l), mass ( m), and time ( t). A combination of these three dimensions suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are called derived dimensions and the units they are measured in are called derived units. [A] = l2. (Note that "dimensions of variable x" is symbolized as [x].) You can find these dimensions by looking at the formula for the area of a square A = s 2 , where s is the length of a side of the square. Clearly [s] = l. Plugging this into the equation gives [A] = [s]2 = l2. For example, area A has derived dimensions Part B Find the dimensions [V ] of volume. Express your answer as powers of length ( l), mass ( m), and time ( t). Hint 1. Equation for volume You have likely learned many formulas for the volume of various shapes in geometry. Any of these equations will give you the dimensions for volume. You can find the dimensions most easily from the volume of a cube V = e3, where e is the length of the edge of the cube. ANSWER: [V ] = l3 Correct Part C Find the dimensions [v] of speed. Express your answer as powers of length ( l), mass ( m), and time ( t). Hint 1. Equation for speed Speed v is defined in terms of distance d and time t as v= Therefore, d . t [v] = [d]/[t]. Hint 2. Familiar units for speed You are probably accustomed to hearing speeds in miles per hour (or possibly kilometers per hour). Think about the dimensions for miles and hours. If you divide the dimensions for miles by the dimensions for hours, you will have the dimensions for speed. ANSWER: [v] = lt−1 Correct The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This means that dimensions as speed. ∆v, which comes from subtracting two speeds, has the same It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be incorrect. This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base dimensions. (As in the preceeding example, l + t is not a valid dimension for a physical quantitiy.) A valid dimension will only involve the product or ratio of powers of the base dimensions (e.g. m2/3 l2 t−2 ). Part D Find the dimensions [a] of acceleration. Express your answer as powers of length ( l), mass ( m), and time ( t). Hint 1. Equation for acceleration In physics, acceleration a is defined as the change in velocity in a certain time. This is shown by the equation a = ∆v/∆t. The ∆ is a symbol that means "the change in." ANSWER: [a] = lt−2 Correct Consistency of Units In physics, every physical quantity is measured with respect to a unit. Time is measured in seconds, length is measured in meters, and mass is measured in kilograms. Knowing the units of physical quantities will help you solve problems in physics. Part A Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by the equation F= where F is the magnitude of the gravitational attraction on either body, Gm1 m2 , r2 m1 and m2 are the masses of the bodies, r is the distance between them, and G is the gravitational constant. In SI units, the units of force are kg ⋅ m/s , the units of mass are kg, and the units of distance are m. For this equation to have consistent units, the units of G must be which of the following? 2 Hint 1. How to approach the problem To solve this problem, we start with the equation F= Gm1 m2 . r2 For each symbol whose units we know, we replace the symbol with those units. For example, we replace m1 with kg. We now solve this equation for G. ANSWER: kg3 m⋅s 2 kg⋅s2 m3 m3 kg⋅s2 m kg⋅s2 Correct Part B One consequence of Einstein's theory of special relativity is that mass is a form of energy. This mass-energy relationship is perhaps the most famous of all physics equations: E = mc2 , where m is mass, c is the speed of the light, and E is the energy. In SI units, the units of speed are m/s. For the preceding equation to have consistent units (the same units on both sides of the equation), the units of E must be which of the following? Hint 1. How to approach the problem To solve this problem, we start with the equation E = mc2 . For each symbol whose units we know, we replace the symbol with those units. For example, we replace m with kg. We now solve this equation for ANSWER: E. kg⋅m s kg⋅m2 s2 kg⋅s2 m2 kg⋅m2 s Correct To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose units we know, we replace the symbol with those units. For example, we replace m with kg. We now solve this equation for the units of the unknown variable. Problem 1.24 Convert the following to SI units: Part A 5.0in Express your answer to two significant figures and include the appropriate units. ANSWER: 0.13 m Correct Part B 54ft/s Express your answer to two significant figures and include the appropriate units. ANSWER: m 16 s Correct Part C 72mph Express your answer to two significant figures and include the appropriate units. ANSWER: m 32 s Correct Part D 17in2 Express your answer to two significant figures and include the appropriate units. ANSWER: 1.1×10−2 m2 Correct Problem 1.55 The figure shows a motion diagram of a car traveling down a street. The camera took one frame every 10 s. A distance scale is provided. Part A Make a position-versus-time graph for the car. ANSWER: Incorrect; Try Again ± Moving at the Speed of Light Part A How many nanoseconds does it take light to travel a distance of 4.40km in vacuum? Express your answer numerically in nanoseconds. Hint 1. How to approach the problem Light travels at a constant speed; therefore, you can use the formula for the distance traveled in a certain amount of time by an object moving at constant speed. Before performing any calculations, it is often recommended, although it is not strictly necessary, to convert all quantities to their fundamental units rather than to multiples of the fundamental unit. Hint 2. Find how many seconds it takes light to travel the given distance Given that the speed of light in vacuum is 3.00 × 108 m/s, how many seconds does it take light to travel a distance of 4.40km ? Express your answer numerically in seconds. Hint 1. Find the time it takes light to travel a certain distance How long does it take light to travel a distance r? Let c be the speed of light. Hint 1. The speed of an object The equation that relates the distance s traveled by an object with constant speed v in a time t is s = vt. ANSWER: r⋅c r c c r Correct Hint 2. Convert the given distance to meters Convert d = 4.40km to meters. Express your answer numerically in meters. Hint 1. Conversion of kilometers to meters Recall that 1 km = 103 m. ANSWER: 4.40km = 4400 m Correct ANSWER: −5 1.47×10 s Correct Now convert the time into nanoseconds. Recall that ANSWER: 1.47×104 ns Correct Score Summary: Your score on this assignment is 84.7%. You received 50.84 out of a possible total of 60 points. 1 ns = 10−9 s. Assignment 2 Due: 11:59pm on Wednesday, February 12, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 2.6 Part A The figure shows the position-versus-time graph for a moving object. At which lettered point or points: Is the object moving the slowest? Is the object moving the fastest? Is the object at rest? Drag the appropriate items to their respective bins. ANSWER: Correct Part B At which lettered point or points is the object moving to the negative direction? ANSWER: A B C D E Correct Conceptual Question 2.7 The figure shows the position-versus-time graph for a moving object. At which lettered point or points: Part A Is the object moving the fastest? ANSWER: A B C D E F Correct Part B Is the object speeding up? ANSWER: A B C D E F Correct Part C Is the object moving to the left and turning around? ANSWER: Correct Kinematic Vocabulary One of the difficulties in studying mechanics is that many common words are used with highly specific technical meanings, among them velocity, acceleration, position, speed, and displacement. The series of questions in this problem is designed to get you to try to think of these quantities like a physicist. Answer the questions in this problem using words from the following list: A. position B. direction C. displacement D. E. F. G. H. I. J. K. coordinates velocity acceleration distance magnitude vector scalar components Part A Velocity differs from speed in that velocity indicates a particle's __________ of motion. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: Correct Part B Unlike speed, velocity is a __________ quantity. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: Correct Part C A vector has, by definition, both __________ and direction. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: Correct Part D Once you have selected a coordinate system, you can express a two-dimensional vector using a pair of quantities known collectively as __________. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: Correct Part E Speed differs from velocity in the same way that __________ differs from displacement. Enter the letter from the list given in the problem introduction that best completes the sentence. Hint 1. Definition of displacement Displacement is the vector that indicates the difference of two positions (e.g., the final position from the initial position). Being a vector, it is independent of the coordinate system used to describe it (although its vector components depend on the coordinate system). ANSWER: Correct Part F Consider a physical situation in which a particle moves from point A to point B. This process is described from two coordinate systems that are identical except that they have different origins. The __________ of the particle at point A differ(s) as expressed in one coordinate system compared to the other, but the __________ from A to B is/are the same as expressed in both coordinate systems. Type the letters from the list given in the problem introduction that best complete the sentence. Separate the letters with commas. There is more than one correct answer, but you should only enter one pair of comma-separated letters. For example, if the words "vector" and "scalar" fit best in the blanks, enter I,J. ANSWER: Correct The coordinates of a point will depend on the coordinate system that is chosen, but there are several other quantities that are independent of the choice of origin for a coordinate system: in particular, distance, displacement, direction, and velocity. In working physics problems, unless you are interested in the position of an object or event relative to a specific origin, you can usually choose the coordinate system origin to be wherever is most convenient or intuitive. ⃗ Note that the vector indicating a displacement from A to B is usually represented as r BA Part G Identify the following physical quantities as scalars or vectors. ANSWER: ⃗ . = r B⃗ − r A Correct Problem 2.4 The figure is the position-versus-time graph of a jogger. Part A What is the jogger’s velocity at t = 10 s? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 1.3 m s Answer Requested Part B What is the jogger’s velocity at t = 25 s? Express your answer to two significant figures and include the appropriate units. ANSWER: v= 0 m s Correct Part C What is the jogger’s velocity at t = 35 s? Express your answer to two significant figures and include the appropriate units. ANSWER: v = -5.0 m s Correct Analyzing Position versus Time Graphs: Conceptual Question Two cars travel on the parallel lanes of a two-lane road. The cars’ motions are represented by the position versus time graph shown in the figure. Answer the questions using the times from the graph indicated by letters. Part A At which of the times do the two cars pass each other? Hint 1. Two cars passing Two objects can pass each other only if they have the same position at the same time. ANSWER: A B C D E None Cannot be determined Correct Part B Are the two cars traveling in the same direction when they pass each other? ANSWER: yes no Correct Part C At which of the lettered times, if any, does car #1 momentarily stop? Hint 1. Determining velocity from a position versus time graph The slope on a position versus time graph is the "rise" (change in position) over the "run" (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. ANSWER: A B C D E none cannot be determined Correct Part D At which of the lettered times, if any, does car #2 momentarily stop? Hint 1. Determining velocity from a position versus time graph The slope on a position versus time graph is the "rise" (change in position) over the "run" (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. ANSWER: A B C D E none cannot be determined Correct Part E At which of the lettered times are the cars moving with nearly identical velocity? Hint 1. Determining Velocity from a Position versus Time Graph The slope on a position versus time graph is the “rise” (change in position) over the “run” (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. ANSWER: A B C D E None Cannot be determined Correct Problem 2.6 A particle starts from 10m at t0 = 0 and moves with the velocity graph shown in the figure. Part A Does this particle have a turning point? ANSWER: Yes No Correct Part B If so, at what time? Express your answer using two significant figures and include the appropriate units. ANSWER: t = 1.0 s Correct Part C What is the object's position at t = 2, 3, 4 s ? Express your answers using two significant figures separated by commas. ANSWER: x2 , x3 , x4 = 10,16,26 m Correct Overcoming a Head Start Cars A and B are racing each other along the same straight road in the following manner: Car A has a head start and is a distance DA beyond the starting line at t = 0. The starting line is at Car A travels at a constant speed vA . Car B starts at the starting line but has a better engine than Car A, and thus Car B travels at a constant speed vB , which is greater than vA . Part A How long after Car B started the race will Car B catch up with Car A? Express the time in terms of given quantities. Hint 1. Consider the kinematics relation Write an expression for the displacement of Car A from the starting line at a time t after Car B starts. (Note that we are taking this time to be t Answer in terms of vA , vB, DA, and t for time, and take x = 0 at the starting line. Hint 1. What is the acceleration of Car A? The acceleration of Car A is zero, so the general formula x(t) = x0 + v0 t + (1/2)at2 has at least one term equal to zero. ANSWER: xA (t) = DA + vA t Hint 2. What is the relation between the positions of the two cars? = 0.) x = 0. The positions of the two cars are equal at time tcatch. Hint 3. Consider Car B's position as a function of time Write down an expression for the position of Car B at time t after starting. Give your answer in terms of any variables needed (use t for time). ANSWER: xB (t) = vB t ANSWER: D tcatch = v −Av B A Correct Part B How far from Car B's starting line will the cars be when Car B passes Car A? Express your answer in terms of known quantities. (You may use tcatch as well.) Hint 1. Which expression should you use? Just use your expression for the position of either car after time t ANSWER: dpass = vvB−DvA B A Correct = 0, and substitute in the correct value for tcatch (found in the previous part). Problem 2.11 The figure shows the velocity graph of a particle moving along the x-axis. Its initial position is x0 particle's (a) position, (b) velocity, and (c) acceleration? Part A Express your answer to two significant figures and include the appropriate units. ANSWER: x = 6.0 m Correct Part B Express your answer to two significant figures and include the appropriate units. ANSWER: vx = 4.0 m s = 2 m at t0 = 0 . At t = 2s , what are the Correct Part C Express your answer to two significant figures and include the appropriate units. ANSWER: m ax = 2.0 s 2 Correct Problem 2.13 A jet plane is cruising at 300m/s when suddenly the pilot turns the engines up to full throttle. After traveling 3.9km , the jet is moving with a speed of 400 m/s. Part A What is the jet's acceleration, assuming it to be a constant acceleration? Express your answer to two significant figures and include the appropriate units. ANSWER: m a = 9.0 s 2 Correct Enhanced EOC: Problem 2.20 A rock is tossed straight up with a velocity of 22 m/s When it returns, it falls into a hole 10 You may want to review ( pages 51 - 54) . m deep. For help with math skills, you may want to review: Quadratic Equations For general problem-solving tips and strategies for this topic, you may want to view a Video Tutor Solution of Time in the air for a tossed ball. Part A What is the rock's velocity as it hits the bottom of the hole? Express your answer with the appropriate units. Hint 1. How to approach the problem Start by drawing a picture of the path of the rock, including its launch point, initial direction, and end point in the hole. Choose a coordinate system, and indicate it on your picture. Where is y = 0 m? What is the positive y direction? What is the position of the launch point and the bottom of the hole? In this coordinate system, what is the sign of the initial velocity and the sign of the acceleration? Calling the launch time t = 0, what is the equation for y as a function of time? What is the y position at the bottom of the hole? This will lead to a quadratic equation for the time t when the rock hits the bottom of the hole. The quadratic equation has two solutions for the time. Not all mathematical solutions make sense physically. Which solution makes sense physically in terms of the picture that you drew at the beginning? Keeping the same coordinate system, what is the velocity in the y direction as a function of time? What is the y velocity when the rock hits the bottom of the hole? ANSWER: v = -26.1 m s Correct Part B How long is the rock in the air, from the instant it is released until it hits the bottom of the hole? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the time the rock was in the air related to the time at which the rock hit the ground in Part A? ANSWER: t = 4.90 s Correct Enhanced EOC: Problem 2.23 A particle moving along the x-axis has its position described by the function x acceleration? You may want to review ( pages 38 - 42) . For help with math skills, you may want to review: Differentiation of Polynomial Functions = ( 2.00 t3 − 5.00 t + 5.00 ) m, where t is in s. At t= 4.00, what are the particle's (a) position, (b) velocity, and (c) Part A Express your answer with the appropriate units. Hint 1. How to approach the problem Evaluate the position at time t= 4.00 s. ANSWER: 113 m Correct Part B Express your answer with the appropriate units. Hint 1. How to approach the problem How do you determine the velocity as a function of time, v(t), from the position, x(t)? What calculus operation do you have to perform? Once you have v(t), how do you determine v at a particular time? ANSWER: m 91.0 s Correct Part C Express your answer with the appropriate units. Hint 1. How to approach the problem How do you determine the acceleration as a function of time, a(t), from the velocity, v(t)? What calculus operation do you have to perform? Once you have a(t), how do you determine the acceleration at a particular time? ANSWER: 48.0 m s2 Correct Problem 2.26 A particle's position on the x-axis is given by the function x = (t2 − 6.00 t + 6.00 ) m, where t is in s. Part A Where is the particle when vx = 4.00m/s ? Express your answer with the appropriate units. ANSWER: 1.00 m Correct Problem 2.30 A particle's velocity is described by the function vx = t2 − 7t + 7 m/s, where t is in s. Part A How many turning points does the particle reach. Express your answer as an integer. ANSWER: 2 Correct Part B At what times does the particle reach its turning points? Express your answers using two significant figures separated by a comma. ANSWER: t1 , t2 = 5.8,1.2 s Correct Part C What is the particle's acceleration at each of the turning points? Express your answers using two significant figures separated by a comma. ANSWER: a1 , a2 = 4.6,-4.6 m/s2 Correct Problem 2.49 A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 35m/s2 for 30s , then runs out of fuel. Ignore any air resistance effects. Part A What is the rocket's maximum altitude? Express your answer to two significant figures and include the appropriate units. ANSWER: h = 72 km Correct Part B How long is the rocket in the air? Express your answer to two significant figures and include the appropriate units. ANSWER: t = 260 s Answer Requested Problem 2.52 A hotel elevator ascends 200 m with maximum speed of 5 m/s . Its acceleration and deceleration both have a magnitude of 1.0 m/s2 . Part A How far does the elevator move while accelerating to full speed from rest? Express your answer with the appropriate units. ANSWER: 12.5 m Correct Part B How long does it take to make the complete trip from bottom to top? Express your answer with the appropriate units. ANSWER: 45.0 s Answer Requested Components of Vectors Shown is a 10 by 10 grid, with coordinate axes x and y . The grid runs from -5 to 5 on both axes. Drawn on this grid are four vectors, labeled A ⃗ through D⃗ . This problem will ask you various questions about these vectors. All answers should be in decimal notation, unless otherwise specified. Part A What is the x component of A?⃗ Express your answer to two significant figures. Hint 1. How to derive the component A component of a vector is its length (but with appropriate sign) along a particular coordinate axis, the axes being specfied in advance. You are asked for the component of the x axis, which is horizontal in this problem. Imagine two lines perpendicular to the x axis running from the head (end with the arrow) and tail of axis between the points where these lines intersect is the x component of the origin (because the tail of the vector is at the origin). ANSWER: Ax = 2.5 Correct Part B What is the y component of A?⃗ Express your answer to the nearest integer. ANSWER: Ay = 3 Correct A⃗ that lies along A⃗ down to the x axis. The length of the x A.⃗ In this problem, the x component is the x coordinate at which the perpendicular from the head of the vector hits Part C What is the y component of B⃗ ? Express your answer to the nearest integer. Hint 1. Consider the direction Don't forget the sign. ANSWER: By = -3 Correct Part D What is the x component of C ?⃗ Express your answer to the nearest integer. Hint 1. How to find the start and end points of the vector components A vector is defined only by its magnitude and direction. The starting point of the vector is of no consequence to its definition. Therefore, you need to somehow eliminate the starting point from C,⃗ and another to the tail, with the x component being the difference between x coordinates of head and tail (negative if the tail is to the right of the head). Another way is to imagine bringing the tail of C ⃗ to the origin, and then using the same procedure you used before your answer. You can run two perpendiculars to the x axis, one from the head (end with the arrow) of to find the components of ANSWER: Cx = -2 A⃗ and B⃗ . This is equivalent to the previous method, but it might be easier to visualize. Correct The following questions will ask you to give both components of vectors using the ordered pairs method. In this method, the x component is written first, followed by a comma, and then the y component. For example, the components of A⃗ would be written 2.5,3 in ordered pair notation. The answers below are all integers, so estimate the components to the nearest whole number. Part E In ordered pair notation, write down the components of vector B⃗ . Express your answers to the nearest integer. ANSWER: Bx, By = 2,-3 Correct Part F In ordered pair notation, write down the components of vector D⃗ . Express your answers to the nearest integer. ANSWER: Dx , Dy = 2,-3 Correct Part G What is true about B⃗ and D⃗ ? Choose from the pulldown list below. ANSWER: They have different components and are not the same vectors. They have the same components but are not the same vectors. They are the same vectors. Correct Problem 3.6 Find x- and y-components of the following vectors. Part A r ⃗ = (430m, 60� below positive x − axis) Express your answers using two significant figures. Enter your answers numerically separated by a comma. ANSWER: rx, ry = 210,-370 m Correct Part B v ⃗ = (610m/s, 23� above positive x − axis) Express your answers using two significant figures. Enter your answers numerically separated by a comma. ANSWER: vx , vy = 560,240 m/s Correct Part C a⃗ = (7.3m/s2 , negative y − direction) Express your answers using two significant figures. Enter your answers numerically separated by a comma. ANSWER: ax , ay = 0,-7.3 m/s2 Correct Problem 3.10 Part A Draw B⃗ = −4 ^ı + 4^ȷ. Draw the vector with its tail at the origin. ANSWER: Correct Part B Find the magnitude of B⃗ . Express your answer using two significant figures. ANSWER: B = 5.7 Correct Part C Find the direction of B⃗ . Express your answer using two significant figures. ANSWER: θ B = 45 � above the negative x-axis Correct Part D Draw r ⃗ = (−2.0 ^ı − 1.0^ȷ ) cm. Draw the vector with its tail at the origin. ANSWER: Correct Part E Find the magnitude of r .⃗ Express your answer using two significant figures. ANSWER: r = 2.2 cm Correct Part F Find the direction of r .⃗ ANSWER: θ r = 26.6 � below the negative x-axis Correct Part G Draw v ⃗ = (−10 ^ı − 100^ȷ ) m/s. Draw the vector with its tail at the origin. ANSWER: Correct Part H Find the magnitude of v.⃗ Express your answer using four significant figures. ANSWER: v = 100.5 m/s Correct Part I Find the direction of v.⃗ ANSWER: θ v = 84.3 � below the negative x-axis Correct Part J Draw a⃗ = (20 ^ı + 10^ȷ ) m/s2. Draw the vector with it's tail at the origin. ANSWER: Correct Part K Find the magnitude of a.⃗ ANSWER: a = 22.4 m/s2 Correct Part L Find the direction of a.⃗ ANSWER: θ a = 26.6 � above the positive x-axis Correct Problem 3.14 Let A⃗ = 5 ^ı − 2^ȷ, B⃗ = −2 ^ı + 6^ȷ, and D⃗ = A⃗ − B.⃗ Part A What is the component form of vector ANSWER: D⃗ = 7 ^ı − 8^ȷ D⃗ = −7 ^ı − 5^ȷ D⃗ = 7 ^ı + 8^ȷ D⃗ = 4 ^ı + 5^ȷ Correct Part B What is the magnitude of vector ANSWER: D⃗ ? D⃗ ? D = 10.6 Correct Part C What is the direction of vector D⃗ ? Express your answer using two significant figures. ANSWER: θ = 49 � below positive x-axis Correct Problem 3.15 Let A⃗ = 4 ^ı − 2^ȷ, B⃗ = −3 ^ı + 5^ȷ, and E⃗ = 4A⃗ + 2B.⃗ Part A Write vector E⃗ in component form. ANSWER: E⃗ = 10 ^ı + 2^ȷ E⃗ = ^ı + 10^ȷ E⃗ = −10^ȷ E⃗ = 10 ^ı − 2^ȷ Correct Part B Draw vectors A,⃗ B⃗ , and E.⃗ Draw the vectors with their tails at the origin. ANSWER: Correct Part C What is the magnitude of vector E?⃗ Express your answer using two significant figures. ANSWER: E = 10.0 Correct Part D What is the direction of vector E?⃗ Express your answer using two significant figures. ANSWER: θ = 11 � counterclockwise from positive direction of x-axis Correct Problem 3.24 Part A What is the angle ϕ between vectors E⃗ and F ⃗ in the figure? Express your answer with the appropriate units. ANSWER: ϕ = 71.6 � Correct Part B Use components to determine the magnitude of G⃗ = E⃗ + F .⃗ ANSWER: G = 3.00 Correct Part C Use components to determine the direction of G⃗ = E⃗ + F .⃗ Express your answer with the appropriate units. ANSWER: θ = 90.0 � Correct Score Summary: Your score on this assignment is 91.3%. You received 129.62 out of a possible total of 142 points. Assignment 3 Due: 11:59pm on Friday, February 14, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 2.68 As a science project, you drop a watermelon off the top of the Empire State Building, 320 m above the sidewalk. It so happens that Superman flies by at the instant you release the watermelon. Superman is headed straight down with a speed of 36.0m/s . Part A How fast is the watermelon going when it passes Superman? Express your answer with the appropriate units. ANSWER: m 72.0 s Correct Problem 2.63 A motorist is driving at 20 m/s when she sees that a traffic light 200 again. It takes her 1.0 s to step on the brakes and begin slowing. m ahead has just turned red. She knows that this light stays red for 15 s , and she wants to reach the light just as it turns green Part A What is her speed as she reaches the light at the instant it turns green? Express your answer with the appropriate units. ANSWER: m 5.71 s Correct Conceptual Question 4.1 Part A At this instant, is the particle in the figurespeeding up, slowing down, or traveling at constant speed? ANSWER: Speeding up Slowing down Traveling at constant speed Correct Part B Is this particle curving to the right, curving to the left, or traveling straight? ANSWER: Curving to the right Curving to the left Traveling straight Correct Conceptual Question 4.2 Part A At this instant, is the particle in the following figure speeding up, slowing down, or traveling at constant speed? ANSWER: The particle is speeding up. The particle is slowing down. The particle is traveling at constant speed. Correct Part B Is this particle curving upward, curving downward, or traveling straight? ANSWER: The particle is curving upward. The particle is curving downward. The particle is traveling straight. Correct Problem 4.8 A particle's trajectory is described by x = ( 12 t3 Part A What is the particle's speed at ANSWER: v = 2 m/s t = 0s? − 2t2 ) m and y = ( 12 t2 − 2t) m , where t is in s. Correct Part B What is the particle's speed at t = 5.0s ? Express your answer using two significant figures. ANSWER: v = 18 m/s Correct Part C What is the particle's direction of motion, measured as an angle from the x-axis, at t= 0 s? Express your answer using two significant figures. ANSWER: θ = -90 � counterclockwise from the +x axis. Correct Part D What is the particle's direction of motion, measured as an angle from the x-axis, at Express your answer using two significant figures. ANSWER: θ = 9.7 � counterclockwise from the +x axis. t = 5.0s ? Correct Problem 4.9 A rocket-powered hockey puck moves on a horizontal frictionless table. The figure shows the graph of vx and the figure shows the graph of vy , the x- and y-components of the puck’s velocity, respectively. The puck starts at the origin. Part A In which direction is the puck moving at t = 3s ? Give your answer as an angle from the x-axis. Express your answer using two significant figures. ANSWER: θ = 51 � Correct Part B above the x-axis How far from the origin is the puck at 5s ? Express your answer to two significant figures and include the appropriate units. ANSWER: s = 180 cm Correct Enhanced EOC: Problem 4.13 A rifle is aimed horizontally at a target 51.0m away. The bullet hits the target 1.50cm below the aim point. You may want to review ( pages 91 - 95) . For help with math skills, you may want to review: Quadratic Equations Part A What was the bullet's flight time? Express your answer with the appropriate units. Hint 1. How to approach the problem Start by drawing a picture of the bullet's trajectory, including where it leaves the gun and where it hits the target. You can assume that the gun was held parallel to the ground. Label the distances given in the problem. Choose an x-y coordinate system, making sure to label the origin. It is conventional to have x in the horizontal direction and y in the vertical direction. What is the y coordinate when the bullet leaves the gun? What is the y coordinate when it hits the target? What is the initial velocity in the y direction? What is the acceleration in the y direction? What is the equation y(t) that describes the motion in the vertical y direction as a function of time? Can you use the equation for necessary to include the motion in the x direction? y(t) to determine the time of flight? Why was it not ANSWER: −2 5.53×10 s Correct Part B What was the bullet's speed as it left the barrel? Express your answer with the appropriate units. Hint 1. How to approach the problem In the coordinate system introduced in Part A, what are the x coordinates when the bullet leaves the gun and when it hits the target? Is there any acceleration in the x direction? What is the equation x(t) that describes the motion in the horizontal x direction as a function of time? Can you use the equation for x(t) to determine the initial velocity? ANSWER: m 922 s Correct Introduction to Projectile Motion Learning Goal: To understand the basic concepts of projectile motion. Projectile motion may seem rather complex at first. However, by breaking it down into components, you will find that it is really no different than the one-dimensional motions that you have already studied. One of the most often used techniques in physics is to divide two- and three-dimensional quantities into components. For instance, in projectile motion, a particle has some initial velocity v.⃗ In general, this velocity can point in any direction on the xy plane and can have any magnitude. To make a problem more managable, it is common to break up such a quantity into its x component vx and its y component vy . Consider a particle with initial velocity v ⃗ that has magnitude 12.0 m/s and is directed 60.0 degrees above the negative x axis. Part A ⃗ What is the x component vx of v? Express your answer in meters per second. ANSWER: vx = -6.00 m/s Correct Part B ⃗ What is the y component vy of v? Express your answer in meters per second. ANSWER: vy = 10.4 m/s Correct Breaking up the velocities into components is particularly useful when the components do not affect each other. Eventually, you will learn about situations in which the components of velocity do affect one another, but for now you will only be looking at problems where they do not. So, if there is acceleration in the x direction but not in the y direction, then the x component of the velocity will change, but the y component of the velocity will not. Part C Look at this applet. The motion diagram for a projectile is displayed, as are the motion diagrams for each component. The x-component motion diagram is what you would get if you shined a spotlight down on the particle as it moved and recorded the motion of its shadow. Similarly, if you shined a spotlight to the left and recorded the particle's shadow, you would get the motion diagram for its y component. How would you describe the two motion diagrams for the components? ANSWER: Both the vertical and horizontal components exhibit motion with constant nonzero acceleration. The vertical component exhibits motion with constant nonzero acceleration, whereas the horizontal component exhibits constant-velocity motion. The vertical component exhibits constant-velocity motion, whereas the horizontal component exhibits motion with constant nonzero acceleration. Both the vertical and horizontal components exhibit motion with constant velocity. Correct As you can see, the two components of the motion obey their own independent kinematic laws. For the vertical component, there is an acceleration downward with magnitude g Thus, you can calculate the vertical position of the particle at any time using the standard kinematic equation y direction, so the horizontal position of the particle is given by the standard kinematic equation x = 10 m/s2 . = y0 + v0 t + (1/2)at2. Similarly, there is no acceleration in the horizontal = x0 + v0 t. Now, consider this applet. Two balls are simultaneously dropped from a height of 5.0 m. Part D How long tg does it take for the balls to reach the ground? Use 10 m/s2 for the magnitude of the acceleration due to gravity. Express your answer in seconds to two significant figures. Hint 1. How to approach the problem The balls are released from rest at a height of 5.0 m at time t ground. = 0 s. Using these numbers and basic kinematics, you can determine the amount of time it takes for the balls to reach the ANSWER: tg = 1.0 s Correct This situation, which you have dealt with before (motion under the constant acceleration of gravity), is actually a special case of projectile motion. Think of this as projectile motion where the horizontal component of the initial velocity is zero. Part E Imagine the ball on the left is given a nonzero initial speed in the horizontal direction, while the ball on the right continues to fall with zero initial velocity. What horizontal speed vx must the ball on the left start with so that it hits the ground at the same position as the ball on the right? Remember that the two balls are released, starting a horizontal distance of 3.0 m apart. Express your answer in meters per second to two significant figures. Hint 1. How to approach the problem Recall from Part B that the horizontal component of velocity does not change during projectile motion. Therefore, you need to find the horizontal component of velocity vx such that, in a time tg = 1.0 s, the ball will move horizontally 3.0 m. You can assume that its initial x coordinate is x0 = 0.0 m. ANSWER: vx = 3.0 m/s Correct You can adjust the horizontal speeds in this applet. Notice that regardless of what horizontal speeds you give to the balls, they continue to move vertically in the same way (i.e., they are at the same y coordinate at the same time). Problem 4.12 A ball thrown horizontally at 27m/s travels a horizontal distance of 49m before hitting the ground. Part A From what height was the ball thrown? Express your answer using two significant figures with the appropriate units. ANSWER: h = 16 m Correct Enhanced EOC: Problem 4.20 The figure shows the angular-velocity-versus-time graph for a particle moving in a circle. You may want to review ( page ) . For help with math skills, you may want to review: The Definite Integral Part A How many revolutions does the object make during the first 3.5s ? Express your answer using two significant figures. You did not open hints for this part. ANSWER: n= Incorrect; Try Again Problem 4.26 To withstand "g-forces" of up to 10 g's, caused by suddenly pulling out of a steep dive, fighter jet pilots train on a "human centrifuge." 10 g's is an acceleration of 98 m/s2 . Part A If the length of the centrifuge arm is 10.0m , at what speed is the rider moving when she experiences 10 g's? Express your answer with the appropriate units. ANSWER: m 31.3 s Correct Problem 4.28 Your roommate is working on his bicycle and has the bike upside down. He spins the 60.0cm -diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second. Part A What is the pebble's speed? Express your answer with the appropriate units. ANSWER: m 5.65 s Correct Part B What is the pebble's acceleration? Express your answer with the appropriate units. ANSWER: 107 m s2 Correct Enhanced EOC: Problem 4.43 On the Apollo 14 mission to the moon, astronaut Alan Shepard hit a golf ball with a 6 iron. The acceleration due to gravity on the moon is 1/6 of its value on earth. Suppose he hits the ball with a speed of 13m/s at an angle 50� above the horizontal. You may want to review ( pages 90 - 95) . For help with math skills, you may want to review: Quadratic Equations Part A How much farther did the ball travel on the moon than it would have on earth? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem Start by drawing a picture of the path of the golf ball, showing its starting and ending points. Choose a coordinate system, and label the origin. It is conventional to let x be the horizontal direction and y the vertical direction. What is the initial velocity in the x and y directions? What is the acceleration in the x and y directions on the moon and on the earth? What are the equations for x and y as a function of time, x(t) and y(t), respectively? What is the y coordinate when the golf ball hits the ground? Can you use this information to determine the time of flight on the moon and on the earth? Once you have the time of flight, how can you use the x(t) equation to determine the total distance traveled? Compare the distance traveled on the moon to the distance traveled on the earth . ANSWER: L = 85 m Correct Part B For how much more time was the ball in flight? Express your answer to two significant figures and include the appropriate units. Hint 1. How to approach the problem What is the equation x(t) describing x as a function of time? What is the initial x component of the ball's velocity? How are the initial x component of the ball's velocity and the distance traveled related to the time of flight? What is the difference between the time of flight on the moon and on earth? ANSWER: t = 10 s Correct Problem 4.42 In the Olympic shotput event, an athlete throws the shot with an initial speed of 12 m/s at a 40.0� angle from the horizontal. The shot leaves her hand at a height of 1.8 m above the ground. Part A How far does the shot travel? Express your answer to four significant figures and include the appropriate units. ANSWER: x = 16.36 m Correct Part B Repeat the calculation of part (a) for angles of 42.5 � , 45.0 � , and 47.5 � . Express your answer to four significant figures and include the appropriate units. ANSWER: x(42.5� ) = 16.39 m Correct Part C Express your answer to four significant figures and include the appropriate units. ANSWER: x(45.0� ) = 16.31 m Correct Part D Express your answer to four significant figures and include the appropriate units. ANSWER: x(47.5� ) = 16.13 m Correct Part E At what angle of release does she throw the farthest? ANSWER: 40.0� 42.5� 45.0� 47.5� Correct Problem 4.44 A ball is thrown toward a cliff of height h with a speed of 32m/s and an angle of 60� above horizontal. It lands on the edge of the cliff 3.2s later. Part A How high is the cliff? Express your answer to two significant figures and include the appropriate units. ANSWER: h = 39 m Answer Requested Part B What was the maximum height of the ball? Express your answer to two significant figures and include the appropriate units. ANSWER: hmax = 39 m Correct Part C What is the ball's impact speed? Express your answer to two significant figures and include the appropriate units. ANSWER: v = 16 m s Correct Problem 4.58 A typical laboratory centrifuge rotates at 3600rpm . Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations. Part A What is the acceleration at the end of a test tube that is 10 cm from the axis of rotation? Express your answer with the appropriate units. ANSWER: m a = 1.42×104 s 2 Correct Part B For comparison, what is the magnitude of the acceleration a test tube would experience if dropped from a height of 1.0 m and stopped in a 1.7-ms-long encounter with a hard floor? Express your answer with the appropriate units. ANSWER: m a = 2610 s 2 Correct Problem 4.62 Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is 6.37 × 106 m, and the altitude of a geosynchronous orbit is 3.58 × 107 m( ≈22000 miles). Part A What is the speed of a satellite in a geosynchronous orbit? Express your answer with the appropriate units. ANSWER: v = 3070 m s Correct Part B What is the magnitude of the acceleration of a satellite in a geosynchronous orbit? Express your answer with the appropriate units. ANSWER: m a = 0.223 s2 Correct Score Summary: Your score on this assignment is 89.5%. You received 103.82 out of a possible total of 116 points. Assignment 4 Due: 11:59pm on Wednesday, February 26, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy ± Two Forces Acting at a Point Two forces, F 1⃗ and F 2⃗ , act at a point. F 1⃗ has a magnitude of 9.80N and is directed at an angle of 56.0� above the negative x axis in the second quadrant. directed at an angle of 54.1� below the negative x axis in the third quadrant. F 2⃗ has a magnitude of 5.20N and is Part A What is the x component of the resultant force? Express your answer in newtons. Hint 1. How to approach the problem The resultant force is defined as the vector sum of all forces. Thus, its x component is the sum of the x components of the forces, and its y component is the sum of the y components of the forces. Hint 2. Find the x component of F 1⃗ Find the x component of F 1⃗ . Express your answer in newtons. Hint 1. Components of a vector Consider a vector A⃗ that forms an angle θ with the positive x axis. The x and y components of A⃗ are, respectively, Ax = A cos θ and Ay = A sin θ, where A is the magnitude of the vector. Note that Ax < 0 and Ay > 0 if π2 < θ < π, Ax < 0 and Ay < 0 if π < θ< 3π . 2 Hint 2. Find the direction of F 1⃗ F 1⃗ is directed at an angle of 56.0� above the x axis in the second quadrant. When you calculate the components of F 1⃗ , however, the direction of the force is commonly expressed in terms of the angle that the vector representing the force forms with the positive x axis. What is the angle that F 1⃗ forms with the positive x axis? Select an answer from the following list, where θ = 56.0� . ANSWER: θ 180� − θ 180� + θ 90� + θ ANSWER: -5.48 N Hint 3. Find the x component of F 2⃗ Find the x component of F 2⃗ . Express your answer in newtons. Hint 1. Components of a vector Consider a vector A⃗ that forms an angle θ with the positive x axis. The x and y components of A⃗ are, respectively, Ax = A cos θ and Ay = A sin θ, where A is the magnitude of the vector. Note that Ax < 0 and Ay > 0 if π2 Typesetting math: 100% < θ < π, Ax < 0 and Ay < if π <θ< 3π . 2 Hint 2. Find the direction of F 2⃗ F 2⃗ is directed at an angle of 54.1� below the x axis in the third quadrant. When you calculate the components of F 2⃗ , however, the direction of the force is commonly expressed in terms of the angle that the vector representing the force forms with the positive x axis. What is the angle that F 2⃗ forms with the positive x axis? Select an answer from the following list, where θ = 54.1� . ANSWER: θ 180� − θ θ − 180� −90� − θ ANSWER: -3.05 N ANSWER: -8.53 N Correct Part B What is the y component of the resultant force? Express your answer in newtons. Typesetting math: 100% Hint 1. How to approach the problem Follow the same procedure that you used in Part A to find the x component of the resultant force, though now calculate the y components of the two forces. Hint 2. Find the y component of F 1⃗ Find the y component of F 1⃗ . Express your answer in newtons. Hint 1. Components of a vector Consider a vector A⃗ that forms an angle θ with the positive x axis. The x and y components of A⃗ are, respectively, Ax = A cos θ and Ay = A sin θ, where A is the magnitude of the vector. Note that Ax < 0 and Ay > 0 if π2 < θ < π, Ax < 0 and Ay < 0 if π < ANSWER: 8.12 N Hint 3. Find the y component of F 2⃗ Find the y component of F 2⃗ . Express your answer in newtons. Hint 1. Components of a vector Typesetting math: 100% θ< 3π . 2 Consider a vector A⃗ that forms an angle θ with the positive x axis. The x and y components of A⃗ are, respectively, Ax = A cos θ and Ay = A sin θ, where A is the magnitude of the vector. Note that Ax < 0 and Ay > 0 if π2 < θ < π, Ax < 0 and Ay < 0 if π < ANSWER: -4.21 N ANSWER: 3.91 N Correct Part C What is the magnitude of the resultant force? Express your answer in newtons. Hint 1. Magnitude of a vector Consider a vector A,⃗ whose components are Ax and Ay. The magnitude of A⃗ is −−−−−−− A = √A2x + A2y . Typesetting math: 100% θ< 3π . 2 ANSWER: 9.38 N Correct Enhanced EOC: Problem 5.9 The figure shows acceleration-versus-force graphs for two objects pulled by rubber bands. You may want to review ( pages 127 - 130) . For help with math skills, you may want to review: Finding the Slope of a Line from a Graph Part A m1 What is the mass ratio m ? 2 Express your answer using two significant figures. Typesetting math: 100% Hint 1. How to approach the problem How are the acceleration and the force on an object related to its mass? How is the slope of each line in the figure related to each object's mass? For each line, what two points are easy to measure accurately to determine the slope of line? How is the slope determined from the x and y coordinates of the two points you chose for each line? ANSWER: m1 m2 = 0.36 Correct A World-Class Sprinter World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 m/s2 . Part A How much horizontal force F must a sprinter of mass 54kg exert on the starting blocks to produce this acceleration? Express your answer in newtons using two significant figures. Hint 1. Newton's 2nd law of motion According to Newton's 2nd law of motion, if a net external force Fnet acts on a body, the body accelerates, and the net force is equal to the mass the body: Fnet = ma. ANSWER: F = 810 N Typesetting math: 100% m of the body times the acceleration a of Correct Part B Which body exerts the force that propels the sprinter, the blocks or the sprinter? Hint 1. How to approach the question To start moving forward, sprinters push backward on the starting blocks with their feet. Newton's 3rd law tells you that the blocks exert a force on the sprinter of the same magnitude, but opposite in direction. ANSWER: the blocks the sprinter Correct To start moving forward, sprinters push backward on the starting blocks with their feet. As a reaction, the blocks push forward on their feet with a force of the same magnitude. This external force accelerates the sprinter forward. Problem 5.12 The figure shows an acceleration-versus-force graph for a 600g object. Typesetting math: 100% Part A What must a1 equal in order for the graph to be correct? Express your answer with the appropriate units. ANSWER: m a1 = 1.67 s2 Correct Part B What must a2 equal in order for the graph to be correct? Express your answer with the appropriate units. ANSWER: m a2 = 3.33 s2 Correct Free-Body Diagrams Learning Goal: To gain practice drawing free-body diagrams Whenever face a problem involving forces, always start with a free-body diagram. Typesettingyou math: 100% To draw a free-body diagram use the following steps: 1. Isolate the object of interest. It is customary to represent the object of interest as a point in your diagram. 2. Identify all the forces acting on the object and their directions. Do not include forces acting on other objects in the problem. Also, do not include quantities, such as velocities and accelerations, that are not forces. 3. Draw the vectors for each force acting on your object of interest. When possible, the length of the force vectors you draw should represent the relative magnitudes of the forces acting on the object. In most problems, after you have drawn the free-body diagrams, you will explicitly label your coordinate axes and directions. Always make the object of interest the origin of your coordinate system. Then you will need to divide the forces into x and y components, sum the x and y forces, and apply Newton's first or second law. In this problem you will only draw the free-body diagram. Suppose that you are asked to solve the following problem: Chadwick is pushing a piano across a level floor (see the figure). The piano can slide across the floor without friction. If Chadwick applies a horizontal force to the piano, what is the piano's acceleration? To solve this problem you should start by drawing a free-body diagram. Part A Determine the object of interest for the situation described in the problem introduction. Hint 1. How to approach the problem You should first think about the question you are trying to answer: What is the acceleration of the piano? The object of interest in this situation will be the object whose acceleration you are asked to find. ANSWER: Typesetting math: 100% the floor. For this situation you should draw a free-body diagram for Chadwick. the piano. Correct Part B Identify the forces acting on the object of interest. From the list below, select the forces that act on the piano. Check all that apply. ANSWER: acceleration of the piano gravitational force acting on the piano (piano's weight) speed of the piano gravitational force acting on Chadwick (Chadwick's weight) force of the floor on the piano (normal force) force of the piano on the floor force of Chadwick on the piano force of the piano pushing on Chadwick Correct Now that you have identified the forces acting on the piano, you should draw the free-body diagram. Draw the length of your vectors to represent the relative magnitudes of the forces, but you don't need to worry about the exact scale. You won't have the exact value of all of the forces until you finish solving the problem. To maximize your learning, you should draw the diagram yourself before looking at the choices in the next part. You are on your honor to do so. Part C Typesetting math: 100% Select the choice that best matches the free-body diagram you have drawn for the piano. Hint 1. Determine the directions and relative magnitudes of the forces Which of the following statements best describes the correct directions and relative magnitudes of the forces involved? ANSWER: The normal force and weight are both upward and the pushing force is horizontal. The normal force and weight are both downward and the pushing force is horizontal. The normal force is upward, the weight is downward, and the pushing force is horizontal. The normal force has a greater magnitude than the weight. The normal force is upward, the weight is downward, and the pushing force is horizontal. The normal force and weight have the same magnitude. The normal force is upward, the weight is downward, and the pushing force is horizontal. The normal force has a smaller magnitude than the weight. ANSWER: Typesetting math: 100% Typesetting math: 100% Correct If you were actually going to solve this problem rather than just draw the free-body diagram, you would need to define the coordinate system. Choose the position of the piano as the origin. In this case it is simplest to let the y axis point vertically upward and the x axis point horizontally to the right, in the direction of the acceleration. Chadwick now needs to push the piano up a ramp and into a moving van. at left. The ramp is frictionless. Is Chadwick strong enough to push the piano up the ramp alone or must he get help? To solve this problem you should start by drawing a free-body diagram. Part D Determine the object of interest for this situation. ANSWER: the ramp. For this situation, you should draw a free-body diagram for Chadwick. the piano. Correct Now draw the free-body diagram of the piano in this new situation. Follow the same sequence of steps that you followed for the first situation. Again draw your diagram before you look at the choices Typesetting math: 100% below. Part E Which diagram accurately represents the free-body diagram for the piano? ANSWER: Typesetting math: 100% Typesetting math: 100% Correct In working problems like this one that involve an incline, it is most often easiest to select a coordinate system that is not vertical and horizontal. Instead, choose the x axis so that it is parallel to the incline and choose the y axis so that it is perpendicular to the incline. Problem 5.18 The figure shows two of the three forces acting on an object in equilibrium. Part A Redraw the diagram, showing all three forces. Label the third force F 3⃗ . Draw the force vector starting at the black dot. The location and orientation of the vector will be graded. The length of the vector will not be graded. ANSWER: Typesetting math: 100% Correct Problem 5.25 An ice hockey puck glides across frictionless ice. Part A Identify all forces acting on the object. ANSWER: Typesetting math: 100% Normal force n⃗ ; Gravity ⃗ FG Normal force n⃗ ; Gravity → ⃗ ; Kinetic friction f FG k Tension T ;⃗ Weight Thrust w⃗ → ⃗ Fthrust; Gravity F G Correct Part B Draw a free-body diagram of the ice hockey puck. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Problem 5.26 Your physics textbook is sliding to the right across the table. Part A Identify all forces acting on the object. ANSWER: Typesetting math: 100% → w;⃗ Kinetic friction fk → → Thrust Fthrust; Kinetic friction fk Weight → w⃗; Kinetic friction fk → Normal force n⃗ ; Weight w⃗; Static friction fs Normal force n⃗ ; Weight Correct Part B Draw a free-body diagram of the object. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Enhanced EOC: Problem 5.35 A constant force is applied to an object, causing the object to accelerate at 13m/s2 . You may want to review ( pages 127 - 130) . For help with math skills, you may want to review: Proportions I Proportions II Typesetting math: 100% Part A What will the acceleration be if the force is halved? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the acceleration of an object related to its mass and the force applied? Expressing the acceleration in terms of the force and mass, what happens to the acceleration if the force is halved but the mass remains the same? ANSWER: m a = 6.50 s 2 Correct Part B What will the acceleration be if the object's mass is halved? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the acceleration of an object related to its mass and the force applied? Expressing the acceleration in terms of the force and mass, what happens to the acceleration if the mass is halved but the force remains the same? ANSWER: m a = 26.0 s 2 Correct Part C Typesetting math: 100% What will the acceleration be if the force and the object's mass are both halved? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the acceleration of an object related to its mass and the force applied? Expressing the acceleration in terms of the force and mass, what happens to the acceleration if both the force and mass are reduced by a factor of two? ANSWER: m a = 13.0 s 2 Correct Part D What will the acceleration be if the force is halved and the object's mass is doubled? Express your answer with the appropriate units. Hint 1. How to approach the problem How is the acceleration of an object related to its mass and the force applied? Expressing the acceleration in terms of the force and mass, what happens to the acceleration if the force is decreased by a factor of two and the mass is increased by a factor of two? Check your answer by choosing numerical values of the force and mass, and then halve the force and double the mass. ANSWER: m a = 3.25 s 2 Correct Typesetting math: 100% Problem 5.44 A rocket is being launched straight up. Air resistance is not negligible. Part A Which of the following is the correct motion diagram for the situation described above? Enter the letter that corresponds with the best answer. ANSWER: Correct Part B Draw a free-body diagram. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Score Summary: Your score on this assignment is 99.7%. You received 63.82 out of a possible total of 64 points. Typesetting math: 100% Assignment 5 Due: 11:59pm on Wednesday, March 5, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Conceptual Question 6.13 A hand presses down on the book in the figure. Part A Is the normal force of the table on the book larger than, smaller than, or equal to mg? ANSWER: Equal to mg Larger than mg Smaller than mg Correct Problem 6.2 The three ropes in the figure are tied to a small, very light ring. Two of these ropes are anchored to walls at right angles with the tensions shown in the figure. Part A What is the magnitude of the tension T 3⃗ in the third rope? Express your answer using two significant figures. ANSWER: T3 = 94 N Correct Part B What is the direction of the tension T 3⃗ in the third rope? Express your answer using two significant figures. Typesetting math: 100% ANSWER: θ = 58 � below horizontal Correct The Normal Force When an object rests on a surface, there is always a force perpendicular to the surface; we call this the normal force, denoted by n⃗ . The two questions to the right will explore the normal force. Part A A man attempts to pick up his suitcase of weight ws by pulling straight up on the handle. However, he is unable to lift the suitcase from the floor. Which statement about the magnitude of the normal force n acting on the suitcase is true during the time that the man pulls upward on the suitcase? Hint 1. How to approach this problem First, identify the forces that act on the suitcase and draw a free-body diagram. Then use the fact that the suitcase is in equilibrium, ∑ F ⃗ = 0, to examine how the forces acting on the suitcase relate to each other. Hint 2. Identify the correct free-body diagram Which of the figures represents the free-body diagram of the suitcase while the man is pulling on the handle with a force of magnitude fpull? Typesetting math: 100% ANSWER: A B C D ANSWER: The magnitude of the normal force is equal to the magnitude of the weight of the suitcase. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull. The magnitude of the normal force is equal to the sum of the magnitude of the force of the pull and the magnitude of the suitcase's weight. The magnitude of the normal force is greater than the magnitude of the weight of the suitcase. Correct Part B Typesetting math: 100% Now assume that the man of weight wm is tired and decides to sit on his suitcase. Which statement about the magnitude of the normal force n acting on the suitcase is true during the time that the man is sitting on the suitcase? Hint 1. Identify the correct free-body diagram. Which of the figures represents the free-body diagram while the man is sitting atop the suitcase? Here the vector labeled wm is a force that has the same magnitude as the man's weight. ANSWER: Typesetting math: 100% A B C D ANSWER: The magnitude of the normal force is equal to the magnitude of the suitcase's weight. The magnitude of the normal force is equal to the magnitude of the suitcase's weight minus the magnitude of the man's weight. The magnitude of the normal force is equal to the sum of the magnitude of the man's weight and the magnitude of the suitcase's weight. The magnitude of the normal force is less than the magnitude of the suitcase's weight. Correct Recognize that the normal force acting on an object is not always equal to the weight of that object. This is an important point to understand. Problem 6.5 A construction worker with a weight of 880N stands on a roof that is sloped at 18� . Part A What is the magnitude of the normal force of the roof on the worker? Express your answer to two significant figures and include the appropriate units. ANSWER: n = 840 N Correct Typesetting math: 100% Problem 6.6 In each of the two free-body diagrams, the forces are acting on a 3.0kg object. Part A For diagram , find the value of ax , the x-component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: m ax = -0.67 s 2 Correct Part B For diagram the part A, find the value of ay , the y-component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Typesetting math: 100% ay = 0 m s2 Correct Part C For diagram , find the value of ax , the x-component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: m ax = 0.67 s2 Correct Part D For diagram the part C, find the value of ay , the y-component of the acceleration. Express your answer to two significant figures and include the appropriate units. ANSWER: Typesetting math: 100% ay = 0 m s2 Correct Problem 6.7 In each of the two free-body diagrams, the forces are acting on a 3.0kg object. Part A Find the value of ax , the x component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: m ax = 0.99 s2 Correct Typesetting math: 100% Part B Find the value of ay , the y component of the acceleration in diagram (a). Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 Correct Part C Find the value of ax , the x component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: m ax = -0.18 s 2 Correct Part D Find the value of ay , the y component of the acceleration in diagram (b). Express your answer to two significant figures and include the appropriate units. ANSWER: ay = 0 m s2 Correct Typesetting math: 100% Problem 6.10 A horizontal rope is tied to a 53.0kg box on frictionless ice. What is the tension in the rope if: Part A The box is at rest? Express your answer as an integer and include the appropriate units. ANSWER: T = 0N Correct Part B The box moves at a steady vx = 4.80m/s ? Express your answer as an integer and include the appropriate units. ANSWER: T = 0N Correct Part C The box vx = 4.80m/s and ax = 4.60m/s2 ? Express your answer to three significant figures and include the appropriate units. ANSWER: Typesetting math: 100% T = 244 N Correct Problem 6.14 It takes the elevator in a skyscraper 4.5s to reach its cruising speed of 11m/s . A 60kg passenger gets aboard on the ground floor. Part A What is the passenger's weight before the elevator starts moving? Express your answer to two significant figures and include the appropriate units. ANSWER: w = 590 N Correct Part B What is the passenger's weight while the elevator is speeding up? Express your answer to two significant figures and include the appropriate units. ANSWER: w = 730 N Correct Part C Typesetting 100% What is math: the passenger's weight after the elevator reaches its cruising speed? Express your answer to two significant figures and include the appropriate units. ANSWER: w = 590 N Correct Block on an Incline A block lies on a plane raised an angle θ from the horizontal. Three forces act upon the block: large enough to prevent the block from sliding . F w⃗ , the force of gravity; F n⃗ , the normal force; and F f⃗ , the force of friction. The coefficient of friction is Part A Consider coordinate system a, with the x axis along the plane. Which forces lie along the axes? ANSWER: Typesetting math: 100% F f⃗ only F n⃗ only F w⃗ only F f⃗ and F n⃗ F f⃗ and F w⃗ F n⃗ and F w⃗ F f⃗ and F n⃗ and F w⃗ Correct Part B Which forces lie along the axes of the coordinate system b, in which the y axis is vertical? ANSWER: F f⃗ only F n⃗ only F w⃗ only F f⃗ and F n⃗ F f⃗ and F w⃗ F n⃗ and F w⃗ F f⃗ and F n⃗ and F w⃗ Correct Typesetting math: 100% Usually the best advice is to choose coordinate system so that the acceleration of the system is directly along one of the coordinate axes. If the system isn't accelerating, then you are better off choosing the coordinate system with the most vectors along the coordinate axes. But now you are going to ignore that advice. You will find the normal force, equations, each multiplied by a trigonometric function. F n⃗ , using vertical coordinate system b. In these coordinates you will find the magnitude Fn appearing in both the x and y Part C Because the block is not moving, the sum of the y components of the forces acting on the block must be zero. Find an expression for the sum of the y components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables Fn , Ff , Fw , and θ . Hint 1. Find the y component of F n⃗ Write an expression for Fny , the y component of the force F n⃗ , using coordinate system b. Express your answer in terms of Fn and θ . Hint 1. Some geometry help - a useful angle The smaller angle between F n⃗ and the y-axis is also θ , as shown in the figure. ANSWER: Typesetting math: 100% Fny = Fn cos(θ) Hint 2. Find the y component of F f⃗ Write an expression for Ffy , the y component of the force F f⃗ , using coordinate system b. Express your answer in terms of Ff and θ . Hint 1. Some geometry help - a useful angle The smaller angle between F f⃗ and the x-axis is also θ , as shown in the figure. ANSWER: Ffy = Ff sin(θ) ANSWER: ∑ Fy = 0 = Fn cos(θ) + Ff sin(θ) − Fw Typesetting math: 100% Correct Part D Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables Fn , Ff , Fw , and θ . Hint 1. Find the x component of F n⃗ Write an expression for Fnx , the x component of the force F n⃗ , using coordinate system b. Express your answer in terms of Fn and θ . ANSWER: Fnx = −Fn sin(θ) ANSWER: ∑ Fx = 0 = −Fn sin(θ) + Ff cos(θ) Correct Part E To find the magnitude of the normal force, you must express Fn in terms of Fw since Ff is an unknown. Using the equations you found in the two previous parts, find an expression for Fn involving Fw and θ but not Ff . Hint 1. How to approach the problem Frommath: your100% answers to the previous two parts you should have two force equations (∑ Fy Typesetting = 0 and ∑ Fx = 0). Combine these equations to eliminate Ff . The key is to multiply the equation for the y components by cos θ and the equation for the x components by sin θ , then add or subtract the two equations to eliminate the term Ff cos(θ) sin(θ) . An alternative motivation for the algebra is to eliminate the trig functions in front of is simple to solve for Fn by using the trig identity sin2 (θ) + cos2 (θ) = 1. At the very least this would result in an equation that Fn . ANSWER: Fn = Fw cos(θ) Correct Congratulations on working this through. Now realize that in coordinate system a, which is aligned with the plane, the y-coordinate equation is immediately to the result obtained here for ∑ Fy = Fn − FW cos(θ) = 0, which leads Fn . CONCLUSION: A thoughtful examination of which coordinate system to choose can save a lot of algebra. Contact Forces Introduced Learning Goal: To introduce contact forces (normal and friction forces) and to understand that, except for friction forces under certain circumstances, these forces must be determined from: net Force = ma. Two solid objects cannot occupy the same space at the same time. Indeed, when the objects touch, they exert repulsive normal forces on each other, as well as frictional forces that resist their slipping relative to each other. These contact forces arise from a complex interplay between the electrostatic forces between the electrons and ions in the objects and the laws of quantum mechanics. As two surfaces are pushed together these forces increase exponentially over an atomic distance scale, easily becoming strong enough to distort the bulk material in the objects if they approach too close. In everyday experience, contact forces are limited by the deformation or acceleration of the objects, rather than by the fundamental interatomic forces. Hence, we can conclude the following: The magnitude of contact forces is determined by ∑ F ⃗ = µn (although they can be smaller than this or even zero). ma,⃗ that is, by the other forces on, and acceleration of, the contacting bodies. The only exception is that the frictional forces cannot exceed Normal and friction forces Two types of contact forces operate in typical mechanics problems, the normal and frictional forces, usually designated by components of the overall contact force: n perpendicular to and f parallel to the plane of contact. Kinetic friction when surfaces slide Typesetting math: 100% n and f (or Ffric , or something similar) respectively. These are the When one surface is sliding past the other, experiments show three things about the friction force (denoted fk ): 1. The frictional force opposes the relative motion at the point of contact, 2. fk is proportional to the normal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairly constant over a wide range of speeds. The constant of proportionality is called the coefficient of kinetic friction, often designated µ k . As long as the sliding continues, the frictional force is then fk = µk n (valid when the surfaces slide by each other). Static friction when surfaces don't slide When there is no relative motion of the surfaces, the frictional force can assume any value from zero up to a maximum µs n, where µs is the coefficient of static friction. Invariably, in agreement with the observation that when a force is large enough that something breaks loose and starts to slide, it often accelerates. µs is larger than µk , The frictional force for surfaces with no relative motion is therefore fs ≤ µ s n (valid when the contacting surfaces have no relative motion). The actual magnitude and direction of the static friction force are such that it (together with other forces on the object) causes the object to remain motionless with respect to the contacting surface as long as the static friction force required does not exceed µs n. The equation fs = µs n is valid only when the surfaces are on the verge of sliding. Part A When two objects slide by one another, which of the following statements about the force of friction between them, is true? ANSWER: The frictional force is always equal to µk n. The frictional force is always less than µk n. The frictional force is determined by other forces on the objects so it can be either equal to or less than µk n. Correct Part B Typesetting math: 100% When two objects are in contact with no relative motion, which of the following statements about the frictional force between them, is true? ANSWER: The frictional force is always equal to µs n. The frictional force is always less than µs n. The frictional force is determined by other forces on the objects so it can be either equal to or less than µs n. Correct For static friction, the actual magnitude and direction of the friction force are such that it, together with any other forces present, will cause the object to have the observed acceleration. The magnitude of the force cannot exceed µs n. If the magnitude of static friction needed to keep acceleration equal to zero exceeds µs n, then the object will slide subject to the resistance of kinetic friction. Do not automatically assume that fs = µs n unless you are considering a situation in which the magnitude of the static friction force is as large as possible (i.e., when determining at what point an object will just begin to slip). Whether the actual magnitude of the friction force is 0, less than µs n, or equal to µs n depends on the magnitude of the other forces (if any) as well as the acceleration of the object through ∑ F ⃗ = ma.⃗ Part C When a board with a box on it is slowly tilted to larger and larger angle, common experience shows that the box will at some point "break loose" and start to accelerate down the board. The box begins to slide once the component of gravity acting parallel to the board Fg just begins to exceeds the maximum force of static friction. Which of the following is the most general explanation for why the box accelerates down the board? ANSWER: The force of kinetic friction is smaller than that of maximum static friction, but Fg remains the same. Once the box is moving, Fg is smaller than the force of maximum static friction but larger than the force of kinetic friction. Once the box is moving, Fg is larger than the force of maximum static friction. When the box is stationary, Typesetting math: 100% Fg equals the force of static friction, but once the box starts moving, the sliding reduces the normal force, which in turn reduces the friction. Correct At the point when the box finally does "break loose," you know that the component of the box's weight that is parallel to the board just exceeds µs n (i.e., this component of gravitational force µs n , can no longer oppose it.) For the box to then accelerate, there must be a net force on the box along the board. Thus, the component of the box's weight parallel to the board must be greater than the force of kinetic friction. Therefore the force of kinetic friction µ k n must be less than the force of static friction µs n which implies µ k < µ s , as expected. on the box has just reached a magnitude such that the force of static friction, which has a maximum value of Part D Consider a problem in which a car of mass M is on a road tilted at an angle θ . The normal force Select the best answer. ANSWER: n = Mg n = Mg cos(θ) n= Mg cos(θ) is found using ∑ F ⃗ = M a⃗ Correct The key point is that contact forces must be determined from Newton's equation. In the problem described above, there is not enough information given to determine the normal force (e.g., the acceleration is unknown). Each of the answer options is valid under some conditions (θ = 0, the car is sliding down an icy incline, or the car is going around a banked turn), but in fact none is likely to be correct if there are other forces on the car or if the car is accelerating. Do not memorize values for the normal force valid in different problems--you must determine n⃗ from ∑ F ⃗ = ma.⃗ Problem 6.17 Bonnie and Clyde are sliding a 323kg bank safe across the floor to their getaway car. The safe slides with a constant speed if Clyde pushes from behind with 375N of force while Bonnie pulls forward onTypesetting a rope with 335 N of force. math: 100% Part A What is the safe's coefficient of kinetic friction on the bank floor? ANSWER: 0.224 Correct Problem 6.19 A 10 kg crate is placed on a horizontal conveyor belt. The materials are such that µs = 0.5 and µ k = 0.3. Part A Draw a free-body diagram showing all the forces on the crate if the conveyer belt runs at constant speed. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part B Draw a free-body diagram showing all the forces on the crate if the conveyer belt is speeding up. Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded. ANSWER: Typesetting math: 100% Correct Part C What is the maximum acceleration the belt can have without the crate slipping? Express your answer to two significant figures and include the appropriate units. ANSWER: m amax = 4.9 s 2 Correct Typesetting math: 100% Problem 6.28 A 1100kg steel beam is supported by two ropes. Part A What is the tension in rope 1? Express your answer to two significant figures and include the appropriate units. ANSWER: T1 = 7000 N Correct Part B What is the tension in rope 2? Express your answer to two significant figures and include the appropriate units. ANSWER: Typesetting math: 100% T2 = 4800 N Correct Problem 6.35 The position of a 1.4kg mass is given by x = (2t3 − 3t2 ) m, where t is in seconds. Part A What is the net horizontal force on the mass at t = 0 s? Express your answer to two significant figures and include the appropriate units. ANSWER: F = -8.4 N Correct Part B What is the net horizontal force on the mass at t = 1 s? Express your answer to two significant figures and include the appropriate units. ANSWER: F = 8.4 N Correct Problem 6.39 Typesetting math: 100% A rifle with a barrel length of 61cm fires a 8g bullet with a horizontal speed of 400m/s . The bullet strikes a block of wood and penetrates to a depth of 11cm . Part A What resistive force (assumed to be constant) does the wood exert on the bullet? Express your answer to two significant figures and include the appropriate units. ANSWER: fk = 5800 N Correct Part B How long does it take the bullet to come to rest after entering the wood? Express your answer to two significant figures and include the appropriate units. ANSWER: t = 5.5×10−4 s Correct Problem 6.45 You and your friend Peter are putting new shingles on a roof pitched at 21� . You're sitting on the very top of the roof when Peter, who is at the edge of the roof directly below you, 5.0m away, asks you for the box of nails. Rather than carry the 2.0kg box of nails down to Peter, you decide to give the box a push and have it slide down to him. Part A If the coefficient of kinetic friction between the box and the roof is 0.55, with what speed should you push the box to have it gently come to rest right at the edge of the roof? Express your answer to two significant figures and include the appropriate units. Typesetting math: 100% ANSWER: v = 3.9 m s Correct Problem 6.54 The 2.0 kg wood box in the figure slides down a vertical wood wall while you push on it at a 45 � angle. Part A What magnitude of force should you apply to cause the box to slide down at a constant speed? Express your answer to two significant figures and include the appropriate units. ANSWER: F = 23 N Correct Typesetting math: 100% Score Summary: Your score on this assignment is 98.8%. You received 114.57 out of a possible total of 116 points. Typesetting math: 100%
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