Homework 8 Problems: Rotational Motion 1. A machine part rotates at an angular speed of 0.06 rad/s; its angular speed is then increased to 2.2 rad/s at an angular acceleration of 0.70 rad/s2. a. [5 points] Find the angle through which the part rotates before reaching this final speed. b. [5 points] In general, if both the initial and final angular speed are doubled at the same angular acceleration, by what factor is the angular displacement changed? Why? 2. A car initially traveling at 29.0 m/s undergoes a constant negative acceleration of magnitude 1.75 m/s2 after its brakes are applied. a. [7 points] How many revolutions does each tire make before the car comes to a stop, assuming the car does not skid and the tires have radii of 0.330 m? b. [3 points] What is the angular speed of the wheels when the car has traveled half the total distance? 3. A pail of water is rotated in a vertical circle of radius 1.00 m. a. [3 points] What two external forces act on the water in the pail? Which of the two forces is most important in causing the water to move in a circle? b. [4 points] What is the pailвЂ™s minimum speed at the top of the circle if no water is to spill out? c. [3 points] If the pail with the speed found in part (c) were to suddenly disappear at the top of the circle, describe the subsequent motion of the water. Would it differ from the motion of a projectile? 4. A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers. a. [5 points] If the vehicle has a speed of 20.0 m/s at point A, what is the force of the track on the vehicle at this point? b. [5 points] What is the maximum speed the vehicle can have at point B for gravity to hold it on the track? 5. One end of a cord is fixed and a small 0.500-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 2.00 m, as shown below. When Оё = 20.0В°, the speed of the object is 8.00 m/s. a. [4 points] Find the tension in the string, b. [3 points] Find the tangential and radial components of acceleration. Find the total acceleration. c. [3 points] Is your answer changed if the object is swinging down toward its lowest point instead of swinging up? Explain your answer. Bonus: [7 points] A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane with the cord making a 30вЃ° angle with the vertical. a. Determine the ballвЂ™s speed. b. If, instead the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical? c. If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at which the ball can move?