A1 Physics
Unit 8: Circular Motion and Gravity
Conceptual Physics Circular Motion Outline
Hewitt: Chapter 9, 12 & 13
Exercises: 6.1 Centripetal Acceleration ,6.2 Torque 7 Law of Universal Gravitation
Fill in the Charts completely
Variables introduced or used in chapter:
Quantity
Symbol
Period
Frequency
speed
centripetal acceleration
Centripetal Force
lever arm
Torque
mass
Distance
gravity
Universal Constant of Gravitation
Units
Formula Chart*
*In Class, we will ‘r’ instead of ‘d’
tangential speed (linear speed)
centripetal acceleration
Centripetal Force
Torque
Newton’s Law of Universal Gravity
Gravitational Acceleration
Define the following terms using COMPLETE SENTENCES:
Rotation
Axis
Revolution
Period
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Unit 8: Circular Motion and Gravity
Frequency
Linear speed / Tangential Speed
Rotational Speed
Centripetal Force
Centrifugal Force
1G
Newton’s Law of Universal Gravitation
Inverse Square Law
Perturbation
Force Field
Gravitational Field
Weightlessness
Lunar Eclipse
Neap Tide
Solar Eclipse
Spring Time
Black Hole
Escape speed
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Unit 8: Circular Motion and Gravity
Circular Motion and Newton’s Universal Gravity Homework Problems
Circular Motion
1. What is the acceleration of a piece of dust on an old-fashioned record album, 15 cm from the
center, if the record is spinning at 33.3 rpm?
2. What is the tension in a 0.500 meter rope which carries a 2.50 kg rock in a horizontal circle with a
velocity of 3.00 m/s.
3. Determine the minimum coefficient of friction required to keep a 920.-kg car on a turn with a
radius of 26.8 m. The car is traveling 29.9 m/s and the roadway is level.
4. With what speed (in mi/hr) must you drive your car across the crest of a hill of radius 37.1-m in
order to feel weightless (1.00 m/s = 2.24 mi/hr)?
5. What is the reading of a bathroom scale when a man of mass 72 kg stands on it while riding in an
elevator accelerating upwards at 5.4 m/s2?
6. Use the following information to determine the orbital velocity at treetop level (2m) on the
surface of the moon. Mass of moon = 7.36 x 1022 kg Radius of Moon = 1.74 x 106 m
7. What is the period of rotation of Venus in earth years if it is a distance of 1.08 x 1011meters from
the sun? (The earth-sun distance is 1.5 x 1011m.)
8. Which of the following statements are true of an object moving in a circle at a constant speed?
Include all that apply.
a. The object experiences a force which has a component directed parallel to the direction of
motion.
b. Inertia causes objects to move in a circle.
c. There can be a force pushing outwards on the object as long as the net force in inwards.
d. Because the speed is constant, the acceleration is zero.
e. The acceleration and the net force vector are directed perpendicular to each other.
f. If the net force acting upon the object is suddenly reduced to zero, then the object would
suddenly depart from its circular path and travel tangent to the circle.
g. The acceleration of the object is directed tangent to the circle.
9. A physics teacher ties an eraser to the end of a string and then whirls it in a counter-clockwise
circle. If the teacher lets go of the string, then the eraser hits a
student (or several students) in the classroom. If the string is let go
when the eraser is at point X on the diagram at the right, then
which student(s) in the class will the eraser hit?
Newton’s Gravity
10. Two objects attract each other with a force of gravity (Fgrav) of 36 N. If the distance separating the
objects is doubled, then what is the new force of gravitational attraction?
11. Two objects attract each other with a force of gravity (Fgrav) of 36 N. If the distance separating the
objects is doubled and the masses one of the objects is tripled, then what is the new force of
gravitational attraction?
12. Two objects attract each other with a force of gravity (Fgrav) of 36 N. If the distance separating the
objects is increased by a factor of 4 and the masses of both objects are tripled, then what is the
new force of gravitational attraction?
13. Suppose that a planet was located 12.0 times further from the sun than the earth's distance from
the sun. Determine the period of the planet.
14. Suppose that the acceleration of gravity on the surface of planet X is 12 m/s2. Determine the
acceleration of gravity at a location ...
a. of 2 radii from the center of planet X.
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15.
16.
17.
18.
Unit 8: Circular Motion and Gravity
b. of 4 radii from the center of planet X.
c. of 2 radii from the surface of planet X.
d. on the surface of planet X if the planet mass were twice as large (same radius).
e. on the surface of planet X if the planet mass were one-half as large (same radius).
The acceleration of gravity on the moon is approximately one-sixth the value on earth's surface. If
a person weighs 60.0 N on the moon's surface, what is his/her approximate mass on Earth?
Determine the force of gravitational attraction between a 52.0-kg mother and a 3.0-kg child if
their separation distance is 0.50 meters.
What is the gravitational attraction between the proton and electron in an Hydrogen atom if they
are 5.3 x 10-11 meters apart? (mproton= 1.67 x 10-27 kg; melectron= 9.11x10-31kg)
Use the following information to help determine the orbital velocity and orbital period of a
satellite at a location of 15000 miles above the surface of the earth. Mass of Earth = 5.98 x 1024
kg, Radius of Earth = 6.37 x 106 m
1609 m = 1.00 mi
For Questions #18-21, identify the type of force which causes the following bold-faced objects to travel
along a circular path.
a. gravity
e. friction
19.
20.
21.
22.
b. normal
f. spring
c. tension
g. electrical
d. applied
h. magnetic
An eraser is tied to a string swung in a horizontal circle.
The moon orbits the earth
A car makes a sharp right-hand turn along a level roadway.
A roller coaster car passes through a loop. Consider the car at the bottom of the loop.
23. Which of the following statements are true about gravitational force? Identify all that apply.
a. The gravitational force only acts between very, very massive objects.
b. The gravitational force between an object and the earth is inversely related to the distance
between the object's and the earth's center.
c. The gravitational force can ALWAYS be accurately calculated by multiplying the object
mass by the acceleration of gravity (m•g).
d. The gravitational force acting upon an object is the same as the weight of the object.
e. The gravitational force between two objects is independent of the mass of the smaller of
the two objects.
f. If object A gravitationally attracts object B with a force of X Newtons, then object B will
also gravitationally attract object A with the same force of X Newtons.
g. The doubling of the separation distance (measured from the center) between two objects
will halve the gravitational force between the objects.
h. It an object is placed two earth-radii above the surface of the earth, then the force of
gravitational attraction between the object and the earth will be one-fourth the magnitude
as on earth's surface.
i. Orbiting astronauts do not experience a force of gravity; this explains why they feel
weightless.
24. Which of the following statements are true about the acceleration of gravity? Identify all that
apply.
a. The acceleration of gravity experienced by objects located near to (and far from) from the
earth depends upon the mass of the object.
b. The acceleration of gravity experienced by objects located near to (and far from) from the
earth depends upon the mass of the Earth.
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Unit 8: Circular Motion and Gravity
c. The acceleration of gravity experienced by objects located near to (and far from) the earth
is inversely related to the distance between the center of the object and the center of the
earth.
d. Increasing the mass of an object will increase the acceleration of gravity experienced by
the object.
e. Doubling the distance between an object and the earth's center will decrease the
acceleration of gravity by a factor of four.
f. The acceleration of an orbiting satellite is equal to the acceleration of gravity at that
particular location.
g. If the mass of the Earth were doubled (without an alteration in its radius), then the
acceleration of gravity on its surface would be approximately 20 m/s2.
h. If the mass of the Earth were doubled and the radius of the earth were doubled, then the
two changes would offset each other and the acceleration of gravity on its surface would
still be approximately 10 m/s2.
Satellites
25. Which of the following statements are true about satellites? Identify all that apply.
a. Satellites are falling projectiles.
b. All satellites follow circular paths.
c. The orbital velocity required of a satellite is dependent upon the mass of the satellite; a
more massive satellite would require a greater orbital speed.
d. The orbital velocity of a satellite does not depend upon the mass of the planet around
which it orbits.
e. A high-altitude satellite will require a greater orbital speed than a low-altitude satellite.
f. By definition, a geosynchronous satellite orbits the earth in a perfect circle, maintaining
the same distance above the surface of the earth.
g. Satellites travel faster along their orbital path when they are closest to the earth.
h. The acceleration of a satellite varies inversely with its distance from the center of the
earth. More distant satellites have smaller accelerations.
26. Which of the following statements are true about the motion of planets about the sun? Identify all
that apply.
a. The force of gravity is the only force which acts upon the planets.
b. Their trajectories are highly elliptical.
c. The planets which are furthest from the sun have the greatest period.
d. For any given planet, the speed is greatest when the planet is closest to the sun.
e. The velocity vector is directed tangent to the elliptical path.
f. The net force vector is at all times directed perpendicular to the velocity vector.
g. To keep the planet from escaping the sun's gravitational field, the net force vector is
greatest when the planet is furthest from the sun.
Short Answer
27. Explain how something can be moving at a constant speed yet be accelerating at the same time.
28. How did Newton come up with the idea that the moon is actually "falling" toward the Earth.
29. Distinguish between true- and apparent-weightlessness.
30. Describe the apparent weight of a person in an elevator while upward, accelerating downward,
and not accelerating.
Diagramming and Analysis
31. In the diagram at the right, draw vector arrows (straight lines with arrowheads) which indicate the
following for an object which is moving in a horizontal clockwise circle.
a. the net force at point A.
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Unit 8: Circular Motion and Gravity
b. the acceleration at point B.
c. the velocity at point C.
32. Construct a free-body diagram showing the direction and types of forces acting upon the car at
the top and the bottom of a loop. Be sure to label the forces according to type.
33. Pete Zaria is riding the Cliff Hanger at Great America. Pete enters a cylindrical barrel which
makes 18.5 revolutions every minute. The diameter of the barrel is 6.92 m. Pete's mass is 64.7 kg.
Construct a free-body diagram showing the forces acting upon Pete. Calculate the acceleration
and net force acting upon Pete Zaria. Finally, determine the coefficient of friction between Pete
and the walls that would be required to support Pete's weight.
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A1 Physics
Unit 8: Circular Motion and Gravity
Twirling Stopper Lab
Theory: Any object that moves in a circle does so because a net force is exerted toward the center of the
circle. That force is called the Centripetal force. The centrifugal force is a fiction and does not exist. You
will never hear a physics teacher talking about it since they don’t teach science fiction.
Purpose: Determine the centripetal force applied by the string while a stopper is moved in a circular path
as determined by the spring scale attached to the string. Then a comparison is made of the scale reading
2
to the value calculated from the formula
F m v
c
r
Materials:
Rubber Stopper/Tubing apparatus
Balance
Meter stick
Stopwatch Spring Scale
Procedure:
1. Pick an apparatus and twirl the stopper at constant speed for one minute.
2. Count the number of revolutions made and record.
3. Read the spring scale and record the average magnitude of force during the one minute period.
4. Record the mass of the stopper as determined using a balance.
5. Measure the length of the string and record.
Summarize the Purpose of the Lab: (using your OWN words!) ____________________________
Observations:
Mass of stopper (kg)
Length of string (m)
Number of revolutions per minute
Average spring scale reading (N)
Calculations: Show your work below
Circumference of stopper’s path (m)
C=2πR
Time for one revolution (s)
T=Total Time / # Rev
Speed of stopper (m/s)
V=C/T
Centripetal force (N)
F = mV2/R
Calculate the percent difference between the spring scale reading and your calculated centripetal force.
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =
|𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 − 𝑆𝑝𝑟𝑖𝑛𝑔 𝑆𝑐𝑎𝑙𝑒|
𝑥100
𝑆𝑝𝑟𝑖𝑛𝑔 𝑆𝑐𝑎𝑙𝑒
Analysis:
Which result do you think is more accurate, the spring scale reading or the calculated value? Why? (use
complete sentences)
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Unit 8: Circular Motion and Gravity
Physics
Apparent Weightlessness Lab
PROBLEM:
How does gravity affects an object in free fall?
HYPOTHESIS:
_________________________________________________________
MATERIALS:
2 plastic, foam or paper cups
2 long rubber bands
2 washers or other small mass
DISCUSSION:
Some people believe that because astronauts aboard an orbiting space
vehicle appear weightless, the pull of gravity upon them is zero. This
condition is commonly referred to as “zero-g.” While it is true that they
feel weightless, gravity is acting upon them. It acts with almost the same
magnitude as on the earth’s surface.
masking tape
large paper clip
water
The key to understanding this condition is realizing that both the
astronauts and the space vehicle are in free fall. It is very similar to how
you feel inside of an elevator with a snapped cable! The primary
difference between the runaway elevator and the space vehicle is that the
runaway elevator has not horizontal velocity (relative to the earth’s
surface) as it falls toward the earth, so it eventually hits the earth. The
horizontal velocity of the space vehicle ensures that as it falls toward the
earth, it also moves around the earth. If falls without getting closer to the
earth’s surface. Both cases involve free fall.
PROCEDURE:
1. Knot together two rubber bands to make one long
rubber band. Knot each end around a washer, and tape the washers
to the ends. Bore a small hole about the diameter of a pencil through
the bottom of a Styrofoam or paper cup. Fit the rubber bands through
the hole from the inside. Use a paper clip to hold the rubber bands in
place under the bottom of the cup (see Figure A). Hang the washers
over the lip of the cup. The rubber bands should be taut.
2. Drop the cup from a height of about 2 –m.
a. What happens to the washers?
Figure A
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Unit 8: Circular Motion and Gravity
3. Remove the rubber from the cup and fill the cup ½ full with water,
using your finger as a stopper over the hole. Hold the cup directly
over a sink. Drop the cup into the sink.
a. What happens to the water as the cup falls?
4. Repeat Step 3 for a second cup ½ filled with water with two holes
poked through its sides (Figure B).
a. What happens to the water as the cup falls?
ANALYSIS:
1. Explain why the washers acted as they did in Step
2.
2. Explain why the draining water acted as it did in
Step 3 and 4.
3. Suppose you were standing on a bathroom scale inside an elevator.
Based on your observations in this activity, predict what would
happen to your weight reading when the elevator:
a. Accelerated upward.
b. Accelerated downward at an acceleration less than g
c. Moved upward at a constant speed.
d. Moved downward at a constant speed.
e. Accelerated downward at an acceleration value greater than g.
CONCLUSION:
Re-answer the original problem on this lab using observations to support
your conclusion.
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Unit 8: Circular Motion and Gravity
Review Sheet
Chart of Standards:
Mass of Earth = 5.98 x 1024 kg
Mass of Sun = 1.99 x 1030 kg Mass of Moon = 7.35 x 1022 kg
6
Radius of Earth = 6.38 x 10 m
Radius of Sun = 6.96 x 108 m Radius of the
6
Moon = 1.74 x 10 m
Distance between center of Sun and center of Earth and Moon = 1.50 x 1011 m
Distance between center of Earth and center of Moon = 3.84 x 108 m
1. What is the direction of the centripetal force of an object in uniform circular motion?
b. What is the direction of the centripetal acceleration?
2. What causes a tin can whirled on the end of a string to move in a circular path?
b. If the string breaks, what causes it to move in a straight-line path?
3. An can on a string takes 5 seconds to revolve 3 times. What is its frequency? [ 0.6 Hertz]
4. What is the period of the can? [ 1.67 sec]
5. What is the period and frequency of the Earth’s rotation based on a 24 hour day. [ 86,400 sec, 1.16 x
10-5 Hz]
6. What is the period and frequency of the Earth’s revolution based on 365.25 days a year?[3.16x107sec,
3.2 x10-8 Hz]
7. What is the period and frequency of the horse on a carousel that takes a minute to revolve twice? [30
sec, 0.03 Hz]
8. What is the tangential velocity for the can in problem 3 if the string is 1.4 m long? [5.27 m/s]
9. What is the centripetal acceleration of a plane in a circular path with a radius of 2500 meters which
takes 60 seconds to complete the circle?[27.42 m/s2]
10. What is the centripetal acceleration of an object moving in a horizontal circular path of 20 m radius
with a speed of 18 m/s? [16 m/s2]
11. An automobile travels around a circular track of radius 120 m at a speed of 2.5 m/s. What is the
centripetal acceleration? [0.05 m/s2]
12. A mass of 1.5 kg moves in a circular path with a constant speed of 3 m/s on a horizontal frictionless
surface. The mass is held to the circular path by a light cord 2.4 m long that has one end fixed and the
other end attached to the mass. Calculate the tension in the cord. [5.63 N]
13. An automobile of mass 1500 kg travels in a circular path of radius 20 m on a horizontal road surface
at a speed of 12.5 m/s (around 72 mph). What is the kinetic friction between the tires and the road?
[11,718.75 N]
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Unit 8: Circular Motion and Gravity
14. Determine the centripetal force acting on a 40 kg child who makes 10 revolutions around in the
Cliffhanger in 29.3 s. The radius of the barrel of the ride is 2.9 m. [533.46 N]
15. If a car travels in a circle at a constant speed, what is the direction of the net force on the car?
b. What is the name of the net force for any uniform circular motion problem?
16. A Lincoln Continental and a VW Beatle are making a turn. The Lincoln is four times more massive
than the Beatle. If they make the turn at the same speed, then how do the centripetal forces acting on
the two cars compare? Explain.
17. The Cajun Cliffhanger at Great America is a ride in which occupants line the perimeter of a cylinder
and spin in a circle at a high rate of turning. When the cylinder begins spinning very rapidly, the floor is
removed from under the rider’s feet. What effect does a tripling in speed have upon the centripetal
force? Explain.
18. A 95 kg halfback makes a turn on the football field. The halfback sweeps out a path which is a
portion of a circle with a radius of 12 meters. The halfback makes a quarter of a turn around the circle in
2.1 seconds. Determine the speed, acceleration and net force acting upon the halfback. [8.976 m/s,
6.714 m/s2, 637.835 N]
19. An 1800 kg car rounds a curve of radius 92 m with a speed of 3.0 m/s. If its wheels are on the verge
of slipping on the road, what is the force of friction? [176.087 N]
20. A model airplane is guided in a horizontal circular path by a cord of length 16 m. If it has a constant
speed of 18 m/s and a mass of 200 g, what is the tension in the cord? [4.05 N]
21. What is the velocity of a 0.5 kg rock revolving on a 1.8 m string if the tension force in the string is 22
N [8.9 m/s]
22. What is the mass of a satellite if the Fg is 10,500 N, its velocity is 1,025 m/s and its distance from the
center of the Earth is 6.41 x 106 m? [64,061.87 kg]
23. What is the Fg of the Earth on the moon? [7.63x1028 N]
24. Two tennis balls weigh 1.64 kg and their centers are 0.020 m apart. Solve for the gravitational force
between them. [4.485 x 10-7 N]
25. Two balls have their centers of mass 2 m apart. One has a mass of 5 kg and the other has a mass of 8
kg.
a. What is the gravitational force between them? [6.67 x 10-10 N]
b. The distance between the balls is now cut in half. What is the gravitational force between them?
[2.668 x 10-9 N]
26. What is the force of gravity acting on a 5000 kg space craft when it orbits three earth radii from the
earth’s center? [5.44 x 103 N]
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