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Circular motion and the Law of Universal Gravitation Study Guide
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Universal Law of Gravitation
o Any two objects will attract each other with a gravitational force. This force is directly
proportional to the objects masses and inversely proportional to the square of the
distance between their centers.
Gm1m2
 Fg 
d2
 Directly proportional means that if one variable (in this case, mass) increases, the
corresponding variable (Force) will increase as well. If the mass of one of the
objects doubles, the Force will double. If the mass of one of the objects is halved,
the Force is also halved.
 Inversely proportional means that when one variable increases, the other
decreases by the inverse. Because the distance is squared, the force is changed by
the square of the factor by which the separation distance has changed. For
example, if the distance is decreased by ½, the force is increased by the inverse
squared, or 22 =4 times the original value. If the distance is tripled, the forced will
be ( 1/3)2 =1/9th the original value.
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Circular motion
o Circular motion results in a continuous change of direction, so even though an object’s
speed does not change, its velocity does. When an object undergoes a change in velocity,
it undergoes acceleration.
o In uniform circular motion, the acceleration vector is always directed towards the center
of the circle.
o The velocity vector is always tangent to the circular path.
o Unbalanced forces cause acceleration. The acceleration is always in the direction of the
force, thus, the Centripetal Force vector is always directed toward the center of the
circle.
o Centripetal acceleration directly depends on the magnitude of the velocity squared and
inversely on the radius of the curve.
2
 ac  vr
 If the velocity increases, the acceleration will also increase by the square of the
increase. Ex: If velocity triples, acceleration will increase by 9 times.
 If the radius increases, acceleration decreases. Ex: If the radius is tripled, the
acceleration will decrease by a factor of 1/3.
o Since F=ma, Fc=mac (Fc=Centripetal Force and ac= centripetal acceleration).
mv2
 Fc 
r
o Centripetal force is not the result of circular motion but is provided by interaction with
another object. Centripetal force could be a tension force, friction force, etc.
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A particle is moving in a circular path. If the force on the particle would suddenly vanish (the
string is cut), in which direction would the ball fly off?
Draw an object undergoing circular motion. Label the force, acceleration, and velocity vectors.
Additional Practice Questions
Newton’s Law of Gravitation Questions:
1. Using Newton's Law of Universal Gravitation, what happens to the force if
a. both masses are tripled?
b. the masses are not changed, but the distance of separation is reduced to 2/3 the original distance?
c. the masses are not changed, but the distance of separation is tripled?
d. both masses are doubled, and the distance of separation is tripled?
e. one of the masses is doubled, the other remains unchanged, and the distance of separation is
quadrupled?
2. Two objects gravitationally attract with a force of 18 N. If the distance between the two objects'
centers is doubled, then the new force of attraction is ______ N.
3. Two objects gravitationally attract with a force of 18 N. If the distance between the two
objects'centers is tripled, then the new force of attraction is ______ N.
4. Two objects gravitationally attract with a force of 18 N. If the distance between the two objects'
centers is halved, then the new force of attraction is ______ N.
5. Two objects gravitationally attract with a force of 18 N. If the distance between the two objects'
centers is decreased by a factor of three, then the new force of attraction is ___ N.
6. Two objects gravitationally attract with a force of 18 N. If the distance between their centers is
decreased by a factor of four, then the new force of attraction is ___ N.
7. Two objects gravitationally attract with a force of 18 N. If the mass of one of the objects is doubled
and the distance between their centers is doubled, then the new force of attraction is ______ N.
8. Two objects gravitationally attract with a force of 18 N. If the masses of both of the objects are tripled
and the distance between the two objects' centers is doubled, then the new force of attraction is
______ N.
Circular Motion Questions:
9. A rock is tied to a string and swung in a clockwise, vertical circle.
When the rock is at the position shown, which of the following
indicates the direction of vT (tangential velocity), aC (centripetal
acceleration), and/or FW (weight)? Which represents the greatest
centripetal acceleration?
10. Ball II has twice the mass of Ball I, and both are moving in circles of equal radii at the same speed.
How do the net forces of Ball I and Ball II compare?
11. A car travels around a curved track. It moves twice as fast the second time around the track, as
compared to the first time. What can be concluded about the centripetal force as the car travels
around the track for the second time?
A. The centripetal force is one-fourth the original amount.
B. The centripetal force is one-half the original amount.
C. The centripetal force is twice the original amount.
D. The centripetal force is four times the original amount.
12. An object moves in uniform circular motion. What is true regarding the force on the object?
A. The direction is away from the center, and its magnitude varies.
B. The direction is toward the center, and its magnitude varies.
C. The direction is away from the center, and its magnitude is constant.
D. The direction is toward the center, and its magnitude is constant.
13. An object with a mass of 1.0 kg attached to a string 0.50 meter long. The object is moving at a constant
speed of 5.0 meters per second in a horizontal circular path with center at point O. What is the
magnitude of the centripetal force acting on the object?
14. An object with a mass of 0.5 kilogram is swung by a string in a horizontal circle of radius 1 meter at a
speed of 2 meters per second. The magnitude of the force on the mass is
15. An object with a mass of 0.5 kilogram is swung by a string in a horizontal circle of radius 1 meter at a
speed of 2 meters per second. As the speed of the object increases, the force needed to keep it in a
circular path will
16. Two masses, A and B, move in circular paths as shown in the diagram.
Compare the centripetal acceleration of mass A to that of mass B.
17. As a cart travels around a horizontal circular track, the cart must undergo
a change in
a. velocity
c. speed
b. inertia
d. Weight
18. A convertible car with its top down is traveling at a constant speed around a
circular track as shown in the diagram. When the car is at point A, if a passenger
in the car throws a ball straight up the ball could land at point
a. A
c. C
b. B
d. D
19. A child is riding on a merry-go-round. As the speed of the merry-go-round is
doubled, the magnitude of the centripetal force acting on the child
a. remains the same
c. is halved
b. is doubled
d. is quadrupled