Name
Period Date
The equation for the Law of Universal Gravitation is
Where F is the attractive force between masses m and
1 m separated by distance r . G
2 is the universal gravitational constant (and relates G to the masses and separation as the constant

similarly relates the circumference of a circle to its diameter). By substituting changes in any of the variables into this equation we can predict how the others change. For example, we can see how the force changes if we know how either or both of the masses changes, or how the separation between their centers changes.
Suppose, for example, that one of the masses is somehow doubled. Then substituting
2 m
1
for m
1
in the equation gives
F new
F

G m
1 m
2 r
2

G
2 m
1 m
2 r
2

2


G m
1 m
2 r
2



2 F old
Therefore the force also doubles. Suppose instead that the separation distance is doubled. Then substituting 2 r for r in the equation gives
F new

G
2 m
1 m
 
2
2 
G m
1 m
2
4 r
2

1
4


G m
1 m
2 r
2



1
4
F old
Hence the force is only ¼ as much.
Use this method to solve the following problems. Write the equation and make the appropriate substitutions.
1. If both masses are doubled, what happens to the force?
2. If the masses are not changed, but the distance of separation is reduced to ½ the original distance, what happens to the force?
4. If both masses are doubled, and the distance of separation is doubled, show what happens to the force.

Universal Gravitation Worksheet 4 page 2
5. If one of the masses is doubled, the other remains unchanged, and the distance of separation is tripled, show what happens to the force.
6. Consider a pair of binary stars that pull each other with a certain force. Show what happens to the force if the mass of each star were three times as great when their distance apart is three times as far?
7. Two identical spheres of mass M and radius R have their surfaces separated by a distance 2 R . The force of gravity between them is F . Suppose two other spheres made of the same material have radii 2 R and have their surfaces separated by a distance of 4 R .
a. What is the mass of each of the new spheres? b. What is the new distance of separation? c. What is the new force of gravity acting between the new spheres?
8. An object of weight W on Earth is moved to a newly discovered planet which has a
1 density
2

E
and radius 2 R
E
, where

E
and R
E
are the density and the radius of
Earth, respectively.
a. What is the mass of the new planet compared to the mass of the Earth? b. What would the object weigh on the new planet?