Answer:
We know that the gravitational force F between two masses P and Q, that are separated by a distance R is:
[tex]F = G*\frac{P*Q}{R^2}[/tex]
Where G is the gravitational constant.
a) Mass P is doubled, then we have 2*P instead of P, the new force is:
[tex]F' = G*\frac{(2*P)*Q}{R^2} = 2*(G*\frac{P*Q}{R^2} ) = 2*F[/tex]
b) Now R is doubled, then instead of R, we have 2*R:
[tex]F' = G*\frac{P*Q}{(2*R)^2} = G*\frac{P*Q}{4*R^2} = G*\frac{P*Q}{R^2}*(1/4) = F/4[/tex]
c) Now we replace P by 2*P, and Q by 3*Q
[tex]F' = G*\frac{(2*P)*(3*Q)}{R^2} = 2*3*(G*\frac{P*Q}{R^2} ) = 6*F[/tex]
d) The entire mass of the system is increased by a factor of 4, then both of the individual masses are increased by a factor of 4.
Then we need to replace P by 4*P, and Q by 4*Q.
[tex]F' = G*\frac{(4*P)*(4*Q)}{R^2} = 4*4(G*\frac{P*Q}{R^2} ) = 16*F[/tex]
e) Now we replace R by R/2.
[tex]F' = G*\frac{P*Q}{(R/2)^2} = G*\frac{P*Q}{R^2/4}= 4*G\frac{P*Q}{R^2} = 4*F[/tex]
