Mathematically present the difference in the gravitational force between two objects when the mass of each is made double and the distance between them is made on fourth of their initial distance.
Solution
Let us consider, two bodies, A and B have the masses ‘\(\rm m_1\)’ and ‘\(\rm m_2\)'. The distance between them from their center is ‘d’. Then the force of gravitation ‘F’ between them is given by;
\(\rm F\)= $ \frac{Gm_1m_2}{d^2} $------(i)
Condition (a)→ If the mass of both the bodies is doubled i.e. \(\rm m_1\) is doubled to (\(\rm 2m_1\)) and \(\rm m_2\) is doubled to (\(\rm 2m_2\)) keeping the distance constant then equation (i) becomes:
\(\rm F_1\)= $ \frac{G2m_12m_2}{d^2} $
or\(\rm F_1\)= $ \frac{4Gm_1m_2}{d^2} $------(ii)
from equations (i) and (ii)
\(\rm F_1\)=\(\rm 4F\)
Hence, the force of gravitation between two bodies will be increased by four times if the mass of both of the bodies is doubled.
Similarly,
Condition(b)→If the distance between two bodies is made one fourth by keeping the mass of the bodies constant, Then the force of gravitation ‘\(\rm F_2\)’ between them is given by;
\(\rm F_2\)= $ \frac{Gm_1m_2}{(1/4d)^2} $------(iii)
or \(\rm F_2\)= $ \frac{16Gm_1m_2}{d^2} $------(iv)
From equations (i) and (iii)
\(\rm F_2\)=\(\rm 16F\)
Hence the force of gravitation between two bodies will be increased by 16 times if the distance between the centers of the bodies is made one fourth.
Now,
\(\rm F_2 \)-\(\rm F_1\) = \(\rm 16F\) - \(\rm 4F\)= \(\rm 12F\)
Hence it can be concluded that gravitational force between two bodies will be more if the distance is made one fourth of initial length than doubling the mass of each body(i.e.16F>4F).
It will be greater by 12 times if the same bodies and initial distance between their centers are taken.