What happens to the force between two objects, if
(i) the mass of one object is doubled?
(ii) the distance between the objects is doubled and tripled?
(iii) the masses of both objects are doubled?
From the relationship,
\(F = \frac{Gm_1m_2}{R^2}\)
(i) If the mass of one object (say body 1) is doubled, then
\(F' = \frac{G\times(2m_1)m_2}{R^2}\)= \(\frac{2 \times Gm_1m_2}{R^2}\)
= 2F
Thus, the gravitational force between the two objects gets doubled.
(ii) If the distance between the two objects is doubled, then
\(F" = \frac{Gm_1m_2}{2R^2}\)= \(\frac{1}{4}\) x \(\frac{ Gm_1m_2}{R^2}\) = \(\frac{1}{4}\)F = \(\frac{F}{4}\)
Thus, the gravitational force between the two objects becomes one-fourth.
If the distance between the two objects is tripled, then
\(F" = \frac{Gm_1m_2}{(3R)^2}\)= \(\frac{1}{9}\) x \(\frac{ Gm_1m_2}{R^2}\) =\(\frac{1}{9}\)x F = \(\frac{F}{9}\)
Thus, the gravitational force between the two objects becomes one-ninth.
(iii) If the masses of both the objects are doubled, then
\(F"' = \frac{G\times (2m_1)\times (2m_2)}{R^2}\)= \(\frac{4 Gm_1m_2}{R^2}\) = 4 F
Thus, the gravitational force between the two objects becomes 4 times.