Question

What happens to the gravitational force between two objects if the distance and mass between them is increasedby four times

Answer 1

Let there be a  body of mass m₁ and mass m₂ at a distance of r.

Then there Force of Gravitation = Gm₁m₂/r²

Now, if mass of both the bodies becomes 4 times of its initial masses and the the distance between them is increased to 4 times, then,

Force = G × 4m₁ × 4m₂/(4r)²

∴ Force = 16/16 Gm₁m₂/r²

∴ Force = Gm₁m₂/r²

Therefore, Force remains the same.

Hope it helps.

Answer 2

we know from Newton's law of gravitation.
gravitational force acts between two bodies is directly proportional to product of masses and inversely proportional to separation between them. e.g., $F=\frac{Gm_1m_2}{r^2}$....(1)

where G is proportionality constant known as gravitational constant.

now, if the distance between them is increased 4 times and also masses of both of them increased by 4 times.

then, $F'=\frac{G(4m_1)(4m_2)}{(4r)^2}$

or, $F'=\frac{16Gm_1m_2}{16r^2}$

or, $F'=\frac{Gm_1m_2}{r^2}$.....(2)

from equations (1) and (2), it is clear that force remains constant.