Question

If the distance between two bodies is increased 4 times by what factor should the mass of the bodies be altered so that the gravitational force between them remains the same?

Answer 1

we know that F=GMm/R²
f=GMm/(4R)²                     (f and F are different)
If f=F, 
GMm/R²=GxMm/(4R)²       (let x be the increase in mass)
divide GMm/R² from both sides,
1=x/16
x=16
therefore it should be multiplied 16 times.
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Answer 2

Explanation:

The gravitational force between two masses is given by :

$F=G\dfrac{m_1m_2}{r^2}$

G is universal Gravitational constant

r is distance between masses

If the distance between two bodies is increased 4 times then let us suppose that the mass is altered by x times so that the gravitational force between them remains the same. So, according to given condition,

$G\dfrac{m_1m_2}{r^2}=G\dfrac{m_1m_2x}{(4r)^2}\\\\\dfrac{1}{r^2}=\dfrac{x}{16r^2}\\\\x=16$

So, the mass of the bodies must be altered by 16 times so that the gravitational force between them remains the same.