If the distance between two masses be increased by a factor of 6, what factor would the mass of one of them have to be altered to maintain the same gravitational force? Would this be an increase or decrease in the mass?

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Solution

According to the Newton's universal law of gravitation,
Case 1 : $F_{1} = \frac{G m_{1} M_{1}}{R_{1}^{2}}$
Case 2 : $F_{2} = \frac{G m_{2} M_{2}}{R_{2}^{2}}$
But it is given that $R_{2} = 6 R_{1}$
$F_{2} = \frac{G m_{2} M_{2}}{(6 R_{1})^{2}} \Rightarrow F_{2} = \frac{G m_{1} M_{1}}{(6 R_{1})^{2}} \Rightarrow F_{2} = \frac{G m_{1} M_{1}}{36 (R_{1})^{2}} \Rightarrow F_{2} = \frac{F_{1}}{36}$
So, the masses of one of them have to be altered to 36 times, to maintain the same gravitational force because the gravitational force is directly proportional to the product of the masses.

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