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If the distance between masses of two objects is increased by five times, by what factor would the mass of one of them have to be changed to maintain the same gravitational force? Would there be an increase or a decrease in the same?

Vidyadip · Accepted Answer

According to the Newton's universal law of gravitationCase: 1
$\text{F}_1=\frac{\text{Gm}_1\text{M}_1}{\text{R}^2_1}$Case: 2
$\text{F}_2=\frac{\text{Gm}_2\text{M}_2}{\text{R}^2_2}$ But it is given that $R_2 = 5R_1$_ $\text{F}_2=\frac{\text{Gm}_2\text{M}_2}{(5\text{R}_1)^2}$
$\Rightarrow\text{F}_2=\frac{\text{Gm}_1\text{M}_1}{(5\text{R}_1)^2}$
$\Rightarrow\text{F}_2=\frac{\text{Gm}_1\text{M}_1}{(25\text{R}_1)^2}$
$\Rightarrow\text{F}_2=\frac{\text{F}_1}{25}$ From the above equation we can say that if the distance between masses of two objects is increased by five times the gravitational force would be decreased to $25th$ part of the previous gravitational force. So, the masses of one of them have to be altered to $25$ times of the previous case, to maintain the same gravitational force because the gravitational force is directly proportional to the product of the masses.Note:
Even if the product of the masses of them is altered to $25$ times of the previous case and if the distance between the masses is increased for five times gravitational force remains unchanged. Because the gravitational force is directly proportional to the product of the masses and is inversely proportional to the square of the distance between the masses.

Time (s)	$0$	$2$	$4$	$6$	$8$	$10$
Speed (m/s)	$4$	$8$	$12$	$16$	$20$	$24$

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