Chapter 13, Problem 61A

By what factor does the force of gravity between two planets change when masses remain the same but the distance between them is increased by four?

To determine

Factor of changing force of gravity.

Answer to Problem 61A

The resulting force of gravity between the planets is 1/16 times the original force.

Explanation of Solution

Given:

Mass of both planets remains same.

Distance between the planets is increased by four times.

Formula Used:

From Newton’s law of gravitation,

  $F=\frac{G{m}_1{m}_2}{r^2}$

  $F$ is force of gravity, $G$ is gravitational constant, $m_1$ and $m_2$ are mass of two objects and $r$ is distance between the two objects.

Calculation:

Let the initial force of gravity is,

  $F=\frac{G{m}_1{m}_2}{r^2}$

As, mass of both planets remains same.

So, $m_1\text{'}=m_1$ and $m_2\text{'}=m_2$

And distance between the planets is increased by four times.

So, $r\text{'}=4r$

Therefore, the resulting force of gravity between the planets is

  $\begin{array}{l}F\text{'}=\frac{G{m}_1\text{'}{m}_2\text{'}}{{\left(r\text{'}\right)}^2}=\frac{G\left(m_1\right)\left(m_2\right)}{{\left(4r\right)}^2}\\ =\frac{G{m}_1{m}_2}{r^2}\times \frac{1}{16}\\ =F\times \frac{1}{16}\left(\text{Since,}F=\frac{G{m}_1{m}_2}{r^2}\right)\\ \Rightarrow F\text{'}=\frac{F}{16}\end{array}$

It means the force is reduced.

Conclusion:

Therefore, the resulting force of gravity between the planets is 1/16 times the original force.