Chapter 13, Problem 57A

Find the change in the force of gravity between two objects when both masses are doubled and the distance between them is also doubled.

To determine

Change in the force of gravity.

Answer to Problem 57A

The resulting force of gravity between the planets is unchanged.

Explanation of Solution

Given:

Mass of both planets is doubled.

Distance between the planets is also doubled.

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 is doubled.

So, $m_1\text{'}=2m_1$ and $m_2\text{'}=2m_2$

And distance between the planets is also doubled.

So, $r\text{'}=2r$

Therefore, the resulting force of gravity between the planets is

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

It means the force is unchanged.

Conclusion:

Therefore, the resulting force of gravity between the planets is unchanged.