Chapter 13, Problem 62A

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

To determine

Factor of changing force of gravity.

Answer to Problem 62A

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

Explanation of Solution

Given:

Mass of both planets remains same.

Distance between the planets is reduced 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 reduced by four times.

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

Therefore, the resulting force of gravity between the planets is

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

It means the force is increased.

Conclusion:

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