Chapter 13, Problem 8A

How does the force of gravity between two bodies change when the distance between them is doubled?

To determine

The change in force of gravity between the two bodies for the given change in distance.

Answer to Problem 8A

The new force will reduces to one-fourth of the original value.

Explanation of Solution

Given:

The distance between the two objects is doubled.

Formula used:

If $A$ and $B$ are the two objects having masses $m_A$ and $m_B$ respectively which are separated by a distance $r$ , then the expression for gravitational force is given by,

  $F\text{ = G}\frac{m_A{m}_B}{r^2}$$\left(1\right)$

Where $\text{ G}$ is gravitational constant, $\text{ G}=6.674\times {10}^{-11}\text{ N}\cdot {\text{m}}^2/{\text{kg}}^2$

Calculation:

The force of gravity between the two bodies when the distance between them is doubled can be obtained by substituting $r=2r$ in equation $\left(1\right)$ ,

  $\begin{array}{c}F\text{ = G}\frac{m_A{m}_B}{{\left(2r\right)}^2}\\ =\text{G}\frac{m_A{m}_B}{4r^2}\\ =\frac{1}{4}\left(\text{G}\frac{m_A{m}_B}{r^2}\right)\\ =\frac{1}{4}F\end{array}$

Conclusion:

The force of gravity between the two bodies reduces to $\frac{1}{4}F$ when the distance between them is doubled.