Chapter 13, Problem 53A

Equate your weight mg to Newton’s equation for gravitational force, $G\frac{mM}{R^2}$

where M is the mass of Earth and R is Earth’s radius. Show that acceleration of free fall is $g=\frac{GM}{R^2}$

To determine

To show:The acceleration of free fall is $g=\frac{GM}{R^2}$ .

Explanation of Solution

Given:

Consider the weight of the person is denoted by $mg$ .

Formula used:

The Newton’s equation for gravitational force is,

  $F_{\text{g}}=G\frac{mM}{R}$

Here, $M$ is the mass of the Earth, $G$ is the gravitational constant, and $R$ is the radius of the Earth.

The expression for the weight $\left(W\right)$ of the person as follows:

  $W=mg$

Here, $m$ is the mass of the person and $g$ is the acceleration due to gravity of Earth.

Calculation:

Consider a person falls freely on the surface of the Earth. The expression for the weight $\left(W\right)$ of the person is,

  $W=mg$ ….. (1)

The Earth exerts a gravitational pull on the person. The Newton’s equation for gravitational force is,

  $F_{\text{g}}=G\frac{mM}{R}$ ….. (2)

To find the expression for the acceleration due to gravity of Earth, equate weight of the person with the weight of the person.

  $\begin{array}{l}W=F_{\text{g}}\\ mg=G\frac{mM}{R}\\ g=G\frac{M}{R}\end{array}$

Thus, the acceleration of free fall is shown as $g=\frac{GM}{R^2}$ .

Conclusion:

Thus, the acceleration of free fall is shown as $g=\frac{GM}{R^2}$ .