Chapter 14, Problem 44A

An orbiting satellite of mass m is pulled toward Earth by a force ma. Equate ma to the force in Newton’s equation for universal gravitation and show that the satellite’s acceleration is a $a=\frac{GM}{d^2}$

To determine

To show:The acceleration of the satellite is $a=\frac{GM}{d^2}$ .

Explanation of Solution

Given:

Mass of orbiting satellite is m .

Satellite is pulled towards Earth by a force equal to ma .

Formula used:

The Newton’s equation for gravitational force is,

  $F_{\text{g}}=G\frac{mM}{d}$

Here, $m$ is the mass of the satellite, $M$ is the mass of the Earth, $G$ is the gravitational constant, and $d$ is the distance between the satellite and the centre of the Earth.

Calculation:

The expression for the gravitation force acting on the satellite is,

  $F_{\text{g}}=G\frac{mM}{d}$ ….. (1)

As per Newton’s Second Law of Motion,

  $F=ma$ ….. (2)

To find the acceleration of the satellite, equate equation (1) and (2).

  $\begin{array}{l}F=F_{\text{g}}\\ ma=G\frac{mM}{d}\\ a=\frac{GM}{d^2}\end{array}$

Conclusion:

Thus, the acceleration of the satellite is shown as $a=\frac{GM}{d^2}$ .