Chapter 14, Problem 49A

To determine

Equation for the time taken by the satellite to completely orbit Earth.

Answer to Problem 49A

The time period of satellite is,

  

Explanation of Solution

Given:

Speed of satellite in a circular orbit V , radial distance r , Mass of Earth M , Time period T

and circumference of Earth is $d=2\pi r$ , Gravitational constant G .

Formula used:

Speed of satellite is,

  $V=\sqrt{\frac{GM}{r}}$ ……(1) General equation for speed is,

  $V=\frac{d}{T}$ …..(2)

Calculation:

To find the time, equate equation (1) and (2),

$\begin{array}{c}\frac{d}{T}=\sqrt{\frac{GM}{r}}\\ \frac{2\pi r}{T}=\sqrt{\frac{GM}{r}}\\ 2\pi r=T\sqrt{\frac{GM}{r}}\\ 4{\pi }^2{r}^2=T^2\frac{GM}{r}\\ T^2=\frac{r}{GM}4{\pi }^2{r}^2\\ T=\sqrt{\frac{r}{GM}4{\pi }^2{r}^2}\\ =\sqrt{\frac{4{\pi }^2{r}^3}{GM}}\\ =2\pi \sqrt{\frac{r^3}{GM}}\end{array}$

Conclusion:

Therefore, the time period of satellite takes to complete one orbit is,

$T=2\pi \sqrt{\frac{r^3}{GM}}$