Question

Which graph best represents the variation of radius r of a circular orbit of a satellite against its time period T?

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C 3

Consider a satellite of mass $M$ revolving in a circular orbit around the earth, which is located at the center of its orbit. If a satellite is at height $h$ above earths surface the radius of the orbit $r=R_e$ th where $R_e$ is radius of earth. The gravitational force between $M_e$ and $M$ provides the centripetal force for circular motion

$\frac{{GM}_{e}m}{{(R_e+h)}^2}=\frac{{mV}^2}{R_e+h}$

or $V^2=\frac{{GM}_e}{R_e+h}V=\sqrt{\frac{{GM}_e}{R_e+h}}$

Hence orbital velocity depends on height of the satellite above earth's surface. Time period if satellite is time taken to complete one revolution.

$T=\frac{2\pi r}{V}=2\pi (R_e+h)$

$\sqrt{R_e+h}/{GM}_e$

$T^2=\frac{{4\pi }^2{(R_e+h)}^3}{{GM}_e}$ where $r=R_e+h$

So, option (C) $-3$ is time.