Question

Two satellites of earth, $S_1$ and $S_2$ , are moving in the same orbit. The mass of $S_1$ is four times the mass of $S_2$ . Which one of the following statements is true ?

• A. The time period of $S_1$ is four times that of $S_2$
• B. The potential energies of earth and satellite in the two cases are equal
• C. $S_1$ and $S_2$ are moving with the same speed
• D. The kinetic energies of the two satellites are equal

Answer 1

Answer: (C)

The satellite of mass $m$ is moving in a circular orbit of radius $r$.
$\therefore$ Kinetic energy of the satellite, $K=\frac{GMm}{2r}\dots$ (i)
Potential energy of the satellite, $U=\frac{-GMm}{r}\dots$ (ii)
Orbital speed of satellite, $v=\sqrt{\frac{GM}{r}}\dots$ (iii)
Time-period of satellite, $T={\left[\left(\frac{4{\pi }^2}{GM}\right)r^3\right]}^{1/2}\dots$(iv)
Given $m_{S_1}=4m_{S_2}$
Since $M,r$ is same for both the satellites $S_1$ and $S_2$
$\therefore$ From equation (ii), we get $U\propto m$
$\therefore \frac{U_{S_1}}{U_{S_2}}=\frac{m_{S_1}}{m_{S_2}}=4$
or, $U_{S_1}=4U_{S_2}.$
From (iii), since $v$ is independent of the mass of a satellite, the orbital speed is same for both satellites $S_1$ and $S_2$.
Hence option (b) is correct.
From (i), we get $K\propto m$
$\therefore \frac{K_{S_1}}{K_{S_2}}=\frac{m_{S_1}}{m_{S_2}}=4$
or, $K_{S_1}=4K_{S_2}$
Hence option (c) is wrong.
From (iv), since $T$ is independent of the mass of a satellite, time period is same for both the satellites $S_1$ and $S_2$.