Initially a satellite of 100 kg is in a circular orbit of radius $1.5 R_{\mathrm{E}}$. This satellit

Question

Initially a satellite of 100 kg is in a circular orbit of radius $1.5 R_{\mathrm{E}}$. This satellite can be moved to a circular orbit of radius $3 R_E$ by supplying $\alpha 	imes 10^6 \mathrm{~J}$ of energy The value of $\alpha$ is $\_\_\_\_$ . (Take Radius of Earth $R_E=6 	imes 10^6 \mathrm{~m}$ and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

Professor PC Thomas Classes · Answer

The total energy of a satellite in a circular orbit of radius $r$ is $E = -\frac{GMm}{2r}$. The energy required to move it from $r_1 = 1.5 R_E$ to $r_2 = 3 R_E$ is $\Delta E = E_2 - E_1 = \frac{GMm}{2} \left(\frac{1}{1.5R_E} - \frac{1}{3R_E}ight) = \frac{GMm}{2R_E} \left(\frac{2}{3} - \frac{1}{3}ight) = \frac{GMm}{6R_E}$. Substituting $GM = gR_E^2$, we get $\Delta E = \frac{mgR_E}{6}$. Putting the values: $\Delta E = \frac{100 	imes 10 	imes 6 	imes 10^6}{6} = 1000 	imes 10^6 	ext{ J}$. Thus, $\alpha = 1000$.