Question
Initially a satellite of 100 kg is in a circular orbit of radius $1.5R_{E}$. This satellite can be moved to a circular orbit of radius $3R_{E}$ by supplying $\alpha \times 10^{6}\text{ }J$ of energy. The value of $\alpha$ is $\_\_\_\_$ .
(Take Radius of Earth $R_{E} = 6 \times 10^{6}\text{ }m$ and $g = 10m/s^{2}$ )
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Solution
Energy of a satellite in a circular orbit is given as $E = \frac{- {GM}_{E}m}{2r};r =$ radius of circular orbit Required energy to be supplied $= \Delta E = E_{f} - E_{i}$
$$\begin{matrix} \Delta E & \ = \left( \frac{- {GM}_{E}\text{ }m}{2\left( 3R_{E} \right)} \right) - \left( \frac{- {GM}_{E}\text{ }m}{2\left( 1.5R_{E} \right)} \right) \\ & \ = \frac{{GM}_{E}\text{ }m}{6R_{E}} \end{matrix}$$
Now, $g = \frac{{GM}_{E}}{R_{E}^{2}} \Rightarrow \frac{{GM}_{E}}{R_{E}} = {gR}_{E}$
$$\begin{matrix} & \begin{matrix} \therefore\Delta E & \ = \frac{1}{6}{gmR}_{E} \\ & \ = \frac{1}{6} \times 10 \times 100 \times 6 \times 10^{6} \\ & \ = 1000 \times 10^{6} \end{matrix} \\ & \alpha = 1000 \end{matrix}$$
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