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  4. A satellite of mass m is orbiting the earth in a circular orbit at a height 2R above earth surface where R is the radius of the earth. If it starts losing energy at a constant rate β , then it will fall on the earth surface after the time t=(GMm/nβR). Assuming that the satellite is approximately in a circular orbit at all times, find the value of n.

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Q. A satellite of mass $m$ is orbiting the earth in a circular orbit at a height $2R$ above earth surface where $R$ is the radius of the earth. If it starts losing energy at a constant rate $\beta ,$ then it will fall on the earth surface after the time $t=\frac{GMm}{nβR}.$ Assuming that the satellite is approximately in a circular orbit at all times, find the value of $n.$

NTA AbhyasNTA Abhyas 2020Gravitation

Solution:

$E_{i}=-\frac{GMm}{2 \left(3 R\right)},$ $E_{f}=-\frac{GMm}{2 \left(R\right)}$
$\therefore \, βt=\frac{GMm}{R}\left[\frac{1}{2} - \frac{1}{6}\right]$
$βt=\frac{GMm}{3 R}\Rightarrow t=\frac{GMm}{3 βR}$