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/nÎ²R). Assuming that the satellite is approximately in a circular orbit at all times, find the value of n.

Question

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/nÎ²R). Assuming that the satellite is approximately in a circular orbit at all times, find the value of n. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Ei=-(GMm/2 (3 R)), Ef=-(GMm/2 (R)) ∴ Î²t=(GMm/R)[(1/2) - (1/6)] Î²t=(GMm/3 R)⇒ t=(GMm/3 Î²R)

Ei​=−2(3R)GMm​, Ef​=−2(R)GMm​ ∴βt=RGMm​[21​−61​] βt=3RGMm​⇒t=3βRGMm​

$E_{i} = - \frac{GM m}{2 ( 3 R )},$ $E_{f} = - \frac{GM m}{2 ( R )}$
$∴ βt = \frac{GM m}{R} [\frac{1}{2} - \frac{1}{6}]$
$βt = \frac{GM m}{3 R} \Rightarrow t = \frac{GM m}{3 βR}$