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(D) For a satellite of mass $m$ orbiting the Earth of mass $M_e$ in a circular orbit of radius $r$,the gravitational force provides the necessary cent

Vedclass · Accepted Answer

Frequency of revolution varies as $\frac{1}{r^{3/2}}$

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(D) For a satellite of mass $m$ orbiting the Earth of mass $M_e$ in a circular orbit of radius $r$,the gravitational force provides the necessary centripetal force:$\frac{GM_e m}{r^2} = \frac{mv^2}{r} \Rightarrow v = \sqrt{\frac{GM_e}{r}}$$1$. Kinetic Energy $(K.E.)$: $K.E. = \frac{1}{2}mv^2 = \frac{GM_e m}{2r}$. Thus,$K.E. \propto \frac{1}{r}$.$2$. Angular Momentum $(L)$: $L = mvr = m \sqrt{\frac{GM_e}{r}} \cdot r = m\sqrt{GM_e r}$. Thus,$L \propto r^{1/2}$.$3$. Linear Momentum $(P)$: $P = mv = m\sqrt{\frac{GM_e}{r}}$. Thus,$P \propto r^{-1/2}$.$4$. Frequency of revolution $(f)$: According to Kepler's Third Law,$T^2 \propto r^3$,so $T \propto r^{3/2}$. Since $f = \frac{1}{T}$,we have $f \propto r^{-3/2}$ or $f \propto \frac{1}{r^{3/2}}$.Therefore,option $D$ is correct.

$A$ satellite is orbiting the Earth in a circular orbit of radius $r.$ Which of the following statements is correct?

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