Question
The time period of an earth satellite in circular orbit is independent of-
A.
both the mass and radius of the orbit.
B.
radius of its orbit.
C.
the mass of the satellite.
D.
neither the mass of the satellite nor the radius of its orbit.
Answer :
the mass of the satellite.
Solution :
we have, $$\frac{{m{v^2}}}{{R + x}} = \frac{{GmM}}{{{{\left( {R + x} \right)}^2}}}$$
$$x \,\,=$$ height of satellite from earth surface, $$ m\,\,=$$ mass of satellite
$$\eqalign{
& \Rightarrow {v^2} = \frac{{GM}}{{\left( {R + x} \right)}}\,\,or\,\, v = \sqrt {\frac{{GM}}{{R + x}}\,} \cr
& T = \frac{{2\pi \left( {R + x} \right)}}{v} = \frac{{2\pi \left( {R + x} \right)}}{{\sqrt {\frac{{GM}}{{R + x}}\,} }} \cr} $$
which is independent of mass of satellite.