First slide

Gravitaional potential energy

Question

A geo-stationary satellite is orbiting the earth at a height at a above the surface of the earth, R being the radius of the earth. The time period of another satellite at a height of 2.5R from the surface of earth is

Moderate

A

$6 \sqrt{2} hr$
B

6 hr
C

$5 \sqrt{2} hr$
D

10 hr

Solution

The time period of a satellite orbiting at a distance r from the centre of the earth is given by
$T^{2} = \frac{4 π^{2} r^{3}}{{GM}^{2}}$
For two satellites at distances r₁ and r₂, we have
$(\frac{T_{1}}{T_{2}}) = {(\frac{r_{1}}{r_{2}})}^{3 / 2}$
or $T_{2} = T_{1} {(r_{2} / r_{1})}^{3 / 2}$
For geostationary satellite,
T₁ =1day =24hours
and $r_{1} = 6 R + R = 7 R$
For other satellite $r_{2} = 2 \cdot 5 R + R = 3 \cdot 5 R$
$∴ T_{2} = 24 \times (3 \cdot 5 R / 7 R)^{3 / 2} = 24 \times (1 / 2)^{3 / 2}$
$= 6 \sqrt{2} hours$

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