First slide

Motion of planets and satellites

Question

A spherical planet has uniform density $\frac{π}{2} \times 10^{4} kg / m^{3}$ . Find out the minimum period for a satellite in a circular orbit around it in seconds. $(Use G = \frac{20}{3} \times 10^{- 11} \frac{N - m^{2}}{{kg}^{2}}) .$

Moderate

A

7500
B

3000
C

4500
D

6000

Solution

Time period is minimum for the satellites with minimum radius of the orbit i.e. equal to the radius of the planet. Therefore.
$\begin{array}{l} \frac{GMm}{R^{2}} = \frac{{mv}^{2}}{R} \Rightarrow V = \sqrt{\frac{GM}{R}} \\ T_{min} = \frac{2 πR}{\sqrt{\frac{GM}{R}}} = \frac{2 πR \sqrt{R}}{\sqrt{GM}} \end{array}$
$using M = \frac{4}{3} {pR}^{3} \cdot ρ T_{min} = \sqrt{\frac{3 π}{Gρ}}$
$Using values T_{min} = 3000 s$

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