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Gravitation — Physics STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 SciencePhysicsGravitation3 Marks
Question
An artificial satellite revolves around a planet in circular orbit close to its surface. Obtain the formula for period of the satellite in terms of density p and radius R of planet.
✓

Answer

Time period of a satellite revolving around the planet at certain height is given by,
$T=2 \pi \sqrt{\frac{(R+h)^3}{G M}}$
For a satellite very close to the surface of planet, $R + h \approx R$
$\therefore \quad T =2 \pi \sqrt{\frac{ R ^3}{ GM }}$
Density of the planet (assuming it to be uniform) is, $\rho=\frac{M}{V}$
$\begin{array}{ll}\therefore & M =\frac{4}{3} \pi R ^3 \rho \\ \therefore & T =2 \pi \sqrt{\frac{ R ^3}{ G \times \frac{4}{3} \pi R ^3 \rho}} \\ \therefore & T =\sqrt{\frac{4 \pi^2}{ G \times \frac{4}{3} \pi \rho}} \\ \therefore & T =\sqrt{\frac{3 \pi}{ G \rho}}\end{array}$

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