There is a hollow spherical planet of inner radius R1 and outer radius R2 and uniform mass density ρ. If the frequency of revolution of a satellite revolving around the planet in an orbit of radius 5R is given by . Find the value of N.(Given : R1 = R, R2 = 3R)

Engineering

Physics

Keplers Law

Gravitational Force and Field

Artificial Satellites

Question

There is a hollow spherical planet of inner radius R₁ and outer radius R₂ and uniform mass density ρ. If the frequency of revolution of a satellite revolving around the planet in an orbit of radius 5R is given by $f = {[(1 + \frac{1}{N^{2}}) \frac{Gρ}{15 π}]}^{1 / 2}$ . Find the value of N.

(Given : R₁ = R, R₂ = 3R)

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$F_{g} = {mE}_{g} = \frac{{mv}_{0}^{2}}{5 R}$

$\Rightarrow \frac{v_{0}^{2}}{5 R} = \frac{G}{{(5 R)}^{2}} [ρ \frac{4}{3} π {(3 R)}^{3} - ρ \frac{4}{3} {πR}^{3}]$

$\Rightarrow v_{0} = {[\frac{4 {πGρR}^{2}}{3} (\frac{26}{5})]}^{1 / 2}$

$∴ f = \frac{v_{0}}{2 π (5 R)} = {[\frac{26}{125} \frac{Gρ}{3 π}]}^{1 / 2}$

$= {[(1 + \frac{1}{5^{2}}) \frac{Gρ}{15 π}]}^{1 / 2}$