Question

Show that the critical velocity of a body revolving in circular orbit very close to the surface of a planet of radius R and mean density ρ is $\sqrt{\frac{\pi \rho G}{3}}$

Answer 1

we know, minimum velocity required to revolve in a circular orbit around a planet is called critical velocity
e.g., $v_c=\sqrt{\frac{GM}{R}}$
where M is the mass of planet , R is the radius of that planet.
we know, mass = volume × density
so, $M=\frac{4}{3}\pi R^3\times\rho$

so, $v_c=\sqrt{\frac{G\frac{4}{3}\pi R^3\rho}{R}}\\\\=2R\sqrt{\frac{G\pi\rho}{3}}$

hence , critical velocity of a body revolving in circular orbit very close to the surface of a planet of radius R and mean density ρ is $2R\sqrt{\frac{\pi \rho G}{3}}$

Answer 2

Answer:

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