Question

a planet of mass M moves in circular orbit of radius r0 in the gravitational potential vr=-k/r then what is the angular momentum of this planet.

Answer 1

Given potential is $V = -\frac{k}{r}$

Let L be angular momentum then total effective potential will be

$V_0 = V + \frac{L^2}{2Mr^2} = -\frac{k}{r} + \frac{L^2}{2Mr^2}$

Differentiate with respect to r and differentiation will be zero at $r = r_0$

Hence, $\frac{dV}{dr} = \frac{k}{r^2} - \frac{L^2}{Mr^3}$

$\frac{dV}{dr}|_{r=r_0} = \frac{k}{r_0^2} - \frac{L^2}{Mr_0^3} = 0$

Solving it we get,

$L = \sqrt{kMr_0}$