Q. A comet of mass $m$ moves in a highly elliptical orbit around the sun of mass $M$. The maximum and minimum distances of the comet from the centre of the sun are $r_{1}$ and $r_{2}$ respectively. The magnitude of angular momentum of the comet with respect to the centre of sun is
Explanation
(d) The gravitational force of sun on comet is radial, hence angular momentum is constant over the entire orbit. Using law of conservation of angular momentum, at locations $A$ and $B$
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$L = mv_{1}r_{1} = mv_{2}r_{2}$ or $v_{2} = \frac{v_{1}r_{1}}{r_{2}}$ …(i)
Using the principle of conservation of total energy at $A$ and $B$
$\frac{1}{2}mv_{1}^{2} - \frac{GMm}{r_{1}} = \frac{1}{2}m\ v_{2}^{2} - \frac{GMm}{r_{2}}$
or $v_{2}^{2} - v_{1}^{2} = 2GM\left( \frac{1}{r_{2}} - \frac{1}{r_{1}} \right)$ …(ii)
Putting the values from Eq. (i) in Eq. (ii) and solving, we get
$v_{1} = \left\lbrack \frac{2GMr_{2}}{r_{1}\left( r_{1} + r_{2} \right)} \right\rbrack^{1/2}$
$\therefore L = mv_{1}r_{1} = m\left\lbrack \frac{2GMr_{1}r_{2}}{\left( r_{1} + r_{2} \right)} \right\rbrack^{1/2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.