A planet of mass m is moving in an elliptical path about the sun. Its maximum and minimum distances from the sun are $r_{1}$
and $r_{2}$ respectively. If $M_{s}$ is the mass of sun then the angular momentum of this planet about the center of sun will be -

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$\sqrt{\frac{2 {GM}_{s}}{(r_{1} + r_{2})}}$

$2 {GM}_{S} m \sqrt{\frac{r_{1} r_{2}}{(r_{1} + r_{2})}}$

$m \sqrt{\frac{2 {GM}_{s} r_{1} r_{2}}{(r_{1} + r_{2})}}$

$\sqrt{\frac{2 {GM}_{s} m (r_{1} + r_{2})}{r_{1} r_{2}}}$

answer is C.

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Detailed Solution

$Applying the oprinciple of conservation of momentum ,we have m v_{1} r_{1} = m v_{2} r_{2} . . . . . . . . . . . . (1) A p p l y i n g t h e p r i n c i p l e o f c o n s e r v a t i o n o f e n e r g y \frac{1}{2} m {v_{1}}^{2} + (- \frac{G M_{s} m}{r_{1}}) = \frac{1}{2} m {v_{2}}^{2} + (- \frac{G M_{s} m}{r_{2}}) . . . . . . . . . . . (2) a n d L = m v_{1} r_{1} . . . . . . . . . . . . . . . . (3) S o l v i n g w e g e t, L = m \sqrt{\frac{2 G M_{s} r_{1} r_{2}}{r_{1} + r_{2}}}$