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Gravitational Force

A planet of m...

Question

A planet of mass m moves along an ellipse so that perihelion and aphelion distances are $r_{1}$ and $r_{2}$ . Find the angular momentum of the planet.

A

m \sqrt{\frac{G M_{s} r_{1} r_{2}}{r_{1} + r_{2}}}

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B

m \sqrt{\frac{2 G M_{s} r_{1} r_{2}}{r_{1} + r_{2}}}

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C

m \sqrt{G M_{s} (r_{1} + r_{2})}

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D

m \sqrt{2 G M_{s} (r_{1} + r_{2})}

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Solution

The correct option is B $m \sqrt{\frac{2 G M_{s} r_{1} r_{2}}{r_{1} + r_{2}}}$
As there are no external forces, the angular momentum of the planet in the given orbit is constant.
Let the velocity at perihelion is $v_{1}$ and at aphelion is $v_{2}$
Lets assume, $M_{s}$ is the mass of the sun and m is the mass of the planet.
Therefore from the conservation of angular momentum, $m v_{1} r_{1} = m v_{2} r_{2}$
$\Rightarrow v_{2} = \frac{v_{1} r_{1}}{r_{2}}$
Conservation of energy:
P.E+K.E=constant
$\Rightarrow - \frac{G m M_{s}}{r_{1}} + \frac{1}{2} m {v_{1}}^{2} = - \frac{G m M_{s}}{r_{2}} + \frac{1}{2} m {v_{2}}^{2}$
$\Rightarrow \frac{G M_{s}}{r_{2}} - \frac{G M_{s}}{r_{1}} = \frac{1}{2} {(\frac{v_{1} r_{1}}{r_{2}})}^{2} - \frac{1}{2} {v_{1}}^{2} \Rightarrow \frac{G M_{s} (r_{1} - r_{2})}{r_{1} r_{2}} = \frac{1}{2} (\frac{{r_{1}}^{2} - {r_{2}}^{2}}{{r_{2}}^{2}}) {v_{1}}^{2} \Rightarrow {v_{1}}^{2} = \frac{2 G M_{s} r_{2}}{r_{1} (r_{1} + r_{2})} \Rightarrow v_{1} = \sqrt{\frac{2 G M_{s} r_{2}}{r_{1} (r_{1} + r_{2})}}$
Therefore, angular momentum is $\Rightarrow m v_{1} r_{1} = m \sqrt{\frac{2 G M_{s} r_{1} r_{2}}{(r_{1} + r_{2})}}$

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