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A planet of mass m revolves in elliptica...

A planet of mass `m` revolves in elliptical orbit around the sun of mass `M` so that its maximum and minimum distance from the sun equal to `r_(a)` and `r_(p)` respectively. Find the angular momentum of this planet relative to the sun.

Text Solution

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Using conservation of angular momentum
`m upsilon_(p)r_(p) = m upsilon_(a) r_(a)`
As velocitties are perpendicular to the radius vectors at apogee and perigee.
`rArr upsilon_(p) r_(p) = upsilon_(a) r_(a)`
Using conservation of energy,
`-(GMm)/(r_(p)) + (1)/(2) m upsilon_(p)^(2) = (-GMm)/(r_(a)) + (1)/(2) m upsilon_(a)^(2)`
BY solving, the above equations,
`upsilon_(p) = sqrt((2GMr_(a))/(r_(p)(r_(p) + r_(a))))`
`L = m upsilon_(p) r_(p) = m sqrt((2GMr_(p)r_(a))/((r_(p) + r_(a))))`
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