A planet of mass `m` moves along an ellipse around the sun so that its maximum and minimum distance from the sun are equal to `r_(1)` and `r_(2)` respectively. Find the angular momentum of this planet relative to the centre of the sun. mass of the sun is `M`.

Options

A. `msqrt((2GMr_(1)r_(2))/((r_(1)+r_(2))^(2)))`
B. `msqrt((2GMr_(1)^(2)r_(2)^(2))/((r_(1)+r_(2))^(2)))`
C. `msqrt((2GMr_(1)^(2)r_(2)^(2))/((r_(1)+r_(2)))`
D. `msqrt((2GMr_(1)r_(2))/((r_(1)+r_(2)))`

Answer

B

Explanation

Applying conservation of angular momentum and mechanical energy at `P` and `P.`
`mv_(1)r_(1)sin90^(@)=mv_(2)r_(2)sin90^(@)` or `v_(2)=((r_(1))/(r_(2)))v_(1)`
`1/2m((r_(1)^(2))/(r_(2)^(2))-1)v_(1)^(2)=GMm(1/(r_(2))-1/(r_(1)))`
or `v_(1)^(2)((r_(1)+r_(2))/(r_(2)))=(2GM)/(r_(1)) v_(1)=sqrt((2GMr_(2))/(r_(1)(r_(1)+r_(2))))`
angular momentum of planet, `L=mv_(1)r_(1)` or `L=msqrt((2GMr_(1)r_(2))/((r_(1)+r_(2)))`