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Question: A planet of mass m moves along an ellipse so that perihelion and aphelion distances are r<sub>1</sub...

A planet of mass m moves along an ellipse so that perihelion and aphelion distances are r1 and r2. Find the angular momentum of the planet.

A

mGMSr1r2r1+r2\sqrt{\frac{GM_{S}r_{1}r_{2}}{r_{1} + r_{2}}}

B

m2GMSr1r2r1+r2\sqrt{\frac{2GM_{S}r_{1}r_{2}}{r_{1} + r_{2}}}

C

mGMS(r1+r2)\sqrt{GM_{S}\left( r_{1} + r_{2} \right)}

D

m2GMS(r1+r2)\sqrt{2GM_{S}\left( r_{1} + r_{2} \right)}

Answer

m2GMSr1r2r1+r2\sqrt{\frac{2GM_{S}r_{1}r_{2}}{r_{1} + r_{2}}}

Explanation

Solution

mv1r1 = mv2r2 or v12 = v22(r2r1)2\left( \frac{r_{2}}{r_{1}} \right)^{2}

Using energy conservation −GmMSr1+mv122\frac{- GmM_{S}}{r_{1}} + \frac{mv_{1}^{2}}{2}

= −GmMSr2+mv222\frac{- GmM_{S}}{r_{2}} + \frac{mv_{2}^{2}}{2}

or −GMSr1+−GMSr2=v222−v222(r2r1)2\frac{- GM_{S}}{r_{1}} + \frac{- GM_{S}}{r_{2}} = \frac{v_{2}^{2}}{2} - \frac{v_{2}^{2}}{2}\left( \frac{r_{2}}{r_{1}} \right)^{2}

or v2 = 2GMSr1r2(r1+r2)\sqrt{\frac{2GM_{S}r_{1}}{r_{2}\left( r_{1} + r_{2} \right)}}

and L = mv2r2 = m2GMSr1r2r1+r2\sqrt{\frac{2GM_{S}r_{1}r_{2}}{r_{1} + r_{2}}}