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  4. The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is vec L . The magnitude of the areal velocity of the planet is:

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Q. The angular momentum of a planet of mass $M$ moving around the sun in an elliptical orbit is $\vec{ L }$. The magnitude of the areal velocity of the planet is:

JEE MainJEE Main 2021Gravitation

Solution:

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For small displacement ds of the planet its area can be written as
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$d A =\frac{1}{2} rd \ell$
$=\frac{1}{2} rd \sin \theta$
$A \cdot vel =\frac{ d A }{ dt }=\frac{1}{2} r \sin \theta \frac{ d s}{ dt }=\frac{ Vr \sin \theta}{2}$
$\frac{ d A }{ dt }=\frac{1}{2} \frac{ mVr \sin \theta}{ m }=\frac{ L }{2 m }$