Kepler's second law (law of areas) is nothing | vedclass

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(C) The area swept by a planet in a small time interval $dt$ is given by $dA = \frac{1}{2} r \cdot (r d	heta)$.Therefore,the rate of area swept is $\

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Conservation of angular momentum

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(C) The area swept by a planet in a small time interval $dt$ is given by $dA = \frac{1}{2} r \cdot (r d	heta)$.Therefore,the rate of area swept is $\frac{dA}{dt} = \frac{1}{2} r \left(r \frac{d	heta}{dt}ight)$.We know that the angular momentum is $\vec{L} = m r^2 \frac{d\vec{	heta}}{dt}$.Substituting this into the area rate equation,we get $\frac{dA}{dt} = \frac{1}{2m} \left(m r^2 \frac{d	heta}{dt}ight) = \frac{L}{2m}$.According to Kepler's $2^{nd}$ law,a planet sweeps out equal areas in equal intervals of time,meaning $\frac{dA}{dt} = 	ext{constant}$.Since $m$ is constant,$L$ must also be constant.Therefore,Kepler's $2^{nd}$ law is a statement of the conservation of angular momentum.

Kepler's second law (law of areas) is nothing but a statement of

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