The motion of planets in the solar system is | vedclass

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Question

From Kepler's second law of motion, a line joining any planet to the sun sweeps

out equal areas in equal intervals of time.

Let at any insta

Vedclass · Accepted Answer

Angular momentum

Vedclass · Answer

From Kepler's second law of motion, a line joining any planet to the sun sweeps

out equal areas in equal intervals of time.

Let at any instant $t,$ the planet is in position $A.$ Then area swept out be $SA$ is

$\mathrm{dA}=$ area of the curved triangle $SAB$

$\mathrm{d} \mathrm{A}=$ area of the curved triangle $\mathrm{SAB}$

$=\frac{1}{2}(A B 	imes S A)$

$=\frac{1}{2}(r d 	heta 	imes r)=\frac{1}{2} r^{2} d 	heta$

The instantaneous areal speed is

$\left.=\frac{d A}{d t}=\frac{1}{2} r^{2}\left(\frac{d 	heta}{d t}ight)ight)=\frac{1}{2} r^{2} \omega$

Let $J$ be angular momentum, $I$ the moment of inertia and m the mass, then

$J=I \omega=m r^{2} \omega$

$=\frac{d A}{d t}=\frac{J}{2 m}=\mathrm{constant}$

The motion of planets in the solar system is an example of the conservation of

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