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Q.
The motion of planets in the solar system is an example of the conservation of
Gravitation
Solution:
From Kepler’s second law a line joining any planet to the sun sweeps out equal areas in equal times, that is, the areal speed of the planet remains constant.
$dA =$ area of curved triangle $SAB$
$ = \frac{1}{2} (AB \times SA)$
$ = \frac{1}{2} (rd \theta \times r) = \frac{1}{2} r^2 d\,\theta$
The instantaneous areal speed of the planet is
$\frac{dA}{dt} = \frac{1}{2} r^2 \frac{d\theta}{dt} = \frac{1}{2} r^2 \omega$
where $\omega$ is an angular speed. Let $J$ be angular momentum of planet about sun
$J = I \omega = mr^2 \omega$
$\frac{dA}{dt} = \frac{J}{2m}$
From Kepler’s law areal speed is constant, therefore angular momentum $J$ is constant.
Hence, Kepler’s second law is equivalent to conservation of angular momentum.
