Assuming that the earth is revolving around the sun in circular orbit of radius $R$, the angular momentum is…
Assuming that the earth is revolving around the sun in circular orbit of radius $R$, the angular momentum is directly proportional to $\mathrm{R}^{\mathrm{n}}$. The value of ' $n$ ' is
2.
1.5
1
0.5
Solution
We know,
Angular momentum $L=m v R$...(i)
For a planet revolving around the sun, $\frac{\mathrm{mv}^2}{\mathrm{R}}=\frac{\mathrm{GMm}}{\mathrm{R}^2}$
...(C.P.F = Gravitational force)
$v^2=\frac{G M}{R}$
$\mathrm{v}=\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}$...(ii)
Putting equation (ii) into (i)
$\begin{array}{ll}
\therefore & \mathrm{L}=\mathrm{m} \times \sqrt{\frac{\mathrm{GM}}{\mathrm{R}}} \times \mathrm{R}=\mathrm{m} \sqrt{\mathrm{GMR}} \\
\therefore & \mathrm{~L} \propto \mathrm{R}^{1 / 2} \\
\therefore & \mathrm{n}=0.5
\end{array}$