Question

The distance between the sun and the earth be r, then the angular momentum of the earth around the sun is proportional to [UP CPMT 2012]

Moderate

Solution

According to Kepler's third law,
$T^{2} \propto r^{3} \Rightarrow T \propto r^{3 / 2}$
We have, $ω = \frac{2 π}{T} \Rightarrow ω \propto r^{- 3 / 2}$
Now, $L = m r^{2} ω \Rightarrow L \propto r^{2} and L \propto ω$
$\Rightarrow L \propto r^{2} \cdot r^{- 3 / 2}$
$∴ L \propto \sqrt{r}$

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