First slide

Gravitational force and laws

Question

Suppose the gravitational force varies inversely as the n^th power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to

Moderate

A

$R^{(\frac{n + 1}{2})}$
B

$R^{(\frac{n - 1}{2})}$
C

$R^{n}$
D

$R^{(\frac{n - 2}{2})}$

Solution

$\begin{array}{l} {mω}^{2} R \propto \frac{1}{R^{n}} \Rightarrow m (\frac{4 π^{2}}{T^{2}}) R \propto \frac{1}{R^{n}} \Rightarrow T^{2} \propto R^{n + 1} \\ ∴ T \propto R^{(\frac{n + 1}{2})} \end{array}$

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