First slide

Motion of planets and satellites

Question

Suppose the gravitational force varies inversely as the $n^{t h}$ 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

$F \propto \frac{1}{R^{n}} - - - - (1) \Rightarrow G r a v i t a t i o n a l f o r c e p r o v i d e s c e n t r i p e t a l f o r c e \Rightarrow F = m ω^{2} R - - - (2) F r o m e q u a t i o n s (1) a n d (2) w e g e t \Rightarrow m ω^{2} R \propto \frac{1}{R^{n}} \Rightarrow m (\frac{4 π^{2}}{T^{2}}) R \propto \frac{1}{R^{n}} (∵ ω = \frac{2 π}{T}) \Rightarrow (\frac{1}{T^{2}}) R \propto \frac{1}{R^{n}} (∵ m, 4, π a r e c o n s \tan t s) \Rightarrow R^{n} R \propto T^{2} \Rightarrow T^{2} \propto R^{n + 1}$
$∴ T \propto R^{(\frac{n + 1}{2})}$

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