A particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the $n^{th}$ power of $R$ . If the period of rotation of the particle is $T,$ then

T \propto R^{(n + 1) / 2}

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T \propto R^{n / 2}

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T \propto R^{3 / 2}

for any

n

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T \propto R^{\frac{n}{2} + 1}

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Solution

The correct option is A $T \propto R^{(n + 1) / 2}$
$F = \frac{k}{R^{n}} = m ω^{2} R$
$ω^{2} \propto \frac{1}{R^{n + 1}} \Rightarrow T = \frac{2 π}{ω}$
So, $T \propto R^{\frac{n + 1}{2}}$

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A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the nth power of R. If the period of rotation of the particle is T, then

The correct option is A T∝R(n+1)/2F=kRn=mω2R ω2∝1Rn+1⇒T=2πω So, T∝Rn+12

A particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the $n^{th}$ power of $R$ . If the period of rotation of the particle is $T,$ then

The correct option is A $T \propto R^{(n + 1) / 2}$
$F = \frac{k}{R^{n}} = m ω^{2} R$
$ω^{2} \propto \frac{1}{R^{n + 1}} \Rightarrow T = \frac{2 π}{ω}$
So, $T \propto R^{\frac{n + 1}{2}}$