(B) The centripetal force required for circular motion is provided by the central force,which is inversely proportional to $R^n$: $m\omega^2 R \propto

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(B) The centripetal force required for circular motion is provided by the central force,which is inversely proportional to $R^n$: $m\omega^2 R \propto \frac{1}{R^n}$ Since $m$ is constant,we have $\omega^2 R \propto R^{-n}$,which simplifies to $\omega^2 \propto R^{-(n+1)}$. Taking the square root of both sides,we get $\omega \propto R^{-\frac{n+1}{2}}$. The time period $T$ is related to angular velocity $\omega$ by $T = \frac{2\pi}{\omega}$. Therefore,$T \propto \frac{1}{\omega} \propto R^{\frac{n+1}{2}}$.

Class 11 Physics 2 Motion in a Plane Dynamics of circular Motion (Centrifugal force) and Pendulum and Motion on Curved path

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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 $n^{th}$ power of $R$. If the period of rotation of the particle is $T$,then

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A
$T \propto R^{\frac{n}{2} + 1}$
B
$T \propto R^{\frac{n+1}{2}}$
C
$T \propto R^{\frac{n}{2}}$
D
$T \propto R^{\frac{3}{2}}$ for any $n$

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2 Motion in a Plane

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Dynamics of circular Motion (Centrifugal force) and Pendulum and Motion on Curved path

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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 $n^{th}$ power of $R$. If the period of rotation of the particle is $T$,then

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