JEE Advanced 2025 · Paper 1 · Physics Q1

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PhysicsSingle correct+3 / -1Question 17 of 48

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$ , inside a ring of radius $R > r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $T = \dfrac{2\pi}{\omega}$ . The correct expression for $\omega$ is ( $g$ is the acceleration due to gravity):

A large ring of radius R (radius drawn as a dashed arrow from its centre) in a vertical plane. A small disk of radius r rests at the bottom of the ring, touching its inner surface; a short double arrow by the disk shows it can move along the ring. A coiled spring of constant k runs along the inside of the ring from the disk's centre up the lower-left arc to a fixed mount on the ring.

\sqrt{\dfrac23\left(\dfrac{g}{R-r} + \dfrac{k}{m}\right)}

\sqrt{\dfrac{2g}{3(R-r)} + \dfrac{k}{m}}

\sqrt{\dfrac16\left(\dfrac{g}{R-r} + \dfrac{k}{m}\right)}

\sqrt{\dfrac14\left(\dfrac{g}{R-r} + \dfrac{k}{m}\right)}

Answer and worked solution

Official answer: A

Let $s$ be the arc displacement of the disk's centre, which moves on a circle of radius $R-r$ . The spring stretches by $s$ .

Kinetic energy. Rolling without slipping, with $I=\frac12mr^2$ :
$KE=\tfrac12mv^2+\tfrac12\cdot\tfrac12mr^2\left(\tfrac vr\right)^2=\tfrac34m\dot s^2.$

Potential energy. For small $s$ the height rise is $\frac{s^2}{2(R-r)}$ , so
$PE=\frac{mg}{2(R-r)}s^2+\frac12ks^2.$

Frequency.
$\omega^2=\frac{\frac{mg}{R-r}+k}{\frac32m}=\frac23\left(\frac g{R-r}+\frac km\right).$

Topics:Simple harmonic motion Rolling motion Spring-mass systems

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