JEE Advanced 2026 · Paper 1 · Physics Q1

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PhysicsSingle correct+3 / -1Question 17 of 48

Consider a large disk of radius $R$ and two smaller disks, each of radius $r=R/50$ , lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation $\Delta\theta$ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities $\omega$ and $2\omega$ while the large disk is held stationary. The time $\tau$ at which the smaller disks are again in contact is:
[Use $\sin(\Delta\theta)=\Delta\theta$ and ignore gravity.]

A large circle of radius R (centre marked, radius drawn horizontally). Two small touching circles of radius r sit on its rim near the top right; dashed lines from the centre to their centres make the small angle Δθ. Curved arrows show the left small disk rotating with ω and the right one with 2ω, rolling in opposite directions.

\tau=51\times\left(2\pi-\dfrac4{51}\right)/\omega

\tau=51\times\left(2\pi-\dfrac2{51}\right)/3\omega

\tau=51\times\left(2\pi-\dfrac4{51}\right)/3\omega

\tau=51\times\left(2\pi-\dfrac2{51}\right)/\omega

Answer and worked solution

Official answer: C

Orbital speed of each small disk. A disk rolling without slipping on the fixed rim has its centre moving with speed $\omega_{\text{spin}}r$ on a circle of radius $R+r=51r$ . Its centre therefore turns about $O$ at
$\Omega=\frac{\omega_{\text{spin}}r}{R+r}=\frac{\omega_{\text{spin}}}{51}.$
So the two centres turn at $\frac{\omega}{51}$ and $\frac{2\omega}{51}$ , in opposite directions: relative rate $\frac{3\omega}{51}$ .

Initial separation. Touching disks have centres $2r$ apart: $(R+r)\Delta\theta=2r$ , so $\Delta\theta=\frac2{51}$ .

Meeting again. They move apart and meet on the other side once their angular gap, $2\pi-\Delta\theta$ the long way round, shrinks back to $\Delta\theta$ . The relative angle covered is $2\pi-2\Delta\theta=2\pi-\frac4{51}$ , so
$\tau=\frac{51}{3\omega}\left(2\pi-\frac4{51}\right).$

Topics:Rolling motion Uniform circular motion

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