JEE Advanced 2024 · Paper 1 · Physics Q9

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PhysicsInteger+4Question 26 of 51

A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$ . If the angular speed at which the large disc rotates is $\omega/n$ , then the value of $n$ is _____.

Perspective view of two horizontal discs. A large disc of mass M and radius R (radius arrow drawn from its centre) has a vertical dashed axis through its centre. A smaller disc of mass M and radius R/2 (radius arrow drawn from its centre) sits on top of the large disc's rim, overlapping its edge on the left; the small disc has its own vertical dashed axis through its centre, which passes through a point on the circumference of the large disc. The two dashed axes are parallel and separated by R.

Your answer

Answer and worked solution

Official answer: 12

The motor's torque is internal and the large disc's axis is fixed, so angular momentum about that axis stays zero.

Let the large disc turn at $\Omega$ and the small disc spin at $\omega$ (relative to the ground). The small disc's centre moves on a circle of radius $R$ with angular speed $\Omega$ :
$\left(\tfrac12 MR^2 + MR^2\right)\Omega + \tfrac12 M\left(\tfrac R2\right)^2\omega = 0 \implies \tfrac32\Omega = -\tfrac18\omega.$

$|\Omega| = \dfrac{\omega}{12}$ , so $n = 12$ .

Topics:Angular momentum and its conservation Moment of inertia

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JEE Advanced 2024 · Paper 1, Physics question 9