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GATE 2025 PH – Question 64

Classical Mechanics · rigid body dynamics: inertia tensor, orthogonal transformations, Euler angles, torque free motion of a symmetric top · 2 marks · Numerical answer

A wheel of mass $4M$ and radius $R$ has mass $3M$ uniformly on its thin rim and point mass $M$ at its center. Massless spokes join them. Its center is connected to a horizontal massless rod of length $2R$, fixed at O. It rolls without slipping as shown, with angular speed $\Omega$ about the vertical axis. If $\mathbf L$ is total angular momentum about O and $|d\mathbf L/dt|=NMR^2\Omega^2$, the value of $N$ (in integer) is _____

Wheel with 3M rim and M central mass on a 2R rod rotating about a vertical axis.

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Correct answer: 6

Explanation

The center moves with speed $2R\Omega$, so the wheel spins at $2\Omega$. Its radial spin angular momentum is $(3MR^2)(2\Omega)$. This vector precesses at $\Omega$, giving $|d\mathbf L/dt|=6MR^2\Omega^2$. The vertical angular-momentum components are constant.