Paper 2, Section II, C

Dynamics and Relativity | Part IA, 2020

An axially symmetric pulley of mass MM rotates about a fixed, horizontal axis, say the xx-axis. A string of fixed length and negligible mass connects two blocks with masses m1=Mm_{1}=M and m2=2Mm_{2}=2 M. The string is hung over the pulley, with one mass on each side. The tensions in the string due to masses m1m_{1} and m2m_{2} can respectively be labelled T1T_{1} and T2T_{2}. The moment of inertia of the pulley is I=qMa2I=q M a^{2}, where qq is a number and aa is the radius of the

The motion of the pulley is opposed by a frictional torque of magnitude λMω\lambda M \omega, where ω\omega is the angular velocity of the pulley and λ\lambda is a real positive constant. Obtain a first-order differential equation for ω\omega and, from it, find ω(t)\omega(t) given that the system is released from rest.

The surface of the pulley is defined by revolving the function b(x)b(x) about the xx-axis, with

b(x)={a(1+x)1x10 otherwise. b(x)= \begin{cases}a(1+|x|) & -1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise. }\end{cases}

Find a value for the constant qq given that the pulley has uniform mass density ρ\rho.

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