First slide

Rolling motion

Question

A cylinder of mass M, radius R is resting on a horizontal platform (which is parallel to x-y plane) with its axis fixed along the y-axis and free to rotate about its axis. The Platform is given a motion in x-direction given by $x = acos ωt$ There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

Difficult

A

$\frac{1}{2} {MRaω}^{2}$
B

${MRaω}^{2}$
C

$2 {MRaω}^{2}$
D

${MRaω}^{2} \cos ωt$

Solution

The motion of platform is described by equation
$\begin{aligned} x = acos ωt \\ ∴ velocity = \frac{dx}{dt} = - aωsin ωt \\ and acceleration = \frac{d^{2} x}{{dt}^{2}} = - {aω}^{2} \cos ωt \end{aligned}$
SO, $a_{max} = {aω}^{2}$
Now, $τ_{max} = {Iα}_{max} = I (\frac{a_{max}}{R})$
$= (\frac{1}{2} {MR}^{2}) (\frac{{aω}^{2}}{R}) = \frac{1}{2} {MRaω}^{2}$

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