Question

A cylinder of mass M and radius R is resting on a horizontal platform (which is parallel to the 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 the x-direction given by `x =A cos (omega t).` There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is ..................

Answer 1

For platform
`x=A cos omega t`
`v=(dx)/(dt)=-A omega sin omega t`
` a=(dv)/(dt)= -omega^(2) A cos omega t`
Maximum acceleration of platform
`a_(max)=omega^(2)A`.
Friction force acting on cylinder provides torque

`f=Ma_(c.m.)implies a_(c.m)=(f)/(M)`
`fxxR=I_(0)alpha=(1)/(2)MR^(2)alpha`
`alpha=(2f)/(MR)`
`a_(A)=a_(c.m.)+Ralpha=(f)/(M)+(2f)/(M)=(3f)/(M)`
As there is no slipping between cylinder and platform, relative accelertion at point of contact should be zero, i.e.
`a_(A)=a_(max)`
`(3f)/(M)=omega^(2)Aimplies f=(Momega^(2)A)/(3)`
Maximum torque by friction
`tau_(max)=fR=(Momega^(2)AR)/(3)`