A cylinder of mass M and radius R is resting on a horizontal paltform (which is parallel to the x-y plane) with its exis 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 acitng on the cylinder during its motion is ..................

Answer

A, B, C

Explanation

Considering the motion of the platform
`x=A cos omegat`
`rArr (dx)/(dt) =- A omega sin omegat rArr(d^2x)/(dt^2) =- A omega^2 con omegat`
The magnitude of the maximum acceleration of the platform is
`:. |Max acceleration| = A omega^2`
When platform moves a torque acts on the cylinder and the
cylinder rotates about its exis.
Acceleration of cylinder, `a_1 = (f)/(m)`
Torque `pi = fR :. l alpha = fR`
`alpha = (fR)/(I) = (fr)/(MR^2 /2)`
or, `alpha = (2f)/(MR) or Ralpha = (2f)/(M)`
`:. Equivalent linear acceleration (Ralpha = a_2)`
`a_2 = (2f)/(M)`
:. Total linear acceleration,
`a_(max) = a_1 +a_2 = (f)/(M)+(2f)/(M) = (3f)/(M)`
or,`Aomega^2 = (3f)/(M) or , f = (MAomega^2)/(3)`
Thus, maximum torque,
`tau_(max) = fxxR = (MAomega^2R)/(3) = (1)/(3) MARomega^2`