Question

A uniform solid cylinder is given an angular speed  and placed on a rough plate of negligible thickness. The horizontal surface below the plate is smooth. Then the angular speed of the cylinder when it starts pure rolling on the plate will be: [ Assume sufficient length of plate ]

Answer 1

Since friction is acting at the lower most point P, angular momentum conservation can be applied about the lower most point.

Angular momentum initial

Li=MVcmR+IwLi=MVcmR+Iw

=Iw=Iw[since vcm=0vcm=0]

=MR2w2=MR2w2

Angular momentum final

Lf=MVcmR+Iw′Lf=MVcmR+Iw′

w′=VcmRw′=VcmR

∴Lf=MVcmR+MR22VcmR∴Lf=MVcmR+MR22VcmR

=32=32MVcmRMVcmR

Li=LfLi=Lf

3232MVcmR=MR2w2MVcmR=MR2w2

Vcm=Rw3