A uniform plate of mass M stays horizontally symmetrically on two wheels rotating in opposite directions. The separation between the wheels is L. the friction coefficient between each and the plate is mu. Find the time period of oscillation of the plate if it is slightly displaced along its length and released.

A uniform plate of mass M stays horizontally symmetrically on two whee

1.	A uniform plate of mass M stays horizontally symmetrically on two wheels rotating in opposite directions. The separation between the wheels is L. the friction coefficient between each and the plate is mu. Find the time period of oscillation of the plate if it is slightly displaced along its length and released.
Answer» Solution :LET the plate be displaced towards right by X and RELEASED. `N_1+N_2=mg` TAKING torque about A `N_2L=mg(L/2+x)`, `N_2=mg(1/2+x/L)` `N_1=mg(1/2-x/L)` Restoring force `F=-(f_2-f_1)=-mu(N_2-N_1)=-mu(2mgx)/(L)` `a=F/m=-(2mug)/(L)x=-omega^2x` `T=2pi((L)/(2mug))`