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.

Explanation

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))`