Two wheels which are rotated by some external source with constant angular velocity in opposite directions as shown in figure. A uniform plank of mass `M` is placed on it symmetrically. The friction co-efficient between each wheel and the plank is `mu`. find the frequency of oscillations, when plank is slightly displaced along its length and released.

Answer

`2pisqrt((l)/(mug))`

Explanation

If plank is displaced by `x` toward right then

Let `N_(1), N_(2)` and `f_(1), f_(2)` are Normal and friction force at Point `A` and `B` by force balance
`N_(1) + N_(2) = Mg_(1)`
amd Torque balance
`Mg (l + X) = N_(2) xx 2l__(2)`
by equation `(1)` and `(2)`
`N_(1) = (Mg)/(2) - (Mgx)/(2l), N_(2) = (Mg)/(2) + (Mgx)/(2l)`
So `f_(1) = mu((Mg)/(2) - (Mgx)/(2l))`
and `f_(2) = mu((Mg)/(2) + (Mgx)/(2l))`

`F.B.D` of `M`
`f_(2) - f_(1) = Ma`
`mu. (2Mgx)/(2l) = Ma = -(d^(2)x)/(dt^(2)).M`
`rArr (d^(2)x)/(dt^(2)) = (-mugx)/(l)`
Time period `= 2pisqrt((l)/(mug))`