A uniform rod is placed on two spinning wheels as shown in figure. The axes of the wheels are separated by a distance `l=20 cm`, the coefficient of friction between that in this case the rod performs harmonic oscillations. Find the period of these oscillations.

Explanation

In the equilibrium position the C.M. of the rod lies nid way betweent eh two rotating wheels. Let us displace the rod horizontally by some small distance and then release it. Let us depeict the forces acting on the rod when its C.M. is at distance `x` from its equilibrium position (figure). Since there is no net vertical force acting on the rod, Newton's second law gives `:`
`N_(1)+N_(2)=mg ...(1) `
For the translational motion of the rod from the Eqn . `: F_(x)=m w _(cx)`
`kN_(1)-kN_(2)=m ddot(x).....(2)`
As the rod experiences no net torque about an axis perpendiuclar to the plane of the figure through the C.M. of the rod.
`N_(1)((l+x)/(2))=N_(2)((l-x)/(2)) ...(3)`
Solving Eqns. `(1), (2)` and `(3)` simultaneoulsly we get
`ddot(x)=-k (2g)/(l) x`
Hence the sought time period
`T=2pi sqrt((l)/(2kg))=pi sqrt((2l)/(kg))=1.5 s`