Question :

A common notion about shm is that kinetic energy is converted to potential energy, and vice versa as the oscillation continues; the sum of the kinetic and potential energy is a constant. In this example, there seems no involvement of any potential energy; where does the kinetic energy come from and go to ?

A uniform plank of mass m is supported horizontally and symmetrically by two identical rollers of fixed axes and of centers separated by a distance L. The rollers rotate in the opposite directions, always pushing the plank back to the central position, as shown in the above diagrams.

Now, the plank is displaced to the right by x. The normal force at the left roller and that at the right, denoted by N_L and N_R respectively, become unequal. They are (found by the equilibrium of moments)

and , where W = mg is the weight of the plank.

Assume the rollers rotate at a fast speed, so the plank always slides on each of them. The frictional forces by the rollers are kinetic, of magnitudes

and ,

where μ_k is the coefficient of kinetic friction.

Relative to the displacement x, f_L is positive and f_R is negative. The net force is

.

Since a = F/m, we get

.

The number μ_kW/L is positive, so a ∝ –x, satisfying the shm equation a = -ω²x. Finally, we conclude the plank's motion is simple harmonic with angular frequency

.