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 NL and NR respectively, become unequal. They are (found by the equilibrium of moments)
, 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
where μk is the coefficient of kinetic friction.
Relative to the displacement x, fL is positive and fR is negative.
The net force is
Since a = F/m, we get
The number μkW/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