A child of mass m sits on top of a rectangular slab ofmass M=35kg, which in turn

Question

A child of mass m sits on top of a rectangular slab ofmass M=35kg, which in turn rests on the frictionless horizontalfloor at a pizza shop. The slab is attached to a horizontal springwith spring constant k=430 N/m (the other end is attachedto an immovable wall). The coefficient of static friction betweenthe child and the top of the slab is =0.40. The shop owner'sintention is that, when displaced from the equilibrium position andreleased, the slab and child (with no slippage between the two)execute SHM with amplitude A=0.50m. Should there be a weightrestriction for this ride? If so, what is it?

Explanation / Answer

The idea is that you want the max acceleration of theoscillator to be equal to or less than the max acceleration thatfriction can provide, so... .     max acc of friction > maxacc of oscillator . Max acc of oscillator is    A k /(M+m)     where M is 35kg   and   m is the mass of thekid. . Max acc of friction is found by using... .      max force of staticfriction =   m amax .           n =    m amax .          mg   = m amax .      amax = g . So now... .       g >    A k / (M + m)           or .      M + m   >   Ak / g       and .     m >   ( A k / g)   -   M .     m > (0.5 * 430 / 0.40 * 9.80)   - 35   .     m >    19.85 kg . So the mass ofthe child should be at least    19.85 kg. Otherwise, the kid will slip and fall off theride. .      amax = g . So now... .       g >    A k / (M + m)           or .      M + m   >   Ak / g       and .     m >   ( A k / g)   -   M .     m > (0.5 * 430 / 0.40 * 9.80)   - 35   .     m >    19.85 kg . So the mass ofthe child should be at least    19.85 kg. Otherwise, the kid will slip and fall off theride.

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