Classical Mechanics Puzzler

The spring and mass system is the standard example of simple harmonic motion (SHM). The linear restoring force of a spring is given by F = -kx. The equation of motion for a mass m attached to the spring is m = -kx. The solution is x = Asin(2pf t + j) where A (the amplitude) and j (the phase) are arbitrary constants determined by initial conditions. However, f (the frequency of oscillation) is not arbitrary but determined by k and m. Simple harmonic motion has the unique feature that the period (the reciprocal of f) is independent of the amplitude. This can be understood qualitatively with the following argument. The force and hence the acceleration of the mass increases linearly with the distance from equilibrium. Although a larger amplitude means that the mass travels a greater distance during each cycle, the average   velocity during the cycle also increases linearly with amplitude . The period of the cycle remains constant because the two effects exactly cancel one another. Now consider the non-linear restoring given by

F = -kx³ = m

Although the equation of motion is considerably more difficult, a few qualitative aspects of the motion can be surmised.

1. Will the period be independent of the amplitude in the case of the non-linear force? If not, will the period increase or decrease with increasing amplitude? Use the case of SHM for comparison in your argument.
2. Will you be able to apply the principles of conservation of energy to the non-linear spring/mass system? Why or why not?

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