An object in simple harmonic motion is isochronous, meaning that the period of its oscillations is independent of their amplitude. (Contrary to a common assertion, the operation of a pendulum clock is not based on this principle. A pendulum clock operates at fixed, finite amplitude. The gearing of the clock compensates for the anharmonicity of the pendulum.) Consider an oscillator of mass m in one-dimensional motion, with a restoring force F(x) = ‒ cx3, where x is the displacement from equilibrium and c is a constant with appropriate units. The motion of this oscillator is periodic but not isochronous.
a) Write an expression for the period of the undamped oscillations of this oscillator. If your expression involves an integral, it should be a definite integral. You do not need to evaluate the expression.
b) Using the expression of part (a), determine the dependence of the period of oscillation on the amplitude.
c) Generalize the results of parts (a) and (b) to an oscillator of mass m in one-dimensional motion with a restoring force corresponding to the potential energy U(x) = γ|x|α/α, where α is any positive value and γ is a constant.
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