Determine the period of the oscillations as a function of the initial energy E of a particle of mass m moving in a potential energy U=A|x|^n

Question

Determine the period of the oscillations as a function of the initial energy E of a particle of mass m moving  in a potential  energy U=A|x|^n

anonymous · Answer

Can you give a more specific description of the system? Is this a spring? a pendulum? something else? I assume you mean to say the potential energy of the system is given by $$U= A \left| x^n \right|$$but what is x? a length? In general oscillatory motion problems involve writing some equation of motion, i.e. for an SHO
$$mx''+ kx = 0$$ which has the period$$T= 2\pi (k/m)^{-1/2}$$ But the energy equations would seem to imply some conservation equation. Perhaps you could just retype the question from your text (or whatever other source it's from) to make it more intelligible.

anonymous · Answer

You can recover the force (=ma) by taking the negative of the partial derivative of U wrt x, then use an approach similar to the one described above... set up a differential equation and define the square of the angular frequency from coefficients.