Varying Potential Energy and Amplitude in Unusual Harmonic Motion

[ehild]

The only problem I see in ehild's solution to the original problem is that the right-hand side must have the square root of 2k/m.

You are rigth, I forgot the square root from the right-hand side. The correct formula is

\int_0^A{\frac{dx}{\sqrt{A^3-x^3}}}=\int_0^{T/4}{\sqrt{\frac{2k}{m}}dt}.

The OP speaks about a particle performing periodic motion with amplitude A, when its potential energy function is k|x|^3. k is a constant. It asks how the time period varies with the amplitude. There is no words about slopes and gravity, but a particle well might have mass.
That potential energy can come from an electric field or the gravity of a specific mass distribution ... anything. What is the sense to consider the problem as rolling down along a slope? The potential energy is a general concept, does not necessarily mean gravitational energy near the Earth surface.

ehild

 

[darkxponent]

no it would not acceleration is x" rate of rate of change of x coordinate only you are calculating acceleration along the slope

I never wrote a=x''. I was trying to say that the net acceleration comes out different from that of OPs problem. What i was trying to prove was that barryj's and OPs problem are not analogy.

That was ehild's post that recommended the energy approach.

I tried what you did, wound up with mx'' + 3k/m x^2 = 0 and could not solve it either. I tried a "guessed" solution of x = Asin(wt) which was disastrous. Evidently, the oscillations are not pure sinusoids, neither should we expect them to be in view of the fact that the ODE is nonlinear.

This diffrential equation can be solved putting a=v*(dv/dx), and it reduces to the same v-x equation of ehild's. So one can use either method. The result is the same.