# Period of an Oscillating Particle

## Homework Statement

A particle of mass m is oscillating with amplitude A in 1D (without damping). Using conservation of energy, I'm asked to determine the period T with the correct proportionality factor and period of integration.

## Homework Equations

Potential energy: $U(x) = a x^2 + b x^4$ where a > 0 and b ≥ 0
Period: $T(A) \propto$∫$\frac{dx}{\sqrt{E(A) + U(X)}}$ (from conservation of energy)

## The Attempt at a Solution

I scribbled down a few attempts but none of them went anywhere. I don't have the slightest idea where to start so I'm hoping someone can point me in the right direction.

I think that's all the information provided, but feel free to ask if you have any questions. It's my first time posting here, my apologies if I haven't follow proper rules or etiquette.

Thanks

## Answers and Replies

Simon Bridge
Homework Helper
Welcome to PF;
I take it the oscillations are harmonic and though undamped, there may be a driving force?
If it were SHM, then the potential energy terms would be quadratic right?

It looks like you are expected to exploit the relationship you already know about between kinetic and potential energy. i.e. you need to examine what else you know about energy in oscillations - which should net you an appropriate proportionality and a region of integration.

Compare with how it's done for SHM.

That appears to be the case. I found some relevant relations while looking through the notes again.

$F = -\frac{dU}{dx}$

So that would give a driving force of

$F = -2ax -4b x^3 = m\ddot{x}$

The Hamiltonian is also given as

$H = \frac{p^2}{2m} + U(x)$

Would that be interchangeable with the total energy E(A)? In that case I would be able to integrate using just the kinetic energy.

Simon Bridge