Period of an Oscillating Particle

[Sacrilicious]

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

 

[Simon Bridge]

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 http://www.cscamm.umd.edu/people/faculty/tiglio/reviewch14-15.pdf.

 

[Sacrilicious]

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]

Is the Hamiltonian interchangeable with the total energy?
Do you know the relationship between kinetic and potential energy for your oscillator?

Wouldn't F=-dU/dx be the restoring force rather than the "driving force"?
The problem statement only says that the oscillation is un-damped, does not say there is no driving force - I don't know, there may be: is there?

 

[Nickie]

for your help!
I would start by breaking down the problem into smaller parts and identifying the key equations and concepts involved. First, I would start by understanding the concept of oscillating motion and how it relates to energy conservation. I would also review the given potential energy equation and make sure I understand its significance in the problem.

Next, I would use the given equation for period and understand how it relates to the potential energy and the amplitude of the oscillation. This equation involves integration, so I would make sure I understand how to apply this concept to solve the problem.

Then, I would start by setting up the conservation of energy equation, where the total energy of the system is equal to the sum of its kinetic and potential energy. This would allow me to solve for the velocity of the particle as a function of its position.

From there, I would use the given equation for period and plug in the expressions for kinetic and potential energy. I would then use integration to solve for the period, keeping in mind the correct proportionality factor.

If I encounter any difficulties or get stuck, I would consult relevant resources such as textbooks or online tutorials, or seek help from a colleague or instructor. It is also important to double check my calculations and make sure they make physical sense.

Overall, the key to solving this problem is a thorough understanding of the concept of oscillating motion and energy conservation, and the ability to apply these concepts through mathematical equations.