Time period of a harmonic oscillator

[VVS2000]

Homework Statement

Given is the potential energy of the harmonic oscillator: U=a|x|^n, amplititude is A
Find the time period of this harmonic oscillator

Relevant Equations

E=(1/2)m(dx/dt)^2 + a|x|^n

 

[anuttarasammyak]

Your result is written as
$$4A\sqrt{\frac{m}{2E}}\int_0^1 \frac{dx}{\sqrt{1-x^n}}$$
where amplitude A is
$$A=(\frac{E}{a})^{\frac{1}{n}}$$

 

[VVS2000]

Your result is written as
$$4A\sqrt{\frac{m}{2E}}\int_0^1 \frac{dx}{\sqrt{1-x^n}}$$
where amplitude A is
$$A=(\frac{E}{a})^{\frac{1}{n}}$$

No, I have'nt written 4A. It's 4. A is inside the root in the denominator.
But how do you solve that integral?

 

[VVS2000]

Your result is written as
$$4A\sqrt{\frac{m}{2E}}\int_0^1 \frac{dx}{\sqrt{1-x^n}}$$
where amplitude A is
$$A=(\frac{E}{a})^{\frac{1}{n}}$$

No, I have'nt written 4A. It's 4. A is inside the root in the denominator.
But how do you solve that integral?

 

[anuttarasammyak]

I made a replacement of x/A $\rightarrow$ x.

The definite integral is a function of $\Gamma$ functions of $\frac{1}{n}$.

 

[LCSphysicist]

Harmonic oscillator in classical physics are not systems subject to an force/ente proportional to its "displacement"? So n shouldn't be two?
Or it is not a harmonic oscillator?

 

[Steve4Physics]

As noted by LCSphysicist (#6), this is NOT a harmonic oscillator (except when n=2). The word 'harmonic' is specifically used twice. Is there a possibility this is a 'trick' question?

A harmonic oscillator is one which has a restoring given by:
$\vec F = -k\vec x$
If I were answering I would demonstrate that only n=2 gives a harmonic oscillator. (Hint, what is the relationship between $\vec F$ and U?)

Then express in the usual SHM equation format:
$\frac {d^2 x }{dt^2} + \omega^2 x = 0$
You then get the period from $\omega$.

Just a thought.