Period of Oscillation Homework Statement

[Robben]

Homework Statement

Assume that the potential is symmetric with respect to zero and the system has amplitude $a$, show that the period is given by : $T=\sqrt{8m}\int^a_0\frac{dx}{\sqrt{V(a)-V(x)}}.$

Homework Equations

$E = \frac12 m(\frac{dx}{dt})^2+V(x)$

The Attempt at a Solution

For a particle, I know that at $t=0$ if we release it from rest at position $x=a$ we then have $\frac{dx}{dt}=0$ at $t=0$ and thus $E=V(a)$. So when the particle reaches the origin for the first time it has gone through one quarter of a period of the oscillator. Thus, I have to integrate with respect to t from $0$ to $\frac{T}{4}$ and rearrange the equation $E$ for $\frac{dx}{dt}$. But from here I am not sure how to set it up properly to get $T=\sqrt{8m}\int^a_0\frac{dx}{\sqrt{V(a)-V(x)}}.$

 

[Goddar]

Hi. Indeed you already have everything to set up your integral:
If you substitute your value of E in Hamilton's equation, can you separate the variables (functions of x on one side, functions of t on the other)?

 

[Robben]

Hi. Indeed you already have everything to set up your integral:
If you substitute your value of E in Hamilton's equation, can you separate the variables (functions of x on one side, functions of t on the other)?

I have $V(a) -V(x) = \frac12m(\frac{dx}{dt}2) \implies 2\sqrt{V(a)-V(x)} = m\frac{dx}{dt} \implies \implies \sqrt{\frac{2}{m}}dt = \frac{dx}{\sqrt{V(a)-V(x)}}$ but I am confused on how they to get $T$.

 

[Goddar]

What about integrating both sides? You know the limits of the integrals (for t and x) from your first post, so what do you get?

 

[Robben]

When I am integrate I am not sure how they got the $\sqrt{8m}$.

 

[Goddar]

What do you get when you integrate dt from 0 to T/4?

 

[Robben]

What do you get when you integrate dt from 0 to T/4?

We will get $T/4$ if we integrate dt from 0 to T/4.

 

[Goddar]

Ok, then what is (T/4)⋅(2/m)^1/2?
This is just algebra...

 

[Robben]

Ok, then what is (T/4)⋅(2/m)^1/2?
This is just algebra...

Oh wow.. how dumb am I. Thank you very much!

 

[juliocezario30]

Can you show step by step procedure? I am confused

 

[haruspex]

Can you show step by step procedure? I am confused

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