How can the period of oscillation be determined using energy considerations?

[MFAHH]

Homework Statement

Attached

Homework Equations

The Attempt at a Solution

I've managed to do parts i) and ii) with not much bother. But as for iii) then I haven't a clue how to show that the period of oscillation is given by that. I've always been under the impression it is simply given by 2pi*sqrt(m/k), but am now wondering whether that was just an approximation in itself. It's clear that the limits would be xb and xa as they define the entire range of the object's motion in the potential curve.

I have a feeling energy considerations may play a part but am not sure how to go about it.

 

[BvU]

What did you find in part ii) ? Can you use that in part III ?

 

[MFAHH]

What did you find in part ii) ? Can you use that in part III ?

Thanks for the reply. I've attached what I got for ii.

 

[BvU]

I see a $\dot{..}$KE = ... where it seems you already know initial position and speed. Your integrand is 2/v(t) ? If you also know x(t) then you can change integration variable to t, right ?
I don't see the equation of motion, or how you derived these KE and PE expressions, though.