Period of Harmonic Motion

[elevenb]

Homework Statement

A harmonic potential is parameterised as:

$$V(x)=\frac{k}{2}(x-x_{0})^2$$


An object moves within this potential with a total energy E > 0.
(i) Where are the two turning points of the motion xA and xB?

(ii) Write down the equation of motion for the object, and use it to find explicit expressions for the kinetic and potential energies as a function of time. Show that the total energy is constant.

(iii) Show that the period of oscillation is given by:

$$t= \sqrt{2m}\int_{X_{a}}^{X_{b}}\frac{dx}{\sqrt{E-V(x)}}$$

and evaluate this integral for the given potential.

Homework Equations

[/B]
OK so I have worked my way through part (i) and (ii) , but I can not see how that integral is the right one, surely this is just integrating 2/v wrt x? I don't understand how that would get you the period?

The Attempt at a Solution

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[DEvens]

Motivate it in your mind with the idea of distance over time. If it were moving at a constant speed, and you knew Xa and Xb, what would be the period?

Now try it in little short segments. Suppose you have a small distance over which you approximate the speed as constant. What will be the time required to travel this small little distance?

Now, what do you do to add up a bunch of times for small little distances?

 

[elevenb]

Thank you so much, been working for a while and could not see where it came from, but now I do! Thank youuu