- #1

alex.dranoel

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## Homework Statement

Determine the period of oscillation, as a function of the energy, when a particle of mass ##m## moves in a field for which the potential energy is

$$ U = U_0 \tan^2\alpha x.$$

## Homework Equations

The relevant equation is given by the general formula for the period ##T## of the oscillations:

$$ T(E) = \sqrt{2m} \int_{x_1(E)}^{x_2(E)} \frac{dx}{\sqrt{E-U(x)}}$$

where ##x_1(E)## and ##x_2(E)## are roots of ##U(x)=E##, giving the limit of the motion.

## The Attempt at a Solution

This problem is found in

*Mechanics*by Landau and Lifshtiz at $11 in the 3rd edition. It is basically a problem of integration. The first thing I did is to find ##x_2(E)##, which is not complicated:

$$E = U_0 tan^2\alpha x \rightarrow x = \frac{1}{\alpha}\arctan \sqrt{\frac{E}{U_0}}$$

Then given the symmetry of the problem, it is clear that

$$ T(E) = 2\sqrt{2m} \int_{0}^{x_2(E)} \frac{dx}{\sqrt{E-U_0\tan^2\alpha x}}$$

Now I am left with an integral that I didn't manage to compute while the result in the book looks very simple.

Thanks for help