Classical Mechanics - Potential Energy Function

[teme92]

Homework Statement

The potential energy function of a particle of mass m is V(x) = cx/(x²+a²), where c and a are positive constants.

Qualitatively sketch V as a function of x. Find two equilibrium points: identify which is a position of stable equilibrium, and find the period of small oscillations about it.

Homework Equations

PE=0.5mv²=-0.5kx²

The Attempt at a Solution

I think I'm supposed to differentiate and let it equal to zero which gave me:

(c(x²+a²) - 2cx²) / (x²+a²)² = 0
cx² + ca² - 2cx² = 0
x = a

Putting that into the function gave me: V(x) = cx²/2x² = c/2

I don't know if any of this is necessary but it doesn't answer the question. Also when it says sketch do I just sub value of x in? Any help would be greatly appreciated.

 

[Orodruin]

It answers part of the question, i.e., one of the equilibrium points. In order to find the other you must find the second solution to your equation (it exists as it is a second order equation in x).

To know which is stable you must look at the second derivative to deduce which is a min and which is a max.

 

[teme92]

Hey Orodruin thanks for the reply. How do I find the second solution?

 

[Orodruin]

You start from the equation you obtained and solve it for the general case. You have x^2 = a^2. This has two possible solutions.

 

[teme92]

Ah yes sorry, so the other value for x is -a. But this gives the same solution as it gets squared?

 

[teme92]

I just realized I should be subbing a in for x so I get V(x) = ±c/2a. So from this how do I tell where the stable equilbrium is? Also how do I find the period?