Unusual harmonic oscillator.

[Onias]

Homework Statement

A particle of mass m moves (in the region x>0) under a force F = -kx + c/x, where k and c are positive constants. Find the corresponding potential energy function. Determine the position of equilibrium, and the frequency of small oscillations about it.

The Attempt at a Solution

I found the potential energy function by integrating -F(x)dx, and the position of equilibrium by putting F=0. I'm having difficulty even starting the third part, I think I have to do it like one does the damped harmonic oscillator. Any help would be greatly appreciated, thanks in advance.

 

[RoyalCat]

Hehe, nice question!

Here's a big big hint:
Work out the potential energy function while setting your plane of reference at the equilibrium point. Look at the expression you get, and examine what happens when $$x\rightarrow x_{eq}$$

Differentiate the energy function to find the force, and solve for the oscillations.

 

[tomcue]

I would approach this problem by first verifying the given force equation and constants. I would also consider the physical significance of the force and the potential energy function in the context of the problem.

To determine the position of equilibrium, I would set the force equal to zero and solve for x. This would give me the position at which the net force on the particle is zero, indicating a stable equilibrium point.

For the third part, finding the frequency of small oscillations about the equilibrium point, I would use the equation for the period of a simple harmonic oscillator, T = 2π√(m/k), where m is the mass and k is the spring constant. However, in this case, the force is not solely dependent on the spring constant, so I would need to consider the effect of the additional term, c/x, on the frequency of oscillations. This could potentially change the frequency and behavior of the oscillator compared to a typical harmonic oscillator.

To fully understand the behavior of this unusual harmonic oscillator, I would also plot the potential energy function and analyze its shape and critical points. This could provide insight into the motion of the particle and the nature of the oscillations. Additionally, I may consider using numerical methods or simulations to further explore the behavior of the system.