# Mechanics potential energy problem

1. Feb 20, 2006

A small ball of mass M carries a positive charge Q. The ball is glued to the end of a massless stick of length L. The other end of the stick is attached to a frictionless pivot that allows the pendulum to swing in the xy plane. Gravity is in the minus y direction. A constant electric field Eº points in the positive x direction.
1) Find a potential energy as a function of angle θ and find an expression for the equilibrium angle(s). How many are there? Explain
2) Find the angular frequency, ωo, of small oscillations about the stable equilibrium angle, θs.
3) Assume that the pendulum is ocsillating with aplitude θo-θs. Find the work done by the electric field as the pendulum moves from θs-θo to θs+θo.

2. Feb 20, 2006

### Tide

What have you done so far to solve the problem?

3. Feb 20, 2006

absolutely nothing, i have no clue how to even start the problem

4. Feb 20, 2006

### phucnv87

Begin by finding all forces that apply to the system

5. Feb 21, 2006

the only thing i got is that theres a force acting in the positive x-direction from the electric field, and a force in the negative y-direction from gravity. i dont know how to find where the acceleration would be

6. Feb 21, 2006

### phucnv87

I will give you solution to the part 1)
The potential energy of the system

$$W=mgl(1-\cos\theta)-QEl\sin\theta$$

where the origine of poential energy is at the lowest point of the mass
The system gets the equilibrium state when

$$\frac{dW}{d\theta}=0\Longrightarrow tan\theta=\frac{mg}{QE}$$

Now, continue ...

Last edited: Feb 21, 2006
7. Feb 21, 2006

### phucnv87

For part 2), because the amplitude of oscillation is small, so we can use

$$1-cos\theta=2\sin^2\frac{\theta}{2}=\frac{1}{2}\theta^2$$

then using the law of conservation of energy.

Last edited: Feb 21, 2006