Mechanics potential energy problem

[osamapw1nladen]

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.

 

[Tide]

What have you done so far to solve the problem?

 

[osamapw1nladen]

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

 

[phucnv87]

Begin by finding all forces that apply to the system

 

[osamapw1nladen]

the only thing i got is that there's a force acting in the positive x-direction from the electric field, and a force in the negative y-direction from gravity. i don't know how to find where the acceleration would be

 

[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 ...

 

[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.