Frequency of oscillating electric dipole in uniform field without using diff eq

DocZaius

1. The problem statement, all variables and given/known data

Electric dipole makes small angle with uniform electric field. Find the frequency of the oscillation using dipole moment p, moment of inertia I, and field magnitude E.

2. Relevant equations
Torque=I*(angular acceleration)=I*(theta)''
Torque=p*E*sin(theta)


3. The attempt at a solution

What I did was set PE*sin(theta)=I*(theta)'' But then I need to solve a differential equation which is absolutley not supposed to be required for this class.

Is there a way to find this frequency without solving a differential equation?

vela

You forgot a minus sign. When the angle is small, you can use the approximation [itex]\sin \theta \cong \theta[/itex], which gives you
[tex]I\ddot{\theta} = -pE\theta[/tex]
Now compare this to the equation of motion for a simple harmonic oscillator.
[tex]m\ddot{x} = -kx[/tex]
where the angular frequency is given by [itex]\omega =\sqrt{k/m}[/itex].

DocZaius

Aren't you technically solving a differential equation by telling me the angular frequency of a simple harmonic oscillator?

I was thinking of an approach where I would integrate torque over time (rather than angle) to get a change in angular momentum, then use that to determine at what point in its cycle the oscillator is, since when that change is zero (after a non-zero time), whatever time passed to reach zero angular momentum would be half the period. Is this feasible?

Thanks for your reply, btw.

vela

Perhaps. You could try it and see how it works out, but I think you'll run into a problem trying to integrate the righthand side of the equation since you don't know how θ varies as a function of time.

I think what's expected is that you're supposed to recognize the form of the resulting differential equation and from that deduce what the frequency and solutions are. You don't actually have to solve the equation, per se. You just essentially follow a recipe and write down what the answers should be.