- #1

zenterix

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- Homework Statement
- An electric dipole lying in the xy-plane with a uniform electric field applied in the positive x-direction is displaced by a small angle ##\theta## from its equilibrium position. The charges are separated by a distance ##2a##. The moment of inertia of the dipole about the center of mass is ##I_{cm}##.

If the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion. What is the period of oscillation?

- Relevant Equations
- $$\vec{\tau}=-2aE_x\sin{\theta} \hat{k}$$

$$\tau_z=I_{cm}\alpha_z$$

$$\alpha_z=\frac{\tau_z}{I_{cm}}=\frac{-2aE_x\sin{\theta}}{I_{cm}}=-B\sin{\theta}$$

where $$B=\frac{2aE_x}{I_{cm}}$$

This is a differential equation. I think that solving this equation would provide the correct result, but I don't want to go this route.

One route is, if ##\theta## is small, to use the approximation ##\sin{\theta}\approx\theta##.

Then

$$\alpha_z(t)=\theta''(t)=-B\theta(t)$$

I think this is the differential equation representing a simple harmonic motion of an ideal spring with

$$B=\frac{k}{m}$$

Here is a picture of the problem

It is not clear to me how to really prove that the equation for ##\theta(t)## is simple harmonic motion, and what the period of this motion is.

It is not clear to me how to really prove that the equation for ##\theta(t)## is simple harmonic motion, and what the period of this motion is.