Electric dipole in an electric field

[vladimir69]

Homework Statement

An electric dipole (something that has charge +q on one end and charge -q on the other end separated by a distance 2a) is in a uniform horizontal electric field of magnitude E. Initially the electric dipole is aligned horizontally until it is displaced slightly by an angle theta from the horizontal. Show that the electric dipole undergoes simple harmonic motion with frequency given by
$$f=\frac{1}{2\pi}\sqrt{\frac{(m_{1}+m_{2})qE}{2m_{1}m_{2}a}}$$

Homework Equations

$$I\alpha=\tau_{net}$$
$$\omega=2\pi f$$
$$F=qE$$
$$\theta(t)=A\cos(\omega t)$$

The Attempt at a Solution

Here is what I got
$$I=(m_{1}+m_{2})a^2$$
$$(m_{1}+m_{2})a^2\frac{d^2\theta}{dt^2}\approx 2aqE\theta$$
and the frequency I get pops out as
$$f=\frac{1}{2\pi}\sqrt{\frac{2qE}{(m_{1}+m_{2})a}}$$
Can't see where I have gone wrong

 

[ehild]

See the units: the given formula can not be correct. Yours is all right.

ehild

 

[vladimir69]

I thought the units were ok in both the equations

 

[ehild]

Yes, you are right, I misread the formula somehow...

The question is if the dipole rotates around a fixed axis through its centre, so both masses are at a distance "a" from the axis of rotation or it is free and then it rotates around its CM.
In case of the first situation, your formula is right. The formula given by your book is valid for the free dipole. In this case you need the moment of inertia with respect to the CM.

ehild

 

[vladimir69]

Ok I see now, thanks for your help.