Question
I have the answers but I need the explanation in depth.
Consider an electric dipole located in a region with an electric field of magnitude E pointing in the positive y direction. The positive and negative ends of the dipole have charges + q and - q, respectively, and the two charges are a distance D apart. The dipole has moment of inertia I about its center of mass. The dipole is released from angle theta = theta 0, and it is allowed to rotate freely. What is omega max, the magnitude of the dipole's angular velocity when it is pointing along the y axis? Express your answer in terms of quantities given in the problem introduction. Thus omega 0 increases with increasing theta 0, as you would expect. An easier way to see this is to use the trigonometric identity 1 - cos theta = 2 sin2 theta/2 to write omega 0 as 2 sin theta 0/2 If theta 0 is small, the dipole will exhibit simple harmonic motion after it is released. What is the period T of the dipole's oscillations in this case? Express your answer in terms of pi and quantities given in the problem introduction.
Explanation / Answer
Total torque acting around the center of mass
2qEsin(theta) * D/2 = qEDsin(theta)
Now torque = I(alpha) , where alpha is the angular acceleration
qEDsin(theta)/I = (alpha)
From fundamental equation for angular acceleration
dw/dt = (alpha)
=> wdw/d(theta) = qEDsin(theta)/I
=>w^2/2 {from 0 to w} = (qED/I) * [-cos(theta)] {from (theta_o) to 0}
Putting and solving we get
w^2/2 = qED(1 - cos(theta_o))/I
w = sqrt(2qED(1 - cos(theta_o))/I)
Hence proved