- #1
archaic
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- 214
- Homework Statement
- We have two charged balls of masses ##m_1=m_2##, and charges ##+q_1=-q_2##, fixed to the ends of a very light rod of length ##\ell##.
The center of the rod is mounted on a friction-free pivot, and the whole system is then immersed in a uniform electric field of magnitude ##E##.
a) Determine the period of oscillation if the system is disturbed from its initial orientation by a small angle ##\theta##.
b) What if ##m_1<m_2##?
- Relevant Equations
- $$\vec F=q\vec E$$
a) This looks somewhat like a pendulum problem (length ##\ell/2##).
I reasoned there will be a clockwise rotation, and that the acceleration is due to the force of magnitude$$F=-\left[(F_2\sin\theta-m_2g\cos\theta)+(F_1\sin\theta+m_1g\cos\theta)\right]=-(|q_2|+q_1)E\sin\theta=-2qE\sin\theta=ma$$
The way I thought of it is that the acceleration put to bring 2 up is added to the acceleration put to bring 1 down to make the rotation acceleration.
For small angles, I get this differential equation$$\ddot\theta=-\frac{4qE}{m\ell}\theta$$
and if I put ##\theta=A\cos(\omega t+\phi)##, then I would get ##\omega^2=\frac{4qE}{m\ell}##, and, from it, ##T=\pi\sqrt\frac{m\ell}{qE}##.
b) If I use the same reasoning, then I get an exponential solution to my problem, since the force of gravity won't disappear, which makes me think that my solution for a) is wrong.
Any help would be great!