Saketh
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A friend of mine has given me a physics problem to solve. Here it is:
mg\tan{\theta} = \frac{q^2}{4\pi \epsilon_0 x^2}
Since x << \ell, I concluded that \sin{\theta} \approx \tan{\theta}. I rewrote the above force equation with sin instead of tan.
After this, I started doing things randomly. At first, I tried using angular speed with \ell as the radius of rotation, but that seemed unnecessarily complicated.
So I went back and differentiated both sides of the first equation. I don't know if this is correct, but I let \cos{\theta} be 1 since \theta \approx 0. So, using the quotient rule, I had:
mg = \frac{1}{2\pi \epsilon_0}\frac{q^2(\frac{dx}{dt}) - q\frac{dq}{dt}}{x^3}
At this point, since \frac{dx}{dt} = \frac{a}{\sqrt{x}}, I started substituting things in. However, I ended up with charge as a function of time, distance as a function of time, and the time derivative of the charge function all in one equation - unsolvable.
The other way that I tried it was to start from the premise that \ell \frac{d\theta}{dt} = \frac{a}{\sqrt{x}} and go from there. But I don't know if this is the correct way either.
Basically, I have no idea what I'm doing. My friend said this is an easy problem, but I am stumped. Am I approaching this incorrectly?
Thanks for the assistance.
Two small equally charged spheres, each of mass m, are suspended from the same point by silk threads of length \ell. The distance between the spheres x << \ell. Find the rate \frac{dq}{dt} with which the charge leaks off each sphere if their approach velocity varies as v = \frac{a}{\sqrt{x}}, where a is a constant.
I started by writing down the forces:mg\tan{\theta} = \frac{q^2}{4\pi \epsilon_0 x^2}
Since x << \ell, I concluded that \sin{\theta} \approx \tan{\theta}. I rewrote the above force equation with sin instead of tan.
After this, I started doing things randomly. At first, I tried using angular speed with \ell as the radius of rotation, but that seemed unnecessarily complicated.
So I went back and differentiated both sides of the first equation. I don't know if this is correct, but I let \cos{\theta} be 1 since \theta \approx 0. So, using the quotient rule, I had:
mg = \frac{1}{2\pi \epsilon_0}\frac{q^2(\frac{dx}{dt}) - q\frac{dq}{dt}}{x^3}
At this point, since \frac{dx}{dt} = \frac{a}{\sqrt{x}}, I started substituting things in. However, I ended up with charge as a function of time, distance as a function of time, and the time derivative of the charge function all in one equation - unsolvable.
The other way that I tried it was to start from the premise that \ell \frac{d\theta}{dt} = \frac{a}{\sqrt{x}} and go from there. But I don't know if this is the correct way either.
Basically, I have no idea what I'm doing. My friend said this is an easy problem, but I am stumped. Am I approaching this incorrectly?
Thanks for the assistance.
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