How can I understand the general equation of motion for torsional harmonic oscillators? I have a science project due in late February. my science project will be on the motion on torsion balances, a torsional harmonic oscillator that uses only the force of gravity to return to its equilibrium state. I am inexperienced with harmonic oscillators and I need help understanding why its general equation of motion is what it is. An explanation would be highly appreciated.
The source for my findings is here: http://en.wikipedia.org/wiki/Torsion_spring#Torsion_balance
 A: Any dynamic system whose acceleration is negatively proportional to displacement will exhibit simple harmonic motion. Mathematically, harmonic motion is seen in any system with $$ \ddot{x} = -\alpha^2 x $$ where $\alpha$ is a constant (and I am using squares here to make sure the coefficient in front of $x$ is positive). You can see that any solution of the form $x(t) = X \sin(\omega t+\varphi)$ will solve the above equation.
$$ \frac{{\rm d}^2}{{\rm d}t^2} X \sin(\omega t+\varphi) = -\alpha^2 X \sin(\omega t+\varphi) $$
has the solution $\omega = \alpha$.  
So if you can find an equation for torsional motion of the form $\ddot{x}=-\alpha^2 x$, the frequency of oscillation is found by $\omega = \alpha$.
A: The key to harmonic oscillation is starting with an equilibrium position (in your case, where the torsion balance is untwisted), and noticing that any displacement from that equilibrium produces a restoring force that is linearly proportional to the displacement.  In your case, the displacement is the angle of twist, and there is a torque produced by the twisted string that is proportional to the amount of twist.  A torque proportional to twist angle works just like a spring, which produces a force proportional to the stretch in the spring (or compression of the spring, the force always points back toward the equilibrium, that's why you get oscillation). If you write Newton's law, F = ma, for a force that works like that, you find the solution is sinusoidal in time (the displacement oscillates around the equilibrium like a sine wave in time), so that's what we call simple harmonic motion.  You probably won't be able to derive theoretically that the twist of a string produces a torque that works like that, but it is found to be true experimentally.  The nice thing about torsion balances is that they produce a very weak restoring force for a given twist angle (so it's a small constant of proportionality there), so very weak external forces yield easily measurable twist angles once any oscillations have damped out.
Having said all this, I think there are two points you may need to realize.  The first is that you can measure any force with the torsion balance-- gravity is just one force that was measured using it historically, but the balance itself has nothing to do with gravity and would work just fine in deep space.  Also, the harmonic oscillation of the torsion balance is also not important, all that matters is you know what force produces what twist, that's how you use the twist to measure the force.  It just happens to be true that the force is proportional to the twist (if the twist is small, this might not continue to hold for large twists), so you do get harmonic oscillation if you remove the external force, but that's not essential for using the balance for its purpose of measuring weak forces.
