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

• Are you looking for a math based answer, based on matching up the stuff you know about the mass on a spring based system, to compare with your string based system? – user140606 Dec 30 '16 at 22:41

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 $\omega = \alpha$.