Torsional oscillations

In the Cavendish experiment to measure gravity, we had a quartz fiber dangling from a ceiling. Attached to it was some rod with masses on it. The fiber exerts some torque when the rod is displaced from its equilibrium position.

If small angles, you can say the the torque exerted is proportional to the displacement from equilibrium

$$\tau ~=~ -\kappa \theta$$ (1.55)

This is just like $F ~=~ -kx$. $\kappa$ is a constant having to do with the properties of the materials.

So applying $I\alpha ~=~ \tau$

$$I {d^2\theta\over dt^2} ~=~ -\kappa \theta$$ (1.56)

or

$${d^2\theta\over dt^2} ~=~ -{\kappa\over I} \theta$$ (1.57)

Again this is just like 1.33, so we have except of $k/m$ we have here $\kappa / I$.

$$\omega^2 ~=~ {\kappa\over I}$$ (1.58)

So the quartz fiber will oscillate back and forth at this angular frequency.