Torsional Pendulum

In SHM, the restoring force is proportional to the displacement and directed opposite to it:

For a torsional pendulum, the analogous quantities are force $F$ replaced by torque $\tau$, spring constant $k$ replaced by torsion constant $\kappa$, and linear displacement $x$ replaced by angular displacement $\theta$.

Substituting these analogous quantities gives the restoring torque:

The negative sign is required because the torque must oppose the angular displacement $\theta$ to restore the disk toward equilibrium. Expressions such as $-\kappa\theta^2$ or $-\kappa\sqrt{\theta}$ do not provide a restoring torque that is linearly proportional to $\theta$.