A torsional pendulum is a rigid body that oscillates by twisting an elastic support, such as a wire or torsion spring, rather than by swinging under gravity.

Let $\theta$ be the body's angular displacement from its equilibrium orientation. Over the linear elastic range, the support exerts a restoring torque

$$\tau=-\kappa\theta,$$

where $\kappa$ is the torsional stiffness, with SI units $\mathrm{N,m/rad}$. The minus sign indicates that the torque acts toward equilibrium.

If the body's moment of inertia about the rotation axis is $I$, rotational dynamics gives

$$I\ddot\theta=-\kappa\theta.$$

Therefore

$$\boxed{\ddot\theta+\frac{\kappa}{I}\theta=0}.$$

This is the harmonic oscillator equation, so

$$\omega=\sqrt{\frac{\kappa}{I}},$$

and

$$\boxed{T=2\pi\sqrt{\frac{I}{\kappa}}}.$$

Unlike the gravitational pendulum, no small-angle approximation is required if the restoring torque remains proportional to $\theta$. The approximation instead lies in the linear torsional constitutive law itself.

Worked example

A platform has moment of inertia

$$I=0.020,\mathrm{kg,m^2}$$

and is suspended by a wire with torsional stiffness

$$\kappa=0.50,\mathrm{N,m/rad}.$$

Its angular frequency is

$$\omega=\sqrt{\frac{0.50}{0.020}}=5.0,\mathrm{rad/s},$$

so its period is

$$T=\frac{2\pi}{5.0}\approx1.26,\mathrm s.$$

Measuring moment of inertia

The period relation can be rearranged as

$$I=\kappa\left(\frac{T}{2\pi}\right)^2.$$

Thus, if $\kappa$ is known, measuring the torsional oscillation period provides an experimental way to determine the moment of inertia of an attached object.

Conversely, a body with known $I$ can be used to determine $\kappa$.

Different restoring mechanisms, same dynamics

A mass on a translational spring obeys

$$m\ddot x+kx=0,$$

while a torsional pendulum obeys

$$I\ddot\theta+\kappa\theta=0.$$

The physical mechanisms and coordinates differ, but the mathematical structure is identical. Mass corresponds to moment of inertia, displacement to angular displacement, and spring stiffness to torsional stiffness.