Torsion pendulum

A torsion pendulum oscillates by twisting a suspension rather than by swinging its support. A horizontal bar suspended from a thin wire experiences an approximately linear restoring torque $-\kappa \theta$ for small angular twists. With moment of inertia $I$ about the wire and negligible damping, its equation of motion and period of an oscillation areHere $\kappa$ is the torsion constant, and the angle is measured from the untwisted equilibrium.

Twist the bar gently, release it, and time several complete cycles. Adding two equal masses at equal distances from the wire increases $I$ without moving the center of mass off the axis, so the period of an oscillation increases. This makes a torsion pendulum useful for measuring a moment of inertia or calibrating the suspension of a torsion balance.

A rigid pendulum tilted from the upward vertical and driven at fixed azimuthal angular velocity $\Omega$ has effective potential $\mu {\theta }^2/2+mga\cos \theta -m{a}^2{\Omega }^2{\sin }^2\theta /2$. With $\mu =2mga$, its upright equilibrium loses linear stability when ${\Omega }^2$ passes $g/a$. At equality the leading restoring torque is cubic and the potential has a positive quartic minimum. This gives a stable quartic oscillator with zero infinitesimal harmonic frequency, rather than a freely drifting neutral state.

Two identical torsional loads of moment of inertia $I$ hang in series from massless fibres of lengths ${\ell }_1,{\ell }_2$ and common filament torsional stiffness $K_t$. Put $R={\ell }_2/{\ell }_1$ and ${\omega }_0^2=K_t/(I{\ell }_1)$. The elastic energy is $K_t{\varphi }_1^2/(2{\ell }_1)+K_t({\varphi }_2-{\varphi }_1{)}^2/(2{\ell }_2)$ and the kinetic energy is $I({\dot{\varphi }}_1^2+{\dot{\varphi }}_2^2)/2$. Their normal modes have the displayed squared angular frequencies. Writing $\lambda ={\omega }^2/{\omega }_0^2$, their amplitude ratio is $A_2/A_1=1/(1-R\lambda )$. The lower mode twists in phase and approaches a combined inertia $2I$ as $R\to 0$, giving ${\omega }_{-}^2\to {\omega }_0^2/2$. The upper mode approaches the original single-load oscillation as $R\to \infty$, giving ${\omega }_{+}^2\to {\omega }_0^2$.