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A wheel that oscillates about a spindle and regulates a watch through an escapement. In the ideal balance-spring model, a torsion spring supplies torque $-\kappa \varphi$ and the wheel has moment of inertia $I$, giving $I\ddot{\varphi }+\kappa \varphi =0$. The period follows from simple harmonic motion. Real spring, bearing and escapement effects disturb this ideal.

Both instruments regulate a sequence of time intervals with an oscillator. A restoring action opposes displacement, while inertia carries the moving part through equilibrium. Their repeated swings control an escapement, which releases the gear train in steps and supplies enough energy to offset frictional losses.

For a simple pendulum of length $L$,At small amplitude, $\sin \theta \simeq \theta$, giving simple harmonic motion withA real clock pendulum is a physical pendulum. If its pivot moment of inertia is $I_p$ and its centre of gravity is a distance $d$ below the pivot, its small-swing period is $2\pi \sqrt{I_p/(Mgd)}$. Gravity supplies the restoring torque. A controlled swing amplitude and stable dimensions are needed: large swings, thermal expansion and changing effective gravity alter the rate.

A watch's balance wheel turns back and forth about its spindle. A torsion spring supplies an approximately linear restoring torque $-\kappa \varphi$, soThe restoring mechanism is elastic rather than gravitational. This makes the balance-spring system suitable for a portable watch, although disturbances, imperfect balance, changing spring properties and escapement interactions still affect the rate.

Near independence of period from amplitude is isochronism. It is approximate for the simple pendulum: its leading correction is $T/T_0=1+{\theta }_0^2/16+\cdots$ with ${\theta }_0$ in radians. Keeping the oscillations repeatable therefore matters as well as selecting the nominal period.

The derivative of an axial torque with respect to relative end-rotation angle, evaluated at the operating configuration. In a linear torsion spring, $N=K_{\phi }\phi$.

An ideal linear torsion spring exerts a restoring torque proportional to the relative angle between its two ends. Integrating the torque gives its elastic potential energy. A spring connecting two rotating bodies contributes $\kappa ({\theta }_2-{\theta }_1{)}^2/2$ to the potential energy; a spring connected to a fixed frame is obtained by setting one angle to zero. The coefficient $\kappa$ is the torsional spring constant.