Balance wheel ($T\simeq 2\pi \sqrt{I/\kappa }$)

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.