Angular SHM

A mechanical watch keeps time based on the oscillations of a balance wheel (Fig. 14.19). The wheel has a moment of inertia I about its axis. A coil spring exerts a restoring torque tz that is proportional to the angular displacement u from the equilibrium position. We write tz = -ku, where k (the Greek letter kappa) is a constant called the torsion constant. Using the rotational analog of Newton’s second law for a rigid body, gtz = Iaz = I d2 u/dt² , Eq. (10.7), we find

This equation is exactly the same as Eq. (14.4) for simple harmonic motion, with x replaced by u and k/m replaced by k/I. So we are dealing with a form of angular simple harmonic motion. The angular frequency v and frequency f are given by Eqs. (14.10) and (14.11), respectively, with the same replacement:

The angular displacement u as a function of time is given by

where ? (the capital Greek letter theta) plays the role of an angular amplitude. It’s a good thing that the motion of a balance wheel is simple harmonic. If it weren’t, the frequency might depend on the amplitude, and the watch would run too fast or too slow as the spring ran down.

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