= Solution
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 $L$,
$$
\ddot\theta+\frac gL\sin\theta=0.
$$
At small amplitude, $\sin\theta\simeq\theta$, giving <simple harmonic motion> with
$$
\boxed{T\simeq2\pi\sqrt{\frac Lg}.}
$$
A 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 $-\kappa\phi$, so
$$
I\ddot\phi+\kappa\phi=0,\qquad
\boxed{T\simeq2\pi\sqrt{\frac I\kappa}.}
$$
The 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.
\Image[/past-exam-of-the-medicine-course-of-the-university-of-oxford/1864/first-examination/physics-1864-clock-oscillators.png]
{title=Gravitational and elastic restoring torques in clock oscillators}
{description=Original schematic comparing a pendulum with a balance wheel and spiral spring. Equations show ideal small-amplitude models, not the detailed construction of a particular historical clock.}
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.
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