11. The clock built twice as large

A grandfather clock keeps perfect time. A craftsman builds an exact copy scaled up by a factor of two in every linear dimension — pendulum, gear train, drive weights — in the same materials. Does the copy run fast, slow, or true? By how much?

Solution

Slow — everything it does takes $\sqrt{2}$ times longer, and it falls about seven hours behind per day.

No equation of motion needs solving. Gravity is the only force that matters in a weight-driven pendulum clock, and gravitational potential energy near the ground, $U = mgz$, is homogeneous of degree $k = 1$ in the coordinates. Mechanical similarity then settles the clock’s fate: rescale every length by $\alpha$ and each trajectory maps onto a rescaled trajectory of the same system with all times multiplied by $\alpha^{1 - k/2} = \sqrt{\alpha}$. At $\alpha = 2$, every swing of the pendulum, every advance of the gear train, every descent of the drive weight takes $\sqrt{2}$ times longer. The masses never enter — they cancel from every gravity-driven motion — and the gears cannot rescue the clock, because gear ratios are pure numbers and merely count swings that have themselves slowed. A full day of dial time takes $24\sqrt{2}$ real hours, so the copy loses $24\,(\sqrt{2}-1)/\sqrt{2} \approx 7$ hours in every real day.

The one-line law behind this, $t'/t = (l'/l)^{1-k/2}$ for a potential homogeneous of degree $k$, is worth carrying around. For $k = 1$ it is Galileo’s $\sqrt{l}$ pendulum. For $k = 2$ the exponent vanishes: harmonic oscillations of every amplitude take the same time, and that isochronism — the balance spring’s indifference to how hard it is driven — is what a watch lives on. For $k = -1$ it gives $t \propto l^{3/2}$, which is Kepler’s third law before a single orbit has been computed; Problema XII spends it.

Deeper in the notebook: the Classical Mechanics shelf — still being bound; the similarity argument will live there with the rest of the variational machinery.