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 sets the timing in a weight-driven pendulum clock — friction and drag fix the amplitude, which the escapement holds anyway, and the suspension’s elastic torque fades relative to gravity as the build grows — 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 $\alpha^{1 - k/2} = \sqrt{\alpha}$. The copy is of course not the same system — it is $\alpha^3 = 8$ times heavier throughout — but kinetic and gravitational energy are both linear in the masses, so building it of the same materials multiplies the entire Lagrangian by a constant, which changes no equation of motion; this is why exact similarity in one material is part of the problem’s hypothesis, and why the masses never enter. 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 — and the gears cannot rescue the clock, because gear ratios are pure numbers and merely count swings that have themselves slowed. In a real day of 24 hours the dial advances only $24/\sqrt{2} \approx 17$ hours, so the copy loses $24\left(1 - \tfrac{1}{\sqrt{2}}\right) \approx 7$ 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 — with $l$ read as the size of the trajectory, which for the pendulum happens to be the size of the build. 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. (Indifference to amplitude, not to scale: double a balance wheel and its hairspring in the same materials and the stiffness grows as $\alpha^3$ against the wheel’s $\alpha^5$ of inertia, so that copy runs $\alpha = 2$ times slow — a worse fate than the pendulum’s $\sqrt{2}$.) For $k = -1$ it gives $t \propto l^{3/2}$ across any family of similar orbits, which is Kepler’s third law before a single orbit has been computed; that the period turns out to care about the semi-major axis alone, whatever the eccentricity, is one extra $1/r$ degeneracy the scaling cannot supply — and Problema XII spends exactly that.

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