The Mathematics of the Coin Cradle

To successfully pass a primary physical inspection, a counterfeit coin must exactly mimic the Mass (M) and Radius (R) of the target authentic bullion. Consequently, the thickness (h) and the Volume (V) of the counterfeit are strictly dictated by the density (ρ) of the material used.

Given that V = M/ρ and V = πR²h, the dimensional constraints dictate that the thickness and volume of the coin are inversely proportional to its density:

The first verification tier evaluates the mechanical integrity of the coin via acoustic resonance. According to Kirchhoff’s thin plate theory, the resonant frequency (f) of the (2, 0) vibrational mode is governed by the material’s Young’s modulus (E), density (ρ), thickness (h), and radius (R):

Because the counterfeiter must match the target radius (R_fake = R_target), we can substitute the thickness constraint (Equation 1) into the frequency formula. This reveals the isolated material ratio required to pass the acoustic test:

To pass the acoustic evaluation, the counterfeit material must perfectly satisfy:

Counterfeiters frequently utilise Tungsten due to its similar density to Gold. However, because the Young’s Modulus of Tungsten (E ≈ 400 GPa) drastically exceeds that of Gold (E ≈ 79 GPa), the acoustic equality fundamentally fails, yielding a severely divergent and easily detectable frequency.

Should a counterfeiter attempt to circumvent the acoustic lock by alloying a lighter metal to artificially reduce the Young’s Modulus, they must subject the sample to the electromagnetic evaluation.

The first-principles model of the pendulum (see The Physics of the Coin Cradle) makes the eddy-current braking exactly proportional to the metal’s electrical conductivity (σ) and to its thickness (t), times a factor G set by the field geometry: the magnet, the gap (3S − t, with 3 mm gauge-block spacers) and the coin’s diameter through the edge currents. The dial reading (R) is a total travel, and falls roughly as one over the braking:

A fake with the same diameter and mass as the target has thickness t ∝ 1/ρ (Equation 1). To first order, a single-metal fake must therefore match the target’s conductivity per unit density:

If the counterfeiter reduces the density (ρ) to fix the acoustic elasticity, the coin must be thicker to keep its mass. A thicker coin carries more metal and sits closer to the magnet, so it brakes the pendulum more, and for a single metal Equation 7 is broken. Layered fakes are different: they can reach a range of values, as the next section explains.

Historical note. This note originally used an empirical power law fitted to 38 readings on Coin Cradle #0001, D_avg = 488 × 2.90^S / (σ^0.998 V^1.431), which gives the condition ρ^1.431/σ in place of Equation 7. The constants belong to that one unit, and the exponent 1.431 is not a physical law: it absorbed the thickness, the gap and the edge effects that the physics model now treats separately. On the same readings the power law (four fitted numbers) reaches about 14% out of sample; the physics model fits nothing and reaches 12.4%.

The argument above rests on two assumptions:

• the fake is made of a single metal, the same all the way through;
• each test is an exact equality, with no tolerance band.

Neither holds in practice. A layered fake mixes the conductivity and density of its layers, and every real test has a band. A fuller eddy-current analysis of the Coin Cradle, modelled rather than measured (no fake has yet been tested on the device), found:

• a gold-plated tungsten fake of a fine-gold coin separates clearly from the genuine coin;
• an engineered multi-layer fake of a 22-carat coin, such as tantalum with tungsten, can land within one to two noise widths of the genuine reading, and adding rhenium can match it;
• testing both faces of the coin catches most layered fakes, unless the layers are arranged symmetrically.

For a single-metal fake, passing both tests exactly requires solving the following pair of equations at once:

For a single metal, changing one property to satisfy the first equation tends to break the second, which is why the two tests together make simple fakes hard to build. This is an argument, not a proof: it does not cover layered fakes or tolerance bands, and Section 4 shows that an engineered layered fake of a 22-carat coin can come close to passing the eddy-current test.