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If you have the dimensions and material of an object, you can compute both the mass and the normal vibration modes. Just the mass is not enough - a large paper "coin" will have a different fundamental frequency than a small tungsten sphere.

A summary of everything that comes below - the result of several edits, and including a nice interaction with the other answer from alemi:

The relationship between the fundamental frequency of the "ping" of a coin and its mass is given (approximately) by $$m \propto \frac{t\sqrt{E}}{f^{1/3}D}$$ where

$E$ = Young's modulus

$t$ = thickness

$m$ = mass of coin

$D$ = diameter of coin

$f$ = fundamental frequency



Here are the details of how I got there...

If you assume that all "coins" are of the same aspect ratio (ratio of diameter to thickness) and made of the same material, then it is indeed possible to compute the relationship between fundamental frequency and mass. From dimensional analysis, if we assume that frequency is a function of

$\eta$: aspect ratio (dimensionless: initially assumed constant for all, ignored)

$D$: diameter ($\text{m}$)

$\rho$ density ($\text{kg}/\text{m}^3$)

$E$: modulus ($\text{kg}/\text{m s}^2$)

Then the combination of the above that gives us units of $1/s$ is $$f = \text{const} \cdot \frac{1}{D} \sqrt{\frac{E}{\rho}}$$

Combining this with the mass of the object which is proportional with $\rho D^3$, then assuming $\rho$ is constant (so we can take it out of the equation) we get

$$f = \text{const} \cdot m^{-1/3}E^{1/2}\\ m = \text{const} \cdot E^{3/2}f^{-3}$$

In other words - mass decreases with the third power of the frequency for coins with the same material and aspect ratio.

But that is not how US coins work. From the US mint website, I extracted the following:

Aspect coin mass diameter thickness material ratio penny 2.500 19.05 1.52 Zn* 12.53 nickel 5.000 21.21 1.95 Cu-Ni 10.88 dime 2.268 17.91 1.35 Cu-Ni 13.27 quarter 5.670 24.26 1.75 Cu-Ni 13.86 old 1c. 3.11 19.05 1.52 Bronze. 12.53 * copper plated...

So the material isn't always the same, and neither is the aspect ratio. That's going to make it a little bit hard to prove or disprove the relationship.

Still - let's have a shot at it. From the experimental data (@alemi's answer) I read the fundamental frequencies as follows:

penny 12.6 nickel 12.4 dime 12.8 quarter 9.2

Now the interesting two are the quarter and the dime, since they have the same material and the most similar aspect ratio (13.3 vs 13.9, so only 5% difference). From the ratio of their masses (2.500), we would expect the ratio of frequencies to be 0.74 ($2.5^{-1/3}$). And the observed ratio is 0.72. That is really quite close...

Put another way - if you knew the frequencies for the dime and the quarter, and you had to estimate the mass of the quarter from the dime, you would obtain

$$\begin{align}\\ m &= 2.268 * \left(\frac{12.8}{9.2}\right)^3\\ &= 6.11\end{align}$$

which is an error of about 7% or less than 0.5 g. I think that is spectacular given there is a factor 2.5 difference between the dime and the quarter.

Encouraged by this result, I decided to see if I could get agreement for the four coins given their different aspect ratio and material. Since both bronze and Cu-Ni alloys have a wide range of Young's modulus, I had to guess a bit (all values in GPa):

material range (GPa) value (GPa) bronze 96 - 120 110 Cu-Ni 120 - 156 120

Next, I had to deal with the aspect ratio. After thinking about this, it was plausible that a larger aspect ratio (thinner coin) would have a lower frequency, so I decided to see what happened if I made frequency dependent on $1/\eta$. This led to the following "expected frequency" formula:

$$\text{expected} = \frac{\sqrt{E}}{m^{1/3}\eta}$$

Computing this with the new mass for the penny (3.11 g) I obtained the following plot for the relationship for each of the coins:

In this plot, the red stars are the numbers (scaled to fit the same chart) that would have been obtained without taking account of the aspect ratio; the blue circles correspond to the values with the $1/\eta$ relationship taken into account. Clearly this improved the fit. Pretty convincing, given the relatively noisy data...

AFTERTHOUGHT

There are multiple frequencies visible in the sound recordings that @alemi showed. Some of these are easily explained by looking at the multiple modes of a simple circular plate - see for example Waller, 1938 Proc. Phys. Soc. 50 70.

Two images from that publication: first, the vibration modes:

And next, their relative frequencies:

This shows that the first harmonic is 1.7x higher than the fundamental frequency. Looking at the data, we see that is indeed about right: in fact, for the quarter we are even seeing the second harmonic (at 2.3x of the fundamental).

The question of the splitting of the fundamental frequency is a bit trickier. If you have ever played with a empty coffee mug, you may have noticed that when you tap the rim, the pitch changes depending on whether you tap right across from the handle, or at 45 degree offset from there. This is because there are two symmetrical modes - one where the handle is a node, and one where it is an antinode. The latter has a slightly lower frequency.

A similar thing may happen with the nickel: when you look at the image of a pre-2005 Nickel, you will see that there is more material in the north-south and east-west direction. This means that there are two vibration modes: the one with nodes in blue, and the one with nodes in red:

Obviously, when the blue lines are the nodes the frequency will be slightly higher.

LITERATURE

I found a paper where this is discussed in some more detail - broadly it agrees with everything said above, and even came up with very similar values for the frequencies (measured and modeled). You can read it at http://me363.byu.edu/sites/me363.byu.edu/files/Emerson_Steed_CoinIdentification.pdf

Interestingly, the authors were unable to capture the sound of the penny, although their model suggested a frequency close to the one that @alemi measured (13.1 kHz). They showed the first vibration mode as

which is a nice colorful 3D representation of the mode described by the 1938 Waller paper.