Measuring the Density of Pennies

posted on 23 Feb 2014 by guy
last changed 7 Jan 2016

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ages: 14 to 99 yrs
budget: $1.00 to $30.00
prep time: 5 to 120 min
class time: 30 to 120 min

This activity provides a detailed lesson in measurement and measurement errors, which can be adapted to many different age groups, starting with any group that knows a little bit of geometry (specifically, how to calculate the volume of a disk). The focus of the lesson is on the nature of errors and uncertainty (including the definition of standard deviation) and the difference between statistical and systematic errors. The lesson also offers an opportunity to study various types of charts (histograms, scatterplots). More advanced groups can apply basic statistics and calculus to the propagation of errors, with an opportunity for spreadsheet calculations. Worksheets and slides with example histograms and figures are attached.

optional equipment: Digital Scale
subjects: Engineering, Mathematics, Physics
keywords: measurement, histogram, uncertainty, error, systematic, statistics, scatterplot

file attachment(s): 


Special Credit: I was inspired to write this lesson following a talk on the same subject by Dr. Duane Deardoff (U. of North Carolina) at the 2013 summer conference of the American Association of Physics Teachers. All of the brilliant insights are his, and any clumsy mistakes are mine.

the formula

What must you measure in order to determine the density of a penny?

Since the density is the amount of mass per volume, and the volume of a penny can be determined (approximately) from the thickness and diameter, it follows that measurements of the mass, the thickness and the diameter of the penny should be enough to calculate the density. The volume $V$ of a disk is just the thickness times the area of a circle: \begin{equation}V = t\times{\pi D^2\over 4}\end{equation} where $t$ is the thickness and $D$ is the diameter. The density $\rho$ is then \begin{equation}\rho = {m\over V} = {4m\over t\pi D^2}\end{equation} where $m$ is the mass of the penny. This assumes that the penny is a uniform cylinder of metal, an assumption we'll come back to at the end of the lesson in "statistical and systematic errors", below.


Fig. 1: Simple balance scales, accurate to 0.1 grams. Image made available by Lilly_M at Wikimedia Commons under the CC-BY-SA 3.0 license.


Fig. 2: Vernier caliper showing 1) outside jaws, 2) inside jaws, 3) depth probe, 4) main scale cm, 5) main scale inch, 6) Vernier cm, 7) Vernier inch, 8) retainer. Image made available by Joaquim Alves Gaspar at Wikimedia Commons under the CC-BY-SA 3.0 license.


Fig. 3: Micrometer with parts labeled. Image made available by Three-quarter-ten at Wikimedia Commons under the CC-BY-SA 3.0 license.


Fig. 4: A sample of 100 thickness measurements of new pennies using a crude micrometer. The average thickness for this sample is 1.42mm and the RMS of the distribution is 0.07mm.

the measurements

Mass can be measured with a good balance scale (figure 1) that measures to 0.1g, or better yet, a digital jeweler's scale, which measures to 1-10 milligrams. A penny typically weighs between 2.4 and 3.2 grams, depending on the year of manufacture and wear.

The diameter and thickness can be measured with a ruler to a precision of about half a millimeter, maybe a little better. The diameter of a penny is nominally 19.05mm and the thickness is nominally 1.52mm at the rim.1

Of the three measurements (mass, diameter and thickness), which is the most critical for calculating the density?

This is a key question to thinking about measurement uncertainties. Assuming your scale measures mass reasonably well (let's say to a tenth of a gram or better), the most critical measurement is probably the thickness of the penny. The reason is that the measurement uncertainty (about half a millimeter using a ruler) is a large fraction of the thickness of the penny. It amounts to about 30% uncertainty (0.5mm/1.5mm $\approx$ 30%) and will produce about a 30% uncertainty in the density. In contrast, the uncertainty on the measurement of the diameter is only about 5% (0.5mm/19.05mm $\approx$ 5%), which will produce about a 10% uncertainty in the density. See below under "propagation of errors" for more on this issue.

Clearly, a ruler is not the best tool for trying to measure the thickness of a penny. It's not precise enough. A better tool might be a Vernier caliper (figure 2) or a micrometer (figure 3). These devices can easily measure to a precision of 0.05 mm or better.

determining uncertainties (standard deviation)

How do you determine the uncertainty of a measurement?

All measurements are subject to errors. Perhaps there's a limit to how precisely one can read the marks on a micrometer, or maybe there are fluctuations in the readout of a balance scale that depend on temperature and humidity, or perhaps the result of an entire experiment depends on the phase of the moon (not usually, but see this article on the effects of lunar and solar tides on a particle beam experiment at CERN).

Whatever the reason, there are usually a number of uncontrollable factors in making a measurement, which might affect the result. Often, the mark of a good scientist is being able to estimate the uncertainty due to those uncontrollable factors in order to understand the accuracy and reliability of the result.

You might make a reasonable estimate of uncertainty by looking at the precision of your measuring tool. The smallest marks on your ruler are probably a millimeter apart, and you might suppose that you can judge a measurement to half that length by eye (which is how we got to a ruler uncertainty of 0.5mm above). The smallest increments on a micrometer scale probably correspond to a much smaller distance, and you might reasonably assume that the measurement uncertainty is correspondingly smaller.

However, there is an important distinction between "precision" and "accuracy". Your micrometer may have a scale marked in thousandths of a millimeter (high precision), but that does not mean it is correct to a thousandth of a millimeter (highly accurate). In the end, the uncertainty depends not only on your tool, but also on how you read it, and maybe many other factors in the environment. To get a better estimate of the uncertainties in measurements, you might repeat a lot of identical measurements, and plot the distribution of results. Figure 4 shows a plot of the measured thickness of many new pennies (which haven't been worn appreciably and should all be the same thickness) using a crude micrometer. See our lesson on Homemade Micrometer to see how we built the micrometer.

Assuming all the pennies have the same thickness, the different measurements in figure 4 indicate a range of measurement errors, from about 0.2mm lower than average to about 0.2mm higher than average. A typical measurement error would be a little less, and might be a good indication of the uncertainty on one of these measurements.

The standard estimate of measurement uncertainty, known as the "root mean square deviation" ("RMS") or sometimes just the "standard deviation",  aims to take a sort of average of errors in a sample of measurements in order to estimate the typical error. To calculate the RMS, first assume that the average of the measurements is close to the correct "true" value.2 The average, or "mean" of a sample of measurements is: \begin{equation}\overline t=\sum_{i=1}^n {t_i\over n}\end{equation} where $\overline t$ is the average, $t_i$ is an individual measurement, and $n$ is the total number of measurements.

Now take the difference between each measurement and the average; this is the "deviation", $\Delta t_i=t_i - \overline t$. Some measurements will be too high, leading to positive deviations, and some measurements will be too low, leading to negative deviations. In order to handle the positive and negative deviations on an equal footing, we'll use the square of each deviation. Then, in order to estimate a typical square deviation, we take the average (or "mean") of all the square deviations. This average is usually known as the "variance" of a distribution. Finally, since we want an estimate of a typical deviation, not an estimate of a typical squared deviation, we'll take the square root of the whole thing. Reading the formula from the outside in, this calculation gives the "root mean square deviation", usually represented by the Greek letter $\sigma$: \begin{equation}\sigma_t = \sqrt{\sum_{i=1}^n {\Delta t_i^2\over n}} = \sqrt{\sum_{i=1}^n {(t_i-\overline t)^2\over n}}\end{equation}

For the distribution shown in figure 4, the RMS is 0.07mm.

propagation of errors (note for geeks)

Folks who know some differential calculus can determine how the errors on individual measurements of mass ($\Delta m$), diameter ($\Delta D$) and thickness ($\Delta t$) propagate to give an error in calculating the density ($\Delta\rho$). The relation derives from an application of the chain rule for derivatives: \begin{equation}\Delta\rho = {\partial\rho\over\partial m}\Delta m +{\partial\rho\over\partial D}\Delta D + {\partial\rho\over\partial t}\Delta t\end{equation} where ${\partial\rho\over\partial m}$ specifies how the calculated density changes $\partial\rho$ in response to a change in the mass $\partial m$, and similarly for diameter and thickness. Thus, ${\partial\rho\over\partial m}\Delta m$ describes the error in density resulting from an error $\Delta m$ in mass. Assuming all the measurement errors are independent of each other, an estimate of the variance on the density is: \begin{equation}\sigma^2_\rho = \left({\partial\rho\over\partial m}\right)^2\sigma^2_m + \left({\partial\rho\over\partial D}\right)^2\sigma^2_D + \left({\partial\rho\over\partial t}\right)^2\sigma^2_t = \left({4\over t\pi D^2}\right)^2\sigma^2_m + \left({8m\over t\pi D^3}\right)^2\sigma^2_D + \left({4m\over t^2\pi D^2}\right)^2\sigma^2_t\end{equation} and the RMS is: \begin{equation}\sigma_\rho = \sqrt{\sigma^2_\rho}\end{equation}

Students who are comfortable with this type of calculation may wish to calculate the RMS values of the individual mass, diameter and thickness distributions, and use those values to predict the RMS of the density distribution following the equation above. Comparison of the prediction with the actual RMS density calculated from the distribution of density measurements tells you how accurate your predition is. (But see the discussion under "questions to ponder" regarding correlated errors.)


Fig. 5: Plot of measured masses for a sample of 50 pennies.


Fig. 6: Penny mass by year, as measured on a sample of 50 pennies. In 1982, the U.S. penny composition changed from 95% copper and 5% zinc to 97.5% zinc and 2.5% copper. Since copper is considerably more dense than zinc, the net mass of the penny dropped at that time.


Fig. 7: Magnified view of a Lincoln head penny, showing the relief. The penny is about 1.5mm thick at the high points, and about 1.1mm thick at the low points.

statistical versus systematic errors

A "statistical error" is a random error that is not correlated from one measurement to the next. An error of this type is just as likely to be too high as it is to be too low. The effect of statistical errors can be reduced by taking the average of a large sample of measurements — the random statistical errors tend to cancel out in the average. In contrast, a systematic error is generally an error in assumption or technique (an error in the "system"), which may be common to many measurements. If a systematic error affects all measurements in the same way, no amount of data, and no averaging can alleviate the bias.

For example, if a micrometer can be read to high precision, but the micrometer itself is manufactured so that all the measurements taken with it are a little bit low, the distribution of measurements may be narrow, indicating small statistical errors, but the whole distribution may be shifted from the "true" value, indicating some sort of systematic error. In figure 4, the RMS of the thickness measurements is 0.07mm, indicating a fairly small statistical uncertainty, but the center of the distribution is too low (at 1.42mm instead of 1.52mm where it belongs) indicating some kind of systematic error in the measurement process. (In this case, the two faces of the micrometer are not exactly parallel; on one side the micrometer, where we made all the measurements, it measures a little too low and on the other side it measures too high.)

In this lesson there are two good opportunities to study systematic errors. One involves how pennies are manufactured, and the other involves our technique for measuring thickness.

First of all, attentive students may notice that the mass measurements for pennies tend to cluster around two separate values. Some pennies weigh in around 2.5 grams, while others are closer to 3.1 grams (figure 5). There aren't many measurements in between. The reason for this odd behavior is that in 1982 the U.S. penny composition changed from 95% copper and 5% zinc to 97.5% zinc and 2.5% copper.3 Figure 6 shows a scatterplot of mass versus year, demonstrating the change. Since we're trying to measure the density of pennies, it's important to pay attention to the year the pennies were minted, and to keep these two populations separate. Pennies minted prior to 1982 should have a density almost the same as pure copper (8.96 g/cm3) while pennies minted after 1982 should have a density almost the same as pure zinc (7.14 g/cm3). If we assumed we were trying to measure the density of pure copper pennies, then including pennies minted after 1982 in the measurements would amount to a systematic error.

Secondly, even if all the measurements are carefully made and there are no instrumental errors, we are still likely to get the wrong answer for the density. As an example, take the nominal values for a penny as specified by the U.S. mint: the mass is 2.50g (after 1982), the diameter is 19.05mm and the thickness is 1.52mm. If we use these values to calculate the density of a penny, we get: $$\rho = {4m\over t\pi D^2} = {4\times 2.50g\over (0.152cm)\pi(1.905cm)^2} = 5.77 {g\over cm^3}$$ Compare this value to the density of pure zinc at 7.14 g/cm3 and we see we have a problem.

What went wrong? In this case, the large size of the error suggests that it occurs in the most critical of the measurements: the thickness. If we measure the thickness with a flat face micrometer or caliper, we are measuring the thickness of the penny at its thickest point (the rim). However, as figure 7 shows, the thickness at the rim is quite a bit bigger than the thickness at other points across the surface. Although the nominal thickness at the rim is 1.52mm, the average thickness of the penny (the thickness it would be if it were truly a uniform cylinder) is just a little over 1.2mm. By measuring the maximum thickness instead of the average thickness, we made a substantial systematic error. Possible means of getting around this error are considered in "questions to ponder" below.

questions to ponder

  • If I make a plot of measurements and see a variation of results, how do I know what is measurement error and what is real variation in the objects being measured?

It's not easy. If you make a lot of IDENTICAL measurements, and get different results, you can figure that the variation has something to do with the measurement process. For example, if you measure the same penny many times with the same micrometer, but get different results, you might assume the variation is due to the measurement process. Maybe you tighten the micrometer screw a little tighter sometimes, maybe you measure at slightly different points on the surface of the penny, maybe the micrometer is uneven and measures differently on different sides (see figure 4). Alternatively, if you happen to know that the maximum width of a penny when brand new is 1.52mm (but that pennies can be worn thin over time) then even if you measure many different pennies, you might figure that measurement results larger than 1.52mm indicate the size of errors due to measurement, while results less than 1.52mm are probably due to a combination of measurement error and wear.

  • How might you get a better estimate of the true value of a parameter?

Assuming measurement errors are independent of each other, averaging many different measurements will give a better result. Independent measurement errors tend to cancel each other in the average. For instance, if you determine the density of $n$ pennies, and each calculation has an uncertainty of $\sigma_\rho$, then the average of all the measurements will have a smaller uncertainty given by: \[\sigma_{avg} = {\sigma_\rho\over\sqrt{n}}\] To derive this formula, recognize that the variance of a sum of measurements is just the sum of the individual measurements. Since the average is $1\over n$ times the sum, the variance on the average is $1\over n^2$ times the variance of the sum, or $1\over n$ times the variance of an individual measurement. Take the square root to get the standard deviation from the variance.

  • Since the thickness of the penny is not uniform, and the average thickness is hard to measure, is there a better way to calculate the volume of a penny?

Good question. Some folks have suggested measuring the volume of a penny by measuring how much water it displaces. This sounds like a clever idea — it measures the true volume rather than an estimated volume assuming the penny is a uniform cylinder — but it might be hard to measure precisely. The best way to do this would probably be to put many pennies in a small container of water (perhaps a graduated cylinder), and record how much the water level rises. We want to get the surface of the water to rise as much as possible when the pennies go in. Calculate the change in volume (water plus pennies compared to just water) and divide by the number of pennies to get the average volume of a single penny.

teaching notes

It makes sense to start the lesson by asking students what parameters they need to measure in order to figure out the density of a penny. This question leads them quickly through the formula for the volume of a cylinder to identify the three basic measurements involved (mass, diameter and thickness).

I also like to ask how students think they should make the measurements. I let one student measure the diameter and thickness of a penny using a ruler, and then ask the class how accurate the measurements are likely to be. Then I ask which measurement is the most critical in calculating the density. This questions starts the conversation about measurement error and gets students thinking about how measurement errors could affect the final result. Also, it makes them realize a ruler is not a very good tool for measuring the thickness of a penny. We then switch to a Vernier caliper or a micrometer.

Much of the class is well spent just letting students record the date, mass, diameter and thickness of a sample of pennies (20 pennies is usually enough), so that we have enough data to make some interesting plots. Let students enter data directly into a spreadsheet or use the attached worksheet. Use a spreadsheet program (Excel or Google Spreadsheets) to also calculate the volume and density of each penny. Using the spreadsheet program, we then make the following plots. (Google spreadsheets is good for this since it has a built-in histogram option that works on all computers.)

  • histogram of thickness measurements - These usually range from 1.2 to 1.6 mm and should peak at 1.52mm for new unworn pennies, unless there is a bias in the measurement. (See figure 4 for an example of bias.) This plot can be used to introduce the concept of root mean square deviation as an error estimate.
  • histogram of mass measurements - This can be a very interesting plot to make. Assuming you have pennies both older and younger than 1982 in your sample, you should see two peaks in the distribution: one at about 2.5g for zinc pennies minted after 1982, and one at about 3.1g for copper pennies minted before 1982. Students may wonder for a while what could cause this strange behavior. If any students do know about the change in penny composition in 1982, you should ask them how they might verify that hypothesis using the data they have collected.
  • scatterplot of date vs mass - This plot gives proof that pennies changed in 1982 (see figure 6). Precede this plot by telling students about the change in penny composition in 1982, and asking them what plot they might make to verify that claim. They can either make two plots (a histogram of mass for pennies before 1982 and one for pennies after 1982) or they can come up with a way of plotting how mass depends on year.
  • histograms of calculated densities - One plot for pennies older than 1982, another plot for pennies younger than 1982. Compare the center values to the known values for the density of copper (8.96 g/cm3) and the density of zinc (7.14 g/cm3). Very likely the centers of your distributions are too low, which can initiate a discussion about the systematic error in the thickness measurement; see the comments in "statistical and systematic errors" about the problem of measuring the maximum thickness rather than the average thickness.

Students who are familiar with calculus can also be asked to calculate sample RMS deviations from their mass, diameter and thickness histograms, in order to predict what the sample RMS should be for the final density histogram. See the section on "propagation of errors" for this formula.

further resources

The U.S. mint has a description of how coins are made at http://www.usmint.gov/faqs/circulating_coins/index.cfm?action=coins .

The U.S. mint specifications for coins currently in production can be found at http://www.usmint.gov/about_the_mint/?action=coin_specifications .

 

  • 1. http://www.usmint.gov/about_the_mint/?action=coin_specifications
  • 2. Note for geeks: For small data samples, the sample RMS tends to underestimate the true error because the mean of a small sample has smaller deviations from its component measurements than the true value. In the extreme case, a sample of one measurement gives a deviation of zero - the mean is the same as the single measurement - which obviously underestimates the typical deviation from the true value. For sample sizes of 6 measurements or more, the effect is usually smaller than 5%.
  • 3. http://en.wikipedia.org/wiki/U.S._Penny


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