Reporting Density from a Noisy Balance and a Coarse Cylinder
Marco is measuring the density of a small irregular pebble for a lab report. His digital balance has 0.01 g resolution; repeated readings of the dry pebble give 12.34 g, 12.36 g, 12.35 g and 12.34 g, with the last digit flickering. For volume he uses a 50 mL graduated cylinder marked in 2 mL intervals; he reads 20.0 mL before adding the pebble and 24.8 mL after. He taps the cylinder to dislodge air bubbles but does not wait for the foam to clear. He computes 12.35 g divided by 4.8 mL, gets 2.57 g/cm3, and reports 'density = 2.57 g/cm3' with four significant figures. He then compares this with a handbook value of 2.65 g/cm3 for quartz and concludes his pebble is 'probably not quartz.' The teacher points out that the cylinder's smallest markings are 2 mL apart and that the pebble is visibly porous.
1. Justify how many significant figures Marco's reported density can actually support. Show the reasoning using the uncertainty of each reading.
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Mass. The four readings span 12.34-12.36 g, so the mass is 12.35 g with an uncertainty of about 0.02 g. Relative uncertainty: 0.02/12.35 = 0.16 %, which is fine.
Volume. This is where it breaks down. The cylinder is marked every 2 mL, so a careful reader can estimate to about a quarter of a division, roughly 0.5 mL, on each of the two readings. When you subtract two readings, the uncertainties add in quadrature (or simply add, worst case), giving about 0.7 mL, or 1 mL worst case, on a volume of 4.8 mL. That is 15-20 % relative uncertainty.
Rule for the result. Density is a quotient, so relative uncertainties combine: roughly 0.2 % from mass plus 15-20 % from volume, dominated entirely by volume. A 20 % uncertainty on 2.57 means the defensible statement is about 2.6 g/cm3, i.e. two significant figures, not four.
Marco's four-figure answer implies an uncertainty of about 0.01 g/cm3, which is roughly a thousand times better than his apparatus allows. Writing 2.57 does not make the measurement that precise.
2. Calculate the lowest and highest plausible densities consistent with his readings, and explain why the comparison with 2.65 g/cm3 is inconclusive.
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Upper bound. Smallest plausible volume and largest plausible mass: if each reading is off by 0.5 mL, the volume could be as low as 4.8 - 1.0 = 3.8 mL. With mass 12.37 g, density = 12.37/3.8 = 3.26 g/cm3.
Lower bound. Largest volume and smallest mass: volume up to 5.8 mL, mass 12.33 g, gives 12.33/5.8 = 2.13 g/cm3.
So the honest result is 2.6 g/cm3, with a plausible range of roughly 2.1 to 3.3 g/cm3.
The handbook value of 2.65 g/cm3 sits almost exactly in the middle of that range, so his data are entirely consistent with quartz. Saying it is 'probably not quartz' reverses the logic: failing to confirm a value with an imprecise instrument is not evidence against it. This is the difference between 'my measurement is inconsistent with X' and 'my measurement is not precise enough to distinguish X from several alternatives'.
There is a second issue cutting the same way: the pebble is porous and bubbles cling to it, which would make the displaced volume too large and the density too low - a systematic error, not just scatter.
3. Propose two changes that would most reduce the uncertainty, and state roughly how much each would help.
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Change 1 - shrink the volume uncertainty, because it dominates. Move to a 10 mL graduated cylinder marked in 0.1 mL, or better, use an overflow/Archimedes method where the displaced water is weighed (1 g of water = 1 cm3, and a balance reading to 0.01 g gives volume to about 0.01 cm3). Reading to 0.1 mL instead of 0.5 mL per reading cuts the volume uncertainty from around 20 % to around 4 %, which is a five-fold improvement in the final answer and by far the biggest single gain.
Change 2 - deal with the porosity systematically. Boil or vacuum the pebble in water, or soak it until no bubbles emerge, then blot and weigh it submerged using a thread. Otherwise the air pockets bias the volume high and the density low, and no amount of averaging removes a systematic error. Coating the pebble in a thin film of wax of known mass is an alternative, but the coating volume must then be corrected.
A third, cheap improvement: measure ten pebbles together. That multiplies the mass and volume by ten while the absolute reading uncertainties stay the same, cutting their relative size ten-fold.