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Error propagation formulae stem from differential calculus, in which we can treat our measurements as variables and our calculations as functions. Instead of memorizing a long list of formulae (for use in an exam, say), we can simply derive them from first principles. We know from single-variable calculus that the differential...
As a continuation of our lesson on measurements and uncertainties, our activity today had us trying to find the density of Philippine coins. This activity, however, had us learn the concept of error and error propagation in experimentation. The goals of today’s class were to perform error propagation and statistical analysis, as well as use estimated uncertainties for measurements.
The measured data for this activity are all located in the following spreadsheet:
Our task for this activity was to experimentally determine the densities of the 5¢, 10¢, ₱1, and ₱5 coins. We were given basic measuring instruments: a vernier caliper with smallest measurement of 0.02 mm to measure the diameters and thicknesses of the coins and an electronic balance with smallest measurement of 0.1 g to measure the mass of the coins.
We started taking the measurements using the coins and the instruments provided. To measure the diameter as well as the thickness of a coin, we took 3 trials at different places along the coin. This is to take into account that the coin might not be uniform all throughout. Also note that to get the proper measurements of the 5¢ coin, which has an inner hole, we took both the the inner and outer diameters of the coin.
Given that the measuring instrument used was the vernier caliper, with least count of 0.02 mm, the uncertainty for the measurements in each trial was ±0.01 mm. The average of all the trials was what we used as the final measurement for that specific quantity. Since the measurements were averaged, the final uncertainty was equal to the average of all uncertainties used. Since the absolute uncertainty for each of the three trials for one measurement was ±0.01 mm, the average was also equal to ±0.01 mm.
The measurements for diameter were then converted to measurements for radius, as the radius of the coins would be the quantities used in later computations. Since we divided the quantity by 2, we would also have to divide the absolute uncertainties for these measurements. Thus, the uncertainty for the value of the length of the radius was equal to ±0.005 mm.
For the mass of the coin, we used the electronic balance, I took the convention that the uncertainties for the measurements taken here were equal to the lowest count on the electronic scale. On non-electronic measuring devices, the norm is to take half the least count on the instrument’s scale. This uncertainty accounts for the discrepancies a researcher may have while reading the measurement. This method does not particularly work when taking the measurement from an electronic measuring device since the measurement is stated there for you to see, no room for human error. As such, the uncertainty used is instead the least count of the electronic measuring device.
Computations
The relative uncertainties were then taken, as the succeeding computations would involve multiplication, division, and powers. The relative uncertainty for each measurement is equal to the absolute uncertainty divided by the measurement. Since relative uncertainty is expressed as a percentage (%), these quantities were multiplied by 100 to get the relative uncertainty as a percentage.
Relative uncertainties are used for these mathematical processes because using absolute uncertainties decreases the uncertainty. When having uncertainties, the goal is to be the most “pessimistic” or to have the highest value of uncertainty possible. Thus, we would want a larger representation of the uncertainty which is achieved by using the relative uncertainty rather than the absolute uncertainty.
Area of the coin’s face was computed by the formula Area = πr^2 leaving units of sq. mm. It should be taken note that for the 5¢ coin, the appropriate area was taken by finding the area of the larger circle using the outer diameter and subtracting from this the area of the smaller circle that uses the inner diameter. Uncertainty for these quantities is taken by multiplying the relative uncertainty for measurements of the radius by 2. (See Reference 1 for detailed rules on uncertainty propagation in mathematical operations.)
The volume of the coin is taken by multiplying the area of a coin’s face by its thickness. The uncertainty for this quantity is taken by simply adding the relative uncertainty for the area and for the thickness. The resulting volume now has units of cubic mm. This is then converted to cubic cm as this is the standard g/cm^3 is standard in relating the density of an object.
The density is then computed by taking the mass of the coin and dividing it by the total volume of the coin. This results in units of g/cm^3 with uncertainty equal to the sum of absolute uncertainties for the involved measurements.
The results gathered are summarized as the following (to 3 sig figs):
5¢ - (7.22 ± 6.21%) g/cm^3
10¢ - (7.09 ± 4.62%) g/cm^3
₱1 - (7.05 ± 2.19%) g/cm^3
₱5 - (7.51 ± 1.59%) g/cm^3
References:
1. Basic rules for uncertainty calculations. Retrieved on March 15, 2015, from http://web.uvic.ca/~jalexndr/192UncertRules.pdf.
