Lab 6: Propagated Uncertainty

The propagation of uncertainty is how various errors affect overall uncertainty. Previous labs have examined the differences between variance, average deviation, and standard deviation, but have invariably limited error to like variables. A problem occurs, given previous understanding, when multiple sources of error affect a single value. In order to contend with this, calculus is necessary, specifically partial derivatives: We single out each variable in a formula and take their derivatives with respects to the function assuming all other variables are constants, and we multiple each derivative by their respect errors, summing them up in the end.

So, assuming function p(x₁, x₂, ..., x_n):

dp = | (∂p/∂x₁) | dx₁ + | (∂p/∂x₂) | dx₂ + ... + | (∂p/∂x_n) | dx_n

In some cases, this could be tedious to compute, so we have a couple ways to acceptably simplify this. One way is to take ln of both sides and separate complex terms using logarithm rules. The second way is to divide both sides by the original function p, which essentially cancels out all the constants in each term.

We practice propagating uncertainty by calculating something of multiple measurements: the density of various metals. We have 3 small cylinders of metals: steel, copper, and brass, that will act as our objects to be measured.

In order to find density, we need their masses and volumes, and consequently their heights and diameters (because we cannot directly measure radius), applying the formula:

ρ = m / v
ρ = m / [π r² h]
ρ = m / [π (d² / 4) h]
ρ = 4 m / [π d² h]

Measuring the weight is easy. We use a scale to measure the normal force, and spit a value back at us in grams.

We use a caliper to measure the height and diameter. A caliper consists of a wrench-like clasp which takes the object. It has little marks on it, like a ruler, to tell the dimensions in mm, and then then a different set of marks which magically gets tenths of mm accuracy. The first mark from the second set from the left that closely matches a mark on the first set is the number for that figure.

Got all the measurements! (Note: We've since updated some of these to make them more accurate. We read the caliper wrong the first time.)

We'll first find the density of each metal, then propagate their uncertainties. Exciting!

ρ = 4 m / [π d² h]

ρ_steel = 4 (49 g) / [π (1.26)² cm² (5) cm]

ρ_steel = 196 g / 24.938 cm³

ρ_steel = 7.86 g/cm³

ρ_copper = 4 (58.4 g) / [π (1.28)² cm² (5.14) cm]

ρ_copper = 233.6 g / 26.457 cm³

ρ_copper = 8.83 g/cm³

ρ_brass = 4 (80 g) / [π (1.60)² cm² (4.80) cm]

ρ_brass = 320 g / 38.604 cm³

ρ_brass = 8.29 g/cm³

Comparing to ideal densities, according to Wikipedia (http://en.wikipedia.org/wiki/Copper), the density of copper is 8.96 g/cm³. Our value is off by 1.5%. Taking into account measurement errors, this isn't too bad--it's within range of variation. Brass, on the other hand, is an alloy of copper and zinc, making it impossible to find an exact density since we don't know the composition of our cylinder, but a good average seems to be 8.55 g/cm³. This gives us a 3% error. An average density of steel, also an alloy, is 7.85 g/cm³ (http://en.wikipedia.org/wiki/Steel). The error in this case is just over 0.1%.

Our numbers are relatively accurate, but let's propagate the uncertainty and see how reliable our numbers are without comparing them to other peoples' experiments. First, we simplify as much as possible:

dρ = | (∂p/∂m) | dm + | (∂p/∂d) | dd + | (∂p/∂h) | dh
dρ = | 4 / (π d² h) | dm + | -8 m / (π d³ h) | dd + | -4 m / (π d² h²) | dh
dρ / ρ = (dm / m) + (2 dd / d) + (dh / h)

Notice that by dividing our uncertainty by the density ρ, we end up with relative uncertainty. We can get our absolute uncertainty back again by multiplying it by ρ. This is much easier to do since we have already solved for density! So it turns out that:

dρ_steel / ρ_steel = (0.1 g / 49 g) + (2 * 0.01 cm / 1.26 cm) + (0.01 cm / 5.00 cm)
dρ_steel / ρ_steel = (0.002041) + (0.015873) + (0.02)
dρ_steel / ρ_steel = 0.038 = 3.8%
dρ_steel = 0.30 g/cm³

dρ_copper / ρ_copper = (0.1 g / 58.4 g) + (2 * 0.01 cm / 1.28 cm) + (0.01 cm / 5.14 cm)
dρ_copper / ρ_copper = (0.001712) + (0.015625) + (0.001946)
dρ_copper / ρ_copper = 0.019 = 1.9%
dρ_copper = 0.17 g/cm³

dρ_brass / ρ_brass = (0.1 g / 80 g) + (2 * 0.01 cm / 1.60 cm) + (0.01 cm / 4.80 cm)
dρ_brass / ρ_brass = (0.00125) + (0.0125) + (0.002083)
dρ_brass / ρ_brass = 0.016 = 1.6%
dρ_brass = 0.13 g/cm³

Our density with uncertainty is:

ρ_steel = (7.86 ± 0.30) g/cm³ (3.8%)

ρ_copper = (8.83 ± 0.17) g/cm³ (1.9%)

ρ_brass = (8.29 ± 0.13) g/cm³ (1.6%)

For further practice, we will take measurements to solve for an unknown mass, and express it with uncertainty.

Two spring scales are hung from long metal rods, carrying a single weight. The spring scales use displacement to measure the force exerted by the mass m on each rod, and also the tension of the strings connecting the mass. These were set up around our lab room; we picked the nearest station #8. θ₁ and θ₂ represent the angles under the strings and a horizontal line perpendicular to where m is hanged.

We use cell phone app Clinometer to measure the angles of the strings by tilting the cellphone on top of it. The app measures a different angle depending on the side the phone is turned, so we use an analog leveler to correlate. Clinometer measures to within 0.1º, but we can only be certain of ± 1º since the meter falls between 2 lines on the leveler.

And of course, we eyeball the tension forces from the spring scales, however glass distortion makes it hard to get a clear reading, so we can only say that we are confident within ± 0.5 N.

Our measurements are as follows:

F₁ = (7.5 ± 0.5) N
F₂ = (7 ± 0.5) N
θ₁ = 44º
θ₂ = 40º

In order to calculate for the mass, we apply Newton's 2nd Law, summing up the vertical components of the forces, then dividing by the gravity constant:

F_{y TOT} = m g
m = F_{y TOT} / g
m = F_TOT sinθ / g
m = (F₁ sinθ₁ + F₂ sinθ₁ ) / g
m = (7.5 sin44º + 7 sin40º) m kg/s² / 9.8 m/s²
m = 9.71 m kg/s² / 9.8 m/s²
m = 0.99 kg

To propagate the uncertainty, we must be careful to change the units for dθ into radians. The remaining process is the same:

dm = | (∂m/∂F₁) | dF₁ + | (∂m/∂F₂) | dF₂ + | (∂m/∂θ₁) | dθ₁ + | (∂m/∂θ₂) | dθ₂
dm = | sinθ₁ / g | dF₁ + | (sinθ₂ / g) | dF₂ + | F₁ cosθ₁ / g | dθ₁ + | F₂ cosθ₂ / g | dθ₂
dm = [(sin44 + sin40) m kg/s² / 9.8 m/s²] 0.5 + [(7.5 cos44 + 7 cos40) m kg/s² / 9.8 m/s²] 0.017453
dm = [ 0.136474 kg ] 0.5 + [ 1.097690 kg ] 0.017453
dm = 0.087 kg

If we divide the uncertainty by the mass, then we get a little more than 8.8%. Given our relatively uncertain measurements, this isn't too surprising. We express our final result as (0.99 ± 0.087) kg. With this, we have concluded our lab on how to propagate uncertainty. We have seen that loose measurements lead to larger relative uncertainty, and vice versa. We have also considered various ways in which we could simplify the calculation, and how they may not be beneficial in every instance.