phys4as15 bkelder: Lab 6 - March 11, 2015 - Propagated Uncertainty in Measurements

Lab 6:
Brandon Elder
3/11/2015

Propagated Uncertainty in Measurements

Purpose: To learn how to calculate the propagated error in each of our density measurements of three pieces of metal. An additional purpose is to calculate the mass of two unknown objects hanging from spring scales and to determine the propagated uncertainty in the calculated value of the mass.

Fig. 1 Calipers used to measure the length and diameter
of all three masses.

Gathering Data: Obtain a pair of calipers (see Fig. 1) a box of masses. Choose three of the masses and measure the height and diameter of all three. Put together the weights with the measurements in a table (see Fig. 2). After all the measurements are taken, weigh the objects on a scale to get the mass.

Fig. 2 Graph of three different masses. Aluminum, Copper,
and Lead.

Fig. 3 Formula for the
volume of a cylinder.

Fig. 4 Formula used to calculate
density. (D=M/V)

Calculations:

Volume: First, calculate the volume for all three objects. Volume of a cylinder is: (pi * radius squared * height) (see Fig. 3). All calculations were recorded on our white board (see Fig. 2).

Density: The formula for calculating density is (mass / volume) (see Fig. 4).

Fig. 5 Known uncertainties in measurements. the uncertainty
in these calipers is .1 mm because the measurement will
always be between .1 mm increments.

Ok, that was the easy part. Now we have a table of data that has some numbers on it. But we must ask ourselves, how certain are we that these numbers are accurate? Anytime that a measurement is taken there will be some degree of uncertainty. For example, when we take a measurement, the object being measured will most likely fall between two divisions on our scale. The reading will come down to a judgement call on the user's end about which division the object is closer to and this is the uncertainty that is always present in any measurement (see Fig. 5). These uncertainties are called known uncertainties. It is a combination of known uncertainties in measurements that leads to unknown uncertainty in the final result.

Fig. 6 In the top of the picture we see the formula used to calculate density.
Underneath it we see the formula that will be used to calculate "dp" the
uncertainty in density.

Propagated Uncertainty: Propagate means to spread out. When we spread out known uncertainties we create unknown uncertainty. Uncertainty in density calculations comes from three different calculations, the mass, diameter, and the height calculations. We must create an equation for density that we can use to calculate the amount of uncertainty in our density measurement. We know that we have three uncertainties, m, l, and h. We also know that density = m/v. We need an equation that has all of our uncertainties in it. If we substitute the formula used to calculate the volume, we will have density = m/(pi*(d/2)^2*h. This simplifies to (4/pi)*(m/(d^2*h) (See Fig. 6).

Take the derivative of the function for each different variable, m, d, and h, respectively. Then, plug the numbers into the new equations and this will give you the "dp/dm", "dp/dd", and "dp/dh" portions of the equation (middle of Fig. 6). The formulas used to calculate these are in Fig. 7.

Fig. 7 Formulas used from the partial derivatives.

The known uncertainty in mass, diameter, and height are taken form the devices used to measure the numbers. The mass is .1 grams, the diameter is .1 cm, and the height is .01 cm (See Fig. 7a). See Fig. 8 and Fig. 9 for the calculations of all three object's unknown, propagated uncertainties. As you can see from our measurements, the calculated densities plus the propagated uncertainties are all within the ranges of the known values of their densities.

Fig. 7a Entire calculation with all formulas for propagated uncertainties.

Fig. 8 Aluminum and Copper Density with uncertainties.

Fig. 9 Lead's density with propagated uncertainty calculated.

Determination of an Unknown Mass:

Next, we are going to apply our knowledge of uncertainty to an unknown mass hanging from a pair of spring scales (see Fig. 10). We will need to measure the angles and record the spring scale readings (see Fig. 11). Then, we used the measured values to determine the mass of the unknowns. The formula used to determine the mass is written down in Fig 11a. The partial derivatives were taken (See Fig. 12) to calculate the propagated error for each variable: F1, F2, theta1 and theta2. Fig. 13 and 14 has the calculations completed for both unknown masses and the amount of uncertain error.