March 9, 2015 Propagated Uncertainty in Measurement Lab

Objective: Be familiar with the process of calculating the uncertainty in measurements.

Part 1

Copper (left), Lead (middle), and Silver (right)

In order to get a more accurate measured result, we find the uncertainty by looking at each measurements. In our case today, in order to find the uncertainty in density for each of the cylinder, we need to analyze the mass, the height and the diameter. We use the method of partial derivative to help analyze the uncertainty in each measurement.

Here we have three metal cylinders, we want to calculate their densities and uncertainties. We begin by writing out the volume equation for cylinder:

V= π (d/2)^2h

The volume is for finding the density, which is mass divide by volume. Therefore, when we put both equation together for the density of the cylinder, we have:

Density = m/v=m/[π * (d/2)^2 * h]

The three measurements that we are trying to find the uncertainty are mass, diameter and heights. We firstly use caliper to measure the diameter and height and then use the balance to measure the mass. 

A caliper

Using caliper to measure the height

Using caliper to measure the diameter

The following are the measurements we get.

	Aluminum	Steel	Copper
Mass	21.0 g	73.9 g	56.6 g
Height	4.86 cm	4.78 cm	5.01 cm
Diameter	1.43 cm	1.51 cm	1.25 cm

Then we use partial derivative to calculate the uncertainty.

This is the calculation for aluminum. But the general set up are the same for the rest of the cylinders. We begin the partial derivative process by finding the measurements that need to be partial derived (red box). Then we find the partial derivative for each measurement (green box). This is the same for all three cylinders because they have the same shape, the difference is the magnitude of each measurements. After calculating all three measurements, we multiply them by their uncertainty in measurement (purple box). The mass has an uncertainty value of 0.1 g found by looking at the balance. The diameter and heights both have an uncertainty value of 0.01 cm, found by looking at the caliper. Next, multiply the partial derivative of each measurements by its uncertainty value accordingly. For example, the partial derivative of mass would multiply the uncertainty value of mass. After we multiply and add them together, we get a small value relative to the density, this value is our uncertainty in density. Lastly, putting together the calculated density of the cylinder and the uncertainty value of density for the cylinder, we got a range of density between 2.7459 g/cm^3~2.6341 g/cm^3.  The density of aluminum is 2.7 g/cm^3, it is within our range!

Since the calculation works the same for all three cylinders. Here is the result and its true density.

	Calculated Density and density Uncertainty (g/cm^3)	True Density (g/cm^3)
Aluminum	2.60±0.0559	2.70
Steel	8.63±0.144	7.75-8.05
Copper	9.206±1.82	8.96

Part 2

Two stations with unknown mass are set up like the picture above. The tension force on the two wires and the angles can be measure. Our job is to find the weight of the unknown mass, the red bottle, and the uncertainty in the mass.

General calculation of the set up

We approach the problem by drawing a free body diagram (please refers to the red box).  Then we calculate the force in the x and y direction. The forces in the x direction is travail, it is not vital to our goal of finding the mass. We find the mass by looking at the forces on y. After we have an equation for finding mass, we write out a general equation for partial derivative of force 1, force 2, angle 1, and angle 2, with respect to mass. Then we determine the equation of partial derivative using the equation of mass. This is still a general set up since we have not insert any actual value yet.

The calculation for set-up #1

 Here we have set-up #1 calculation, we simply follow the step. One thing that must pay attention is the uncertainty value because the uncertainty value changes when using different measuring equipment. In this case, we use two spring scales that has an uncertainty value of 1 N, we also use a measuring device for angle, the measuring device is 2 degree apart, that means the uncertainty value for the angle is 2. When calculating, one must remember to change the degree to radian because calculus works in radian, not degree.  Lastly, we have an uncertainty value for mass, we put it together with the mass. Then we have the mass with its uncertainty value!

We repeat the same process for set-up#2.

The calculation for set-up #2

Here we have set-up #2, the forces and angles are different; however, the calculation process is still the same. One thing need to pay attention is that the uncertainty value for the force. The uncertainty value of force is 0.5 N, it is different from set-up #1 because we use different kinds of spring scale. Just as in set-up #1, we multiply the partial derivative with the uncertainty value, then we add them up. Eventually we get an uncertainty value of mass.

Results:
Set-up #1: 0.95 kg± 0.1656 kg

Set-up #2: 0.60 kg±0.14 kg

Summary: 

In this lab, we are trying to find the uncertainty value of a measured object. In part 1, we begin by finding the density of three metal cylinders. We measure their heights, diameters and mass. We come up with an equation describing the relationship between the measurements. We then take the partial derivative of these three measurements. The reason we use partial derivative is that when we take the partial derivative, we find out how the mass is being affect when that particular measurement is changing. After we have partial derivative of all three measurements, we multiply them by the uncertainty value of the measuring device. Lastly we add the product of all three partial derivatives and their uncertainty values. Then we get the uncertainty value of the density.

In part 2, we are trying to find out the mass of the unknown object and its uncertainty value in mass. We have a red bottle hanging on two wires. The wires has tension in it, we take down the measurements. The wires also make two angles with the mass, we measure the angles as well. After having all the data, we draw a free body diagram, sum up the forces on x and y direction, then we find the equation for mass. The equation of mass is use to find the partial derivative of each measurements- force 1, force 2, angle 1, angle 2. Then we multiply the partial derivative with its corresponding uncertainty value. Note that the uncertainty value of force is different in each set up because we use different kind of spring scale. Lastly we have a mass and its uncertainty value!