09-March-2015: Propagated Uncertainty

The purpose of this lab was to determine the uncertainty in measuring the density of certain objects, as well as the uncertainty when calculating other unknown values.

Part 1: Copper, Steel, and Aluminum Masses:
For the first part of this lab, we sought out to find the propagated uncertainty when calculating the density of three different cylindrical masses. The three masses where copper, steel, and aluminum.

Procedure:

Fig. 1
No device is flawless. We had to account for the error in each 
instrument as well as place the appropriate amount of significant
figures within our data table. 

To calculate the density of these solid cylinders, we first needed to find their individual masses and volumes. To do this, we used a set of calipers and a mass scale. However, these instruments would not give us EXACT results. Thus we had to account for the uncertainty that came with each instrument. The calipers had an uncertainty of  0.01 cm while the mass scale had an uncertainty of 0.1 g. We had to account for this when we measured the mass, height, and diameter of each cylinder (see Fig. 1).

Once we had found our values, we could then plug them into our density equation. For density, we substituted volume with the actual equation for the volume of a cylinder (see Fig. 2).

Fig. 2
The equation for density was required to be broken down into the variables we had measured.

Once we had this, we had to find a way to account for the uncertainty in our density calculation. After all, if the values that we input into our equation are uncertain, then the value the equation will yield will also be uncertain. To do this we had to take the derivative of our density equation with respect to three variables (mass, height, and diameter). We did this using partial derivatives (see Fig. 3).

Fig. 3
Since density is in terms of three variables, in order to find the derivative of density, we needed to take the partial
derivative of each variable.

Once we had this, we were able to input our values for each cylinder to find their calculated uncertainty. This value would then be recorded together with our calculated density in order to show the level of uncertainty in our findings. For Copper, we found the density to be 9060 kg/m^3 with an uncertainty of 236 kg/m^3. For Steel, we found the density to be 7.60 x 10^3 kg/m^3 with an uncertainty of 149 kg/m^3. Finally, for aluminum we found the density to be 2740 kg/m^3 with an uncertainty of 50.0 kg/m^3 (see Fig. 4).

Fig. 4
Our final density calculations and uncertainty.

Part 2: Finding the Mass of a Hanging Object.

We again used this tool of calculated propagated uncertainty when calculating the mass of two hanging objects. 

Procedure:

Fig. 5
Thankfully, the system was already set up
the day of the lab. We just need to record
we saw.

There were models set up across the classroom with a mass hanging in the middle of some stretched string (see Fig. 5). Our goal was to calculate the mass of the object on the string for two of the set ups, while still accounting for our uncertainty. To do this, we needed four pieces of information from each set up. First, we needed to know the tension within the two parts of the string. To do this, we measured the force scale set up on each string. Second, we measured the angle at which the tension force was being applied. To do this, we used a type of inclinometer to measure the angles (see Fig. 6). Once we had all of our data, we created a free body diagram, as well as net force equations, for each of the setups (see Fig. 7). We found the general expression for the mass of the object to be m=F(1)sin(θ(1))+F(2)sin(θ(2))/g. Now we could come with our combined uncertainty for mass. 

Fig. 6
To use the inclinometer properly, we need to place its base
parallel to the string whose angle we are measuring.

Fig. 7
Creating the free body diagram gave us a clear
indication of what was occurring within the
system.

The Uncertainty:

To find the uncertainty in our calculation of mass, we once again found the derivative of the mass equation. We again did this by partial derivation of all the variables (see Fig. 8). 

Fig. 8
Once again, in order to find the uncertainty in mass, we needed to take the partial derivative of all the variables.

NOTE: When inputting the uncertainty in our measured angle, we converted the degree to radians.

This is because the calculus portion of this lab is based only in radians.

For our mass #1, we calculated the mass to be 0.846kg with an uncertainty of 0.068kg. For our mass #2, we calculated the mass to be 0.933kg with an uncertainty of 0.060kg (see Fig. 9).

Fig. 9
Final calculations for our mass and uncertainty.