3/04/15 Propagated Uncertainty / Unknown Mass Lab 6

Purpose: The objective of part 1 for this lab was to learn how to calculate propagated error in each of our density measurements using calculus and determine if our measurements were within the experimental uncertainty of the accepted values. Part 2 required us to find the mass of an unknown hanging object using tension and angles, and to use the knowledge acquired from part 1 to calculate the uncertainties in our measurements.

Materials:

small aluminum, copper, and iron cylinders.

An electronic balance was used the measure the mass of each aluminum, copper, and iron cylinders

Calipers that allowed us to measure the height and diameter of each cylinder.

Newton spring scales to measure the force of tension created by the hanging mass. And an angle measuring device to measure the angle the string made with the horizontal.









Part 1:

Procedure: Using the electronic balance and calipers, we were able to measure the height, diameter, and mass of each cylinder. By using our values for mass and volume, we calculated the densities of each cylinder as well.

Since our main goal was to calculate propagated error, we needed values for uncertainty in mass, diameter, and height. We then derived an equation for the density of a cylinder and took the derivative with respect to height, mass, and diameter to create an equation for uncertainty in density.

The calculated uncertainty in density for aluminum, copper, and iron

Summary: The accepted values for the density of aluminum, iron, and copper are as follows:
2.7 g/cm^3, 7.874 g/cm^3, and 8.92 g/cm^3. Our experimental values were: 2.9114 +/- .038 g/cm^3, 7.78 +/- .1375 g/cm^3, and 9.187 +/- .181 g/cm^3. 

Part 2

Procedure: A set up was already completed prior to beginning the experiment. We needed to gather measurements for both the spring scale readings and angles the strings made with the horizontal. 

Unknown Mass #7

Unknown Mass #8

A free body diagram and measured values were drawn and gathered as shown. Using Newtons 2nd law an equation was derived using the free body diagram to determine the mass of #8 and #7.

Again as is part 1, we needed values for uncertainty in F1, F2, ɵ1, and ɵ2. We then used our derived equation from Newtons 2nd law and took the derivative with respect to F1, F2, ɵ1, and ɵ2 in order to generate an equation that would give us uncertainty in mass. It was important to note that the uncertainty in F1 and F2 were different for both set ups and to make sure the uncertainty in ɵ1, and ɵ2 were in radians rather than degrees. 

The calculated uncertainty in mass for set up #7 and #8

Summary: In conclusion, the masses of each set up were calculated to be 828.3g (#7) and 918.4g (#8). The uncertainty values were extremely high with #7 being +/- 98.8 g and #8 being +/- 77.6g. This high uncertainty could be due to wrong angle measurements, due to the fact that it was somewhat difficult to measure the angle and not disturb the spring scales and improper spring scale readings. As a group we learned how uncertainty in measurements leads to uncertainty in our final results.