9-Mar-2015. Propagated Uncertainty in Measurements.

Part1: PreLab: Measuring the Density of Metal Cylinders 

Using Density = Mass / Volume, we will calculate the density of three metal cylinders, one aluminum, one steel, and one copper. The mass scale will have an uncertainty of 0.01 since that is how far it measures. We will also used a Caliper ruler that has a measurement capability of 0.01; therefore 0.01 is the uncertainty factor as well. 

Diameter of Aluminum

Diameter of Steel 

Diameter of Copper

Height of Aluminum

Height of Copper

Height of Steel 

Mass of Aluminum 

Mass of Steel

Mass of Copper

Using these values of diameter, height, and mass, we calculated the density deriving a formula from p=m/V.

Derived density formula

We also calculated the propagated error in each of our density measurements. First we recognized the uncertainty of each measuring tool, adding or subtracting the last tenth of the tool's measuring ability. We then calculated the actual range in error by taking partial derivatives of each variable to find how much each level of uncertainty would actually affect the resulting density.

The partial derivatives of each variable.

These calculations represent the derived densities and their uncertainty deviation.

The calculations for finding density of each cylinder

The real density of each are represented in this table:

(Values acquired from Honors of Physical Science)

We determined that although the measurements are not within the experimental uncertainty of the accepted values, they were pretty accurate. 

Steel's experimental density was 7.75g/cm^3 which had a 0.6% error when compared to the accepted value.

Copper's experimental density was 9.22g/cm^3 which had a 3% error when compared to the accepted value. 

Aluminum's experimental density was 2.74g/cm^3 which had 1% error when compared to the accpeted value. 

Part 2: Determination of an unknown mass

In this experiment, there is a bottle hanging from two strings with different tensions of Force on both sides holding an unknown mass for two masses. Our goal is find the mass of each of the two unknowns. This is a statics problem since the mass is in equilibrium, demonstrated in the picture below. 

Picture of Unknown 1

We then set the vertical forces equal to each other to find the variables necessary in finding the mass. 

The variables included force one, force two, theta one, and theta two

We then found each variable using a spring that measured, in Newtons, the force of tension on the string and another tool to find theta. The spring had an uncertainty of 0.5 N and the protractor had a 2 degree uncertainty or an uncertainty of 0.1pie radians. 

Protractor

Spring

We then found the values to each variable of Unknown 1 and Unknown 7. 

Measured Variables

We then used these values to calculate the mass of each unknown and the uncertainty by taking the partial derivatives of each variable from the equation we used to find the masses. 

Mass and mass uncertainty calculations

In the end, the mass of Unknown 1 was experimentally determined to be 0.746 kg plus or minus 0.0902 and the mass of Unknown 7 was experimentally determined to be 0.723 kg plus or minus 0.0879.