Lab 3: Propagated Uncertainty

           Aaron

Max

Fergusson

 

Theory/Introduction:  The purpose of this lab is to calculate the propagated uncertainty from the calculations of two scenarios:

1. Determining the density of metal cylinders by measuring its height, diameter, and mass.
2. Determining the mass of an object that is supported by strings that are attached to their own individual spring scale.

There are two ways to find propagated uncertainty. One method theorizes the worst case scenario. An example of this formula in terms of the propagated uncertainty formula for density would be:

This equation calculates the maximum possible change that could happen to each variable and adds them all up to find the maximum error possible.However, since this equation covers a wide range of possibilities, we will be using the second method which uses this formula:

This method squares the maximum possible error for each variable, adds them all together, then takes the square root of their sum.

 

Apparatus/Experimental Procedure:

1. We measured the length and width of the two cylinders using a Vernier caliper and we measured their mass by using an electronic balance. Since these both have uncertainties in their measurements, they increase the propagated error.

   

   The vernier caliper let’s us approximate a measurement to a hundredth of a centimeter. It does this by using a separate scale called the vernier scale. The number where the mark on the main scale and the mark on the vernier scale meet (mark 6 on the diagram) will give you the digit in the hundredth place, so the measurement would be .96 cm in the diagram.

   Here are our measurements after using the electronic balance and the vernier caliper:

	Height	Diameter	Mass
Cylinder 1	5.10 cm ± 0.01	1.60 cm ± 0.01	27.80 g ± 0.01
Cylinder 2	3.29 cm ± 0.01	1.30 cm ± 0.01	29.10 g ± 0.01

 

2. The forces that were holding up the weight could be seen on their spring scales and the degrees they were angled at were measured using a protractor.

	Force Pulled	Angle Above Horizontal
Spring Scale 1	5.3N ± 0.1	54° ± 1 0.942 rad ± 0.0175
Spring Scale 2	10.5N ± 0.1	10° ± 1 0.175 rad ± 0.0175

 

Analysis:

Now that we have all the variables we need for density and its propagated uncertainty we plugged them in into the equations:

 

We found Metal 1 = 2.71g/cm^3 ±  .0545 and Metal 2 = 6.66 g/cm^3 ± .134. The same concepts to find the propagated uncertainty for density were applied for the unknown mass.

Using a free-body diagram of the mass we were able to make a equilibrium equation: mg=T1sinθ1 + T2sinθ2

From there we solved for mass and proceeded to get the propagated uncertainty (derivative of mass). Mass = .62 kg ± .021