4-March-2015 Propagated Uncertanity in Measuerments

Purpose: An introduction to error calculations, using known density of metals and comparing them in lab measurements. Since the metals density are known, any deviation when checked with our measurements must be do to the accumulated error, for part one. Part two will be determining the mass of two hanging unknown masses, using force gages and the uncertainty in them and the angles to determine the accumulated error.

Theory: My measurements will be close, 10%, to the expected value, and any error will be predicted from the predicted errors that come with certainty in measurement.

Procedure: We will take three metal cylinders of different types (copper, iron, and aluminum) and sizes, measure their mass, diameter, and height with calipers. Calculate the mass and then calculate the propagated error to determine if they are within the experimental uncertainty.

* We determined the mass, height, and diameter of all three metal cylinders, all measurements with a +/- 0.05 level of certainty.

* We calculated the volume and densities

*Determined that the partial derivative of volume with regards to mass as 4/(Pi*h*d^2)

*Determined that the partial derivative of volume with regards to diameter as -8m/(Pi*h*d^3)

*Determined that the partial derivative of volume with regards to height as -4m/(Pi*h^2*d^2)

*The total uncertainty in the density of each cylinder would be determined by adding all the partial derivatives, multiplied by the original uncertainty (P=partial derivative mass *(uncertainty)+ partial derivative of height*(uncertainty)+ partial derivative of the diameter*(uncertainty)

*For iron we got 6.68 g/cm^3 +/-  0.13, and actual 7.87 g/cm^3, error 13%

*For copper we got 8.94 g/cm^3 +/-  0.18 and the actual  8.96 g/cm^3, error 1.8%

*For Aluminum we got 2.81 g/cm^3 +/- 0.05 and the actual was 2.7 g/cm^3, error 6%

Conclusion

* All but one of the metals fell into my expected error, iron.

*I still believe this is a good method for determining density, as the other two metals were very accurate to the expected density.

*Perhaps my sample wasn't pure iron, as checked the periodic table and no element has a similar density to the one I calculated.

Part 2
Theory: The accumulated error will be accounted for in my calculations.

Procedure: Take measurements of two hanging masses, the angles and force gages, and calculate the propagated error for each

Recordings of Mass 1
 

Mass 1	Unknown
Force Gage 1	5.75
Uncertainty	0.1
Angle 1	8.5
Uncertainty	1
Force Gage 2	9.5
Uncertainty	0.3
Angle 2	54
Uncertainty	1

 *Calculating mass with uncertainty 0.871kg +/- 0.0355

Recordings of Mass 2

Mass 2	Unknown
Force Gage 1	7
Uncertainty	0.1
Angle 1	27
Uncertainty	1
Force Gage 2	7.3
Uncertainty	0.3
Angle 2	40
Uncertainty	1

*Calculating mass with uncertainty 0.8031 +/- 0.0454