Physics 4A (9/07/16 )
Propagation of Error
Brett Mccausland
Lab Partner: Olivia Luphan
Purpose
Calculate propagation of error in a experimental calculation (density), and draw conclusions to the accuracy of are calculated propagation of error by seeing if the true density falls within the range of error for each calculation.
Procedure
We used a clinometer,2 cylinders, and a scale as seen in the image below.
With the use of the clinometer we measured the height and diameter of the cylinder.With the use of the scale we measure the mass of each cylinder.
Calculations
The clinometer had an precision of (+/-)0.01 cm and the scale had a precision of (+/-)0.1 g. Are goal was to find the density of the cylinders so we used the formula
and to find the propagation of error of are experimental calculation given the precision of are instruments.In order to do this me needed to take the partial derivative of are original equation for row for each of are uncertain values . Since the volume of a cylinder is 
we plugged that v in for are formula for density and we took the partial derivative for d , h and m. Once we had are partial derivatives we then needed to plug in are values and multiply the partial derivatives by their corresponding measured uncertainty, square them , add them all together, and take their square root.
and to find the propagation of error of are experimental calculation given the precision of are instruments.In order to do this me needed to take the partial derivative of are original equation for row for each of are uncertain values . Since the volume of a cylinder is we plugged that v in for are formula for density and we took the partial derivative for d , h and m. Once we had are partial derivatives we then needed to plug in are values and multiply the partial derivatives by their corresponding measured uncertainty, square them , add them all together, and take their square root.
Conclusion and Data Analysis
As you can see calculated a density of are aluminum cylinder of 2.84 g/cm cubed with a propagation of error of (+/-) 0.02. The actual density of aluminum is 2.70 g/cm cubed which means that the value does not fall within are predicted range of error. We calculated a density for the brass cylinder of 8.17 g/cm cubed with a propagation of error of (+/-) 0.14.The reason being for the brass cylinder having a less precise calculation is that it was much smaller in volume, this shrink in size of the measurement means that the inaccuracy of the instrument is a larger percentage of the entire measurement being made. The actual density of brass is 8.55 g/cm cubed which means it also is not within the range of error that should be occurring. Since the first cylinder is off by +0.14 and the second cylinder is off by -0.28 the error appears to be random. The brass cylinder produced a error of 4.4% and the aluminum cylinder produced a error of 5.1%.There are many unknown as far as what could have gone wrong in the data however the calculations stand correct. The one area I see to be the most probable for producing the error was in the scale, since we never confirmed for ourselves that it was working at the precision it was supposed to be, and it had the lowest precision to begin with. Also when it comes to measuring the diameters and heights each of the group members took turns taking measurements and reading the instrument and it is possible that the instrument moved during a handoff and an incorrect value was recorded or that it was not being read correctly since the markings on the instrument are near microscopic.Had one person done all the measurements I think it would have either eliminated the error or made the error systematic instead of random.
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