3/4 Propagated Uncertainty in Measurements

Purpose: To determine unknown values (such as density or mass) and uncertainties, and then calculate propagated uncertainty of the unknown values.

PART ONE

FIGURE (1)

Procedure:

For the first part of the lab we were given three cylinders and we were to measure the height, diameter, and mass of each one (FIGURE (2)). We measured the height and diameter of each cylinder using a caliper (FIGURE (1)).

Once we found the height, diameter, and mass we could find the volume and density (FIGURE (2)).

FIGURE (2)

After that, it was time to find the propagated uncertainty. First we needed to obtain an equation for density in terms of mass, height, and diameter. The equation we came up with is :

 

As shown in FIGURE (3), we then set up the equation to obtain uncertainty for density using partial derivatives and the uncertainties in mass, diameter, and height. 

FIGURE (3)

Once we obtain the partial derivatives we were able to start plugging in our measured values of mass, diameter, and height (FIGURE (4)).

FIGURE (4)

Conclusion:

For the first cylinder, we calculated a density value of 2.9114 g/cm^3  ± .038. The actual value of Aluminum is 2.7 g/cm^3, which is not within our calculation of uncertainty. This tells us that we made an error somewhere, most likely in one of the measurements. 

For the second cylinder, we calculated a density value of 9.187 g/cm^3 ± .181. The actual value of Copper is 8.96 g/cm^3, which again is not within our calculation of uncertainty. Again, this probably means we made an error in a measurement,

For the third cylinder, we calculated a density value of 7.78 g/cm^3 ± .1375. The actual value of Iron is 7.87 g/cm^3, which is within our calculation of uncertainty. Yay!

PART TWO

Procedure:

For the second part of the lab we were determining an unknown mass and the propagated uncertainty of that mass. The mass we were calculating was hanging from to spring scales at different angels (FIGURE (1) & (2)).We measured the force of each spring scale and the angle. We did this with two different masses. 

FIGURE (1)

FIGURE (2)

  After we took all of the measurements, we set up our equations (FIGURE (3)).

FIGURE (3)

This gave us the equation to find the mass and also our equation to derive so we could find our uncertainty. All that was left was plugging in our data (FIGURE (4) & (5)):

FIGURE (4)

FIGURE (5)

Conclusion:
It looks like our data makes sense. The uncertainties are not that high and each mass is a reasonable number. Although, there is still room for error. When we were taking our data some of the measurements were hard for us to read, which leads to inaccuracy and therefore systematic errors.We also used different types/brands of spring scales. This could mean that the scales are calibrated differently and would also give us inaccuracy on our measurements.  Of course there also could have been random errors, due to the equipment we were using. In order to check and make sure that our data makes sense, we could compare it to other classmates data and see if we are in each others range of uncertainty.