Purpose: The purpose of this lab was understand the meaning of propagated uncertainty and applying the idea by comparing measurements and results from the experiment with accepted values from the scientific community.
Procedure: This was a two part lab, the first part being calculating the propagated uncertainty in our measurements of density of metal cylinders. The second one being the determination of two unknown masses. But before delving into the two experiments, we must first understand the term propagated uncertainty. Propagated uncertainty is how uncertainty in measurements leads to uncertainty in final result. Basically, the more unsure or unspecific you are about your measurements, the even more uncertain you will be about your result. It is sort of a compounded effect of error leading to a sort of range for the possibility of an outcome. Its pretty similar to the idea of significant figures because the name of the game is accuracy.
Part 1: For the first part, we were given three metal cylinders and were asked to use a calipers to measure the cylinders' height and diameter, and a scale to measure density. A caliper is a similar to a ruler but more precise. it can clamp the ends of the object you are measuring or go inside to measure things such as the inside of a tube. The vernier caliper measures down to 1/100th of a cm so the uncertainty in our measurements is just that. The electronic scale we used measured down to the tenth of a gram so that is the level of uncertainty for our mass measurements.
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Lab equipement(scale, masses, caliper) |
We created a data chart showing the height, diameter, and mass of the three metals iron, zinc, and copper.
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Data table of measurements of metal cylinders |
We used the formula density = (4*m)/(pi*diameter^2*height) to find the densities of our metals. We then needed to calculate the range of result by finding our propagated uncertainty. In order to find the propagated uncertainty of our density, we had to find the derivative for each variable within the equation of density first. Then the propagated uncertainty for our density dp = | dp/dm |*dm + | dp/dh |*dh + | dp/dd |*dd. where dm, dh, and dd are the propagated uncertainty of the mass, height, and diameters.
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propagated uncertainty of Fe |
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Propagated uncertainty of Zn |
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Propagated uncertainty of Cu |
Result/Analysis: The actual accepted densities are here. Fe is 7.874 c/cm^3, Zn is 7.14 g/cm^3, and Cu is 8.92 g/cm^3. So we were not within the range for any of the three metals, although the ranges were not that wide. This shows that even with very precise tools, it was difficult for us to get precise measurements.
Conclusion/Uncertainty: The first part of the experiment showed that we as humans are not perfect in measuring objects and that it is important to set a range for uncertainty in order to maintain integrity in all of our experiments. Without having that integrity to say when there is a certain uncertainty in our calculations, our evidence is not strong. In order for our experiment and results to be part of what the scientific community can call evidence we must be accurate in our measuring and experiments.
Part 2: For the second part of the lab, we were asked to determine the mass of two unknown objects.
We had an unknown mass hang on a string that was tied to two separate spring scales. these spring scales, 10N, had to be adjusted before putting the masses on so it would read 0 with no mass hanging. The mass hangs down and we made sure that the springs were held at asymmetrical angles as shown in the lab.
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Lab Apparatus with scales and hanging mass |
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tool that measures angles |
We measured the angles of the string and recorded the readings on the white board. We then estimated the uncertainty in our angle readings and scale readings. We used the measured values to determine the mass of the unknown mass.
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measurements of readings |
Using the formula for forces we calculated the mass with our measurements and also the derivatives in respect to each variable.
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calculation for derivatives of forces and angles |
Plugging in the values, we were able to solve for what the mass should be with the range of propagated uncertainty.
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propagated uncertainty for first unknown mass
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Results/Analysis: Just like the first part of the lab, the second showed that it hard to hit within the range of the propagated error, even with careful measuring techniques. Our calculations of the mass using our values proved that we were off a little from the actual measured mass using a scale. It is also understood that the equipment we used is also not 100% reliable.
Conclusion/Uncertainty: The final conclusion through this two part lab showed just how important it is to solve with a propagated uncertainty to insure quality of our data. It was very hard with our experience and our tools to perfectly match the accepted values but the gap was also not too far off. Just enough that it was outside the range of the propagated uncertainty. Again, the importance of a lab like this is to show just how skewed a final result can be from a few hundredths of a cm or N in prior measurements. Also how our experiments mean nothing to other scholars if we don't follow strict guidelines to insure to quality of our data. The integrity of our experiments determines the quality of our results and how accurate they really are.