Eric Chong
Lab 6: Propagated Uncertainty in Measurements
Lab Partners: Lynel Ornedo and Harvey Thai
Purpose: The goal of this lab is to practice and get acquainted the use of calipers and calculated uncertainty. Calipers are different than what we are used to, so this is our introduction to such an equipment.
Introduction: Up until now, we have never used calipers. This short lab is a means to help us know how to read this precise instrument. Calipers have two scales built into it. The first scale is the main scale, and is read normally like any other scale such as rulers. There is also a smaller second scale, which is slightly more complicated to read, but is not too hard once we knew how to read it. This smaller scale is a means to get precise measurements in the smaller decimal places such as 0.02 cm. The lab contains two different cylinders of different types of material, and we must measure the density of both of them with calculated uncertainty. Calculated uncertainty is essentially carrying over the uncertainty we gained from raw measurements into our results, since whatever we measure cannot be better or worse than what we get out of it.
Procedure: First off, we prepared the calipers and the aluminum and zinc cylinders.
We started with the aluminum cylinder and measured the diameter of its base with the calipers. To do so, we placed the base of the cylinder in the clamp and tightened it until it held the cylinder in place. Then, we read the main measurement normally by looking at where the first of the bottom markings lined up with the main markings. Then, we looked at where the subsequent small markings first lines up with one of the markings. This is important because you may notice that some of the markings do not line up with the main markings as if the calipers were built wrong, and this is intentional. The first instance the small markings line up with the main markings, aside from the very first marking, indicates the hundredths place of the measurement for the diameter. The measurement we got for the diameter of the aluminum cylinder is 1.26 cm. We did the same for its height and got 5.09 cm. Finally, in order to get its density, we need its mass, so we put it on a balance and measured 18.32 g. I will talk more in-depth about uncertainty when I get to the calculations.
The process is the same with the zinc cylinder, and we got 1.32 cm. for its diameter, 3.28 cm. for its height, and 28.90 g for its mass. Here is a table with all of the measurements:
We put plus or minus 0.01 next to our numbers because that is the decimal place that we are least certain of. Since we are pretty much eyeing up the small secondary markings of the calipers, we are uncertain of whether that is the exact measurement. Thus, we have to put the 0.01 in order to indicate that that is where our uncertainty lies.
Next, we manipulated the density equation in order to give us something that is easier to use with our measurements. In order to calculate the uncertainty, we have to do partial derivatives for each variable and put it in its "square root form" in order to give us the uncertainty of the result. Here is what the calculations look like.
Finally, this becomes a matter of just plugging in numbers for each cylinder. Here is the calculation for the aluminum cylinder:
And here is the calculation for the zinc cylinder:
As such, the density of the aluminum and zinc cylinder are 2.887 +/- 0.046 g/cm^3, and 6.439 +/- 0.0995 g/cm^3, respectively. We left the uncertainty of the zinc calculation as 0.0995 and not 0.1 because we wanted to indicate where the uncertainty is. Writing the rounded 0.1 as 0.100 does not help as well, so we made a decision to leave the uncertainty as 0.0995, though that means we probably should have increased the decimal place of the zinc density by one in order to match the uncertainty. Nonetheless, the process is what matters and we now know how to calculate the uncertainty while learning how to use calipers.
Conclusion: Overall, this short lab about calculating the density of the cylinders demonstrates the use of calipers and how we deal with uncertainty in our measurements. We first take the measurement, look at the decimal place that is most uncertain and we calculate the uncertainty based on that using partial derivatives in the square root form. This way, we can look upon measurements and see how precise we are.
Introduction: Up until now, we have never used calipers. This short lab is a means to help us know how to read this precise instrument. Calipers have two scales built into it. The first scale is the main scale, and is read normally like any other scale such as rulers. There is also a smaller second scale, which is slightly more complicated to read, but is not too hard once we knew how to read it. This smaller scale is a means to get precise measurements in the smaller decimal places such as 0.02 cm. The lab contains two different cylinders of different types of material, and we must measure the density of both of them with calculated uncertainty. Calculated uncertainty is essentially carrying over the uncertainty we gained from raw measurements into our results, since whatever we measure cannot be better or worse than what we get out of it.
Procedure: First off, we prepared the calipers and the aluminum and zinc cylinders.
We started with the aluminum cylinder and measured the diameter of its base with the calipers. To do so, we placed the base of the cylinder in the clamp and tightened it until it held the cylinder in place. Then, we read the main measurement normally by looking at where the first of the bottom markings lined up with the main markings. Then, we looked at where the subsequent small markings first lines up with one of the markings. This is important because you may notice that some of the markings do not line up with the main markings as if the calipers were built wrong, and this is intentional. The first instance the small markings line up with the main markings, aside from the very first marking, indicates the hundredths place of the measurement for the diameter. The measurement we got for the diameter of the aluminum cylinder is 1.26 cm. We did the same for its height and got 5.09 cm. Finally, in order to get its density, we need its mass, so we put it on a balance and measured 18.32 g. I will talk more in-depth about uncertainty when I get to the calculations.
The process is the same with the zinc cylinder, and we got 1.32 cm. for its diameter, 3.28 cm. for its height, and 28.90 g for its mass. Here is a table with all of the measurements:
We put plus or minus 0.01 next to our numbers because that is the decimal place that we are least certain of. Since we are pretty much eyeing up the small secondary markings of the calipers, we are uncertain of whether that is the exact measurement. Thus, we have to put the 0.01 in order to indicate that that is where our uncertainty lies.
Next, we manipulated the density equation in order to give us something that is easier to use with our measurements. In order to calculate the uncertainty, we have to do partial derivatives for each variable and put it in its "square root form" in order to give us the uncertainty of the result. Here is what the calculations look like.
Finally, this becomes a matter of just plugging in numbers for each cylinder. Here is the calculation for the aluminum cylinder:
And here is the calculation for the zinc cylinder:
As such, the density of the aluminum and zinc cylinder are 2.887 +/- 0.046 g/cm^3, and 6.439 +/- 0.0995 g/cm^3, respectively. We left the uncertainty of the zinc calculation as 0.0995 and not 0.1 because we wanted to indicate where the uncertainty is. Writing the rounded 0.1 as 0.100 does not help as well, so we made a decision to leave the uncertainty as 0.0995, though that means we probably should have increased the decimal place of the zinc density by one in order to match the uncertainty. Nonetheless, the process is what matters and we now know how to calculate the uncertainty while learning how to use calipers.
Conclusion: Overall, this short lab about calculating the density of the cylinders demonstrates the use of calipers and how we deal with uncertainty in our measurements. We first take the measurement, look at the decimal place that is most uncertain and we calculate the uncertainty based on that using partial derivatives in the square root form. This way, we can look upon measurements and see how precise we are.
No comments:
Post a Comment