Measuring the density of metal cylinders, determining the unknown mass of an object, and examining the propagated uncertainty in measurements.
The Set-Up:
Part 1: Measuring the density of metal cylinders
We were asked to measure the density of three objects made in order of aluminum, steel, and copper. These three objects have a shape of cylinder which means that to calculate the density, we would collect the data of diameter, height, and mass. The vernier calipers and a micrometer were introduced to help us measure the diameter and height. The small balance was also given to measure the mass of three objects. These three devices are shown in the figure 1 and 2.
Figure 1: The micrometer used to measure the height and diameter of three objects.
Figure 2: The small balance used to measure the mass of three objects.
After measuring the height, diameter, and mass of three objects, we recorded the data and put them on the white board (seen in Figure 3).
Figure 3: The data of three objects including the mass, height, and diameter.
When we were done with recording the data, we set up equation to calculate the density as well as the propagated uncertainty in measurement. As we knew these three objects had the shape of cylinders, thus the equation to calculate the density would be:
p=(4/∏)* [m/(d2*h)]
Then we set up an equation to calculate the propagated uncertainty in measurements:
dp= (∂p/∂m)*dm + (∂p/∂d)*dd + (∂p/∂h)dh
To calculate the density of three objects, we just plugged in the data and did algebraically. However, to calculate the propagated uncertainty (dp), we first solved for ∂p/∂m, ∂p/∂d, and ∂p/∂h. Also, as we measured the data before, we also noticed the uncertainty of each device.
dm = ±0.1g , dh = ±0.01cm, dd = ±0.01cm
The next step is to perform calculation. Figure 4 is the result for the density and uncertainty of aluminum, figure 5 is that of steel, and figure 6 is that of copper.
Figure 4: The density and propagated uncertainty of aluminum.
Figure 5: The density and propagated uncertainty of steel
Figure 6: The density and propagated uncertainty of copper.
The Results:
The results we got for the density of aluminum, steel, and copper in order is pAl = 2.69g/cm3 ,
pFe = 8.63g/cm3 , pCu = 9.21g/cm3. The uncertainty in measurement of aluminum, steel, and copper in order is dpAl = -0.05 , dpFe = -0.12 , dpCu=-0.15 . We then compared our results with the accepted value to examine how far we were off. The accepted value for density of aluminum, steel, and copper in order is pAl = 2.79g/cm3 , pFe = 7.86g/cm3 , pCu = 8.96g/cm3. We noticed that for aluminum and copper, our results were close to the accepted value. However, for steel, our result were much higher than the accepted value. The difference between experimental value and accepted value were explained by the propagated uncertainty in measurements as well as human error in measurements.
Part 2: Determination of an unknown mass
The Set-Up:
First of all, we obtained two spring scales: one was 5N and the other was 10N. Adjusting the spring scale to make sure that it read 0 when nothing hung on it, and read to appropriate value when a known weight was hung on it. We set up two clamps onto the edge of a lab table holding two long sticks with a long rod in each. Then we suspended the two spring scales at asymmetric angles and hang the unknown mass on them. We would run two trials with different set of angles and a different hanging mass. Below is the apparatus of the experiment (Figure 7).
Figure 7: The apparatus of the experiment. Two clamps attach to the lab table holding two long sticks with a long rod in each. An unknown mass is hung on the rod.
Data Collecting:
In order to determine the unknown mass of the object, we need to measure the angles and force on the spring scale. We did the experiment two times, and below was our data (Figure 8).
Figure 8: Collecting the data in two different set-ups.
Data Analysis:
Our objective is to figure out the unknown mass of the object. To do so, we needed to relate the unknown mass to our known values including the forces and the angles. We broke forces into two components: Fx and Fy. The mass of the object was related to the component Fy, thus we would set up an equation to show the relationship between Fy and mass. As we got the equation, we would take partial derivative to find the propagated uncertainty in measurements (seen in Figure 9).
Figure 9: Establishing two equations to solve for the unknown mass of object and the propagated uncertainty in measurements.
Since we already had the data, we just plugged into two equations we came up with to solve the problem. For the first set-up, our results are shown in the figure 10 and 11.
 |
| Figure 10: Solving ∂m/∂F1, ∂m/∂F2, ∂m/∂⦶1, ∂m/∂⦶2 |
 |
| Figure 11: The mass and uncertainty for the first trial |
For the second trial, our results are shown in the figure 12 and 13.
 | |
|
 |
| Figure 13: The mass and uncertainty for the 2nd trial. |
The Results:
The result we got for the first trial for the mass and uncertainty in order is m=0.9490kg and dm=0.1656kg. For the second trial, the mass and uncertainty of the object in order was m=0.7928kg and dm=0.1307kg.
Conclusion:
In this lab, we performed two experiments: one was to calculate the density and uncertainty of metal cylinders and the other one was to determine the unknown mass of the object. We were introduced to the concept of partial derivative, thus we used it to calculated the propagated uncertainty in measurements. We also reviewed the formula to calculate the density of objects having the shape of cylinder. Last but not least, we determined the unknown mass based on the relationship to the known values including the angles and forces.
No comments:
Post a Comment