4-Mar-2015: Calculating density

PURPOSE
The purpose of this experiment was to find unknown quantities by calculating them from measured values and to apply our knowledge on finding propagated uncertainties.

PROCEDURES

Figure 1: Scale used to measure mass

Figure 2: Caliper used to measure diameters and heights

We began the first part of this experiment by measuring the dimensions of three cylinders manufactured from three different materials. We measured their masses by using a scale shown in Figure 1. Then, we measured their diameters and heights by using a caliper shown in Figure 2. The measured values are listed below in Figure 3. We listed each material according to their colors.

Figure 3: Measured dimensions of the cylinders

From these measured values, we first found the volume of each cylinder by multiplying the area of its base (area of base = πr², where r = d/2) by its height (h). We then calculated the densities of each cylinder by dividing its mass by its volume. The mathematical process is shown below in Figures 4, 5 and 6.

Figure 4: Calculations of density and propagated error for light gray cylinder

Figure 5: Calculations of density and propagated error for dark gray cylinder

Figure 6: Calculations of density and propagated error of bronze cylinder

Also included in Figures 4,5, and 6, are the propagated uncertainties of the calculated densities. The propagated uncertainties (dρ) were found by taking the partial derivative of the density equation with respect to each of the three variables used in the equation (m,d, and h) and adding them together. For a more detailed explanation on how to find the propagated uncertainties, refer to the lab handout for links to tutorials on propagated uncertainties.

Figure 7: Hanging mass of unknown value

Figure 8: Measuring the angles of each string

Figure 9: Measuring the tension in each string

For the second part of our experiment, we attempted to find the mass of an object by using Newton's second law of motion. First, the object was hung on a string that split into two (Figure 7), which can be treated as such in our calculations . Then, we measured the angle that each string made with the horizontal (Figure 8). In addition, we measured the tension in each string (Figure 9).

Figure 10: Calculation of hanging mass through Newton's second law of motion

From the angles and tensions, we calculated the weight the object by using the first equation illustrated above in Figure 10, underlined in black; the mass was found by dividing each side of the equation by the acceleration due to gravity (g). We formulated this equation from applying Newton's second law of motion in the y-direction. The propagated uncertainty of this calculated mass was found by using the second equation shown in Figure 10, underlined in purple.

CONCLUSION

This lab was helpful in strengthening our ability to solve for propagated uncertainties by applying our theoretical knowledge to the real-world. Furthermore, it showed us that when taking multiple measurements in a single experiment, the room for error increases by a significant amount. It was interesting to find the experimental densities of the three metals and compare them to the actual values. We learned that the light gray cylinder was made of aluminum. We found that the light gray material had a density of 2.745 g/cm^3 with a propagated uncertainty of 0.057 g/cm^3. Since aluminum's density is 2.700 g/cm^3, our results appear to be very accurate. Moreover, the dark gray material had a density of 7.678 g/cm^3 with a propagated uncertainty of 0.135 g/cm^3. This particular cylinder turned out to be iron, which has a density of 7.87 g/cm^3. Finally, we found bronze-colored metal's density to be 8.952 g/cm^3 with a propagated uncertainty of 0.174 g/cm^3. This cylinder was made up of copper, which has a density of 8.96 g/cm^3. This was the closest of the three measurements we made. Based on our very accurate calculations, we believe that this experiment was successful.