The purpose of this experiment was to become more aware of uncertainty and its importance by calculating the propagated uncertainty in measurements of density and forces on a static system.
Apparatus:
The first portion of the experiment involved finding the densities of three metal cylinders with differing dimensions and different types of metals. A vernier caliper was used to measure the dimensions of the cylinders. A vernier caliper was used instead of a ruler because it is more accurate, decreasing uncertainty in the measurements of dimension. The uncertainty of a caliper is 0.01 cm, whereas the uncertainty of a ruler is 0.1 cm. A vernier caliper, as shown in the figure below, contains an additional 0.01 cm marking. When a measurement is performed with a caliper, it is read by looking for the 0.01 cm marking that best aligns with one of the 0.1 cm markings. That 0.01 cm marking is the hundredths place value for the measurement. This is why the uncertainty in a vernier caliper is one digit less than a ruler.
The vernier caliper works by adjusting the length between its "arms" to fit the dimension of the object that is measured. Doing so gives no uncertainty for the tenths place but only for the hundredths place.
The second portion of the experiment involved determining an unknown hanging mass by measuring the forces that were acting on the mass. The magnetic protractor, shown above, was used to measure the angle of the attached ropes (force vectors) from the horizontal. This apparatus is used by placing it parallel to the slope of the interest, which then gives the reading of the angle of the slope.
Procedure (Part 1):
Figure 4: Measurements of the dimensions and masses of the three cylinders
Using the vernier caliper, the dimensions of each of the above cylinders were measured and recorded. In addition, the masses of these cylinders were measured using a balance that has an uncertainty of 0.1 g. Once the dimensions were measured, the density of each cylinder was calculated. The top cylinder was numbered 1, the middle cylinder 2, and the bottom cylinder 3. Cylinder 1 contains a density of 4.26 g/cm3. The density of cylinder 2 was calculated to be 2.89 g/cm3. Lastly, the density of cylinder 3 was found to be 9.22 g/cm3.
Figure 5: Diagram to solve for the volume and density of a cylinder
The volume of the cylinder was found by squaring half the diameter of the cylinder, followed by multiplying the height of the cylinder and the value of pi, as seen in the above figure. Then, the density was found by taking the measured mass, in grams, and dividing it by the volume of the cylinder, as seen above.
Abstract (Part 1):
Figure 6: Uncertainty in density partial derivative equation
The uncertainty in density was found by taking the partial derivatives of all the variables of density, then multiplying them by the their uncertainty and summing them together (as seen above). Since density is a function of 3 variables: m, d, and h ( where m = mass, h = height, and d = diameter), three partial derivatives each with respect to one variable was made. A partial derivative is taken by treating the variable it is with respect to as a variable and everything else (including other variables) as a constant. Doing so gives you the three partial derivative equations shown in the picture below:
Figure 7: Partial derivatives of density with respect to each of its variables
Using these equations and plugging their values into the equation shown in Figure 6, the uncertainty of density can be found. The uncertainty in the measurement of mass (dm) is 0.1 g, whereas the uncertainties in the measurements of height and diameter (dh and dd, respectively) are 0.01 cm.
Figure 9: The densities and uncertainties in density of cylinders 1 and 2 and finding the uncertainty in density of cylinder 2
Figure 10: Finding the uncertainty in density of cylinder 3 and the values of density and uncertainty in density of cylinder 3.
Using the values of the densities and their uncertainties of the cylinders, the identity of the metal of cylinder 2 was found to be Aluminum (d = 2.70 g/cm^3). The metal that makes up cylinder 3 was identified to be copper (d = 8.96 g/cm^3). The identity of the metal of cylinder 1 was not successfully identified due to having a small density yet a color different than the lighter metals. Some possibilities that could have caused the calculated densities to not be as close to the densities of metals include:
1) The cylinders were alloys (a mixture of two metals) yet were assumed to be only one metal and compared to pure metals
2) Rusting of the metals might have caused their mass to increase, resulting in a larger than actual density.
3) The cylinders might not be completely solid on the inside (i.e. the cylinders have some empty space inside), reducing the mass and therefore obtaining a smaller density.
The uncertainties of the densities ranged from 1% to 2% of the density values, which correlated strongly with the uncertainties in the used instruments.
Procedure (Part 2):
The setups shown below in Figures 11 and 12 were already present before this experiment. The tensions on each of the strings in each setup (displayed by the 10 N spring scales) were measured as well as the angles of the strings from the horizontal. The instrument used for the measurements of the angles was the magnetic protractor shown in Figure 3. The uncertainty in the angle measurements is 2 degrees, and the uncertainty in the tension measurements in 0.5 N.
For the setup of unknown mass # 1, the tension in the left string was recorded to be 4.8 N, and the tension in the right string was 6.4 N. The angle of the left string was measured as 26 degrees and the angle of the right string was measured to be 46 degrees.
Figure 12: Setup for hanging unknown mass number 8
For the setup of unknown mass # 8, the tension in the left string was recorded to be 10.5 N, and the tension in the right string was 7.2 N. The angle of the left string was measured as 48 degrees and the angle of the right string was measured to be 10 degrees.
Abstract (Part 2):
Once all of the tensions and angles were measured, free body diagrams were drawn for each of these two setups, as shown below:
Figure 13: Free body diagrams, the mass equation, and solving for the masses of the unknowns in the above two setups
Summing the forces in the y-direction, the unknown masses can be found as a function of the two tensions, the angles, and gravity. The x-direction need not be summed since there is no unknown variable or mass component in the x-direction. Used the derived formula above for mass, the unknown masses were calculated.
However, since there was uncertainty in the measurements of the tensions and angles, there must be uncertainty in the calculated masses. We can find the uncertainty by taking partial derivatives of the above derived equation for mass with respect to each of the 4 variables: The tension of the first string, the tension of the second string, the angle of the first string, and the angle of the second string. Once the partial derivatives with respect to each variable were found, they can be multiplied each by the uncertainty of the measurement itself (i.e. 2 degrees (in radians) for angle and 0.5 N for tension). The degrees must be changed to radians since partial derivatives is a calculus operation. The partial derivatives are shown in the upper part of the figure below:
Figure 14: Derived partial derivative equations for the uncertainty in the calculated unknown masses / solving for uncertainty in unknown mass # 1
Using these partial derivative equations, the uncertainties in the masses of the unknowns of the two hanging mass setups were calculated by plugging the values into the formulas, then summing up the value for each of the four formulas. The uncertainty in the measurements of the angles and tension forces are already included in the partial derivative equations (dF and dθ).
Figure 15: Solving for uncertainty in unknown mass number 8
The uncertainties in the masses ranged from 10% to 15% of the calculated masses, which was expected since the uncertainties in the instruments used (added up) were roughly in the same range.
Conclusion:
The purpose of this experiment was achieved in that the importance of taking into account uncertainty of any measurements was learned. In addition, the propagated uncertainties of the densities and unknown masses were successfully calculated. Most, if not all, of the uncertainties of this experiment were found in the uncertainties of the instruments and the measurements of masses, tensions, and dimensions. Most of the error in this experiment was found in the uncertainties of the instruments and measurements, especially in the second portion of the experiment. One major error in the first portion was the assumption that the cylinders were fully solid, had the shape of a perfect cylinder, and were composed of pure metal of one kind. The identity of the metal that composed the first cylinder was not found due to the density being small yet the qualitative features being different than the lighter metals. The identity of the second cylinder was found to be aluminum, and the identity of the third cylinder was found to be copper.
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