04-Mar-2015 Lab 6: Density of Metal Cylinders

PURPOSE:  To calculate the propagated error after finding the density of three metals from height, diameter, and mass, and assessing whether the found densities falls within the accepted errors.  Then, to calculate the mass of an unknown object from spring force and angles, find the propagated error, and assessing whether the found mass falls within the accepted error.

MATERIALS:

For Part 1:

• Aluminum, steel, and copper samples
• Mass Scale
• Vernier caliper - used to measure height and diameter of various metals

Vernier Caliper - used to measure external dimensions of an object

For Part 2:

• Angle protractor 
• 2 spring scales - suspended from 2 different points and attached to the same hanging mass

PROCESS:

Part 1

• We first measured the height, diameter, and mass of 3 metals to calculate the density of each using the formula for density:
• Light gray = aluminum
• Dark grey = steel
• Bronze = copper

3 Metals from left to right: aluminum, steel, copper

Table containing 3 properties of metal dimensions

Equation used to calculate density

To find the error in each component of density (mass, diameter, and height), we took the partial derivative of density with respect to each component.  This tells us how much an uncertainty in any of these three affects the density.  That partial derivative is then multiplied by a numerical value of how uncertain we thought our measurements could be off by.  For mass, we chose an uncertainty of 0.1 grams; for diameter, we chose an uncertainty of 0.01 cm, and for height, we also chose an uncertainty of 0.01 cm (because for diameter and height measurements we used the same tool).

Equations of uncertainty for mass, diameter, and height.

Using the above calculations, the following photos show calculations for each of the 3 metals.

Calculations for the propagated uncertainty in aluminum.

Calculations for the propagated uncertainty in steel

Calculations for the propagated uncertainty in aluminum.

Results:

• Aluminum: 2.745 g/cm^3 +/- 0.057
• Steel: 7.678 g/cm^3 +/- 0.135
• Copper: 8.952 g/cm^3 +/- 0.158

From the internet, we found the following results for the actual densities of our metals:

• Aluminum: 2.7 g/cm^3
• Steel: 7.850 g/cm^3
• Copper: 8.96 g/cm^3

Our results show that aluminum and copper fell under the accepted values for measured density, but steel was outside the accepted value by 0.037 g/cm^3.  One source said that the density of steel could range based on its alloying constituents.  

Part 2:

For this next part of the lab, our goal was to determine the mass of an object based on the angles which it hung at and a spring force, to calculate uncertainty, and then see if the actual mass fell within our measurements.

We began by writing out an equation for the mass hanging from two tension forces at two angles.  Under that equation we also wrote out what the formula for propagated uncertainty in the mass would be by taking the partial derivative of the mass with respect to each component (Force 1, Force 2, Theta 1, and Theta 2):

The experiment was already set up for each group, with two spring scales suspended from poles a certain distance apart and connected by string.  We made sure to calibrate the springs by adjusting them to read zero when nothing was hung on it.  We then placed an object of unknown mass to hang in the center of the string, causing the string and springs to angle downwards (see following photo):  

Spring Scale - measures force in Newtons

Protractor used to measure the angle of each string

We performed this on two masses: "Weight #3" and "Weight #1."  The following data is what we collected (Note: we had to convert degrees to radians):

Then, we took this data and calculated forces in each direction (free body diagram shown).  Under the free body diagram is the formula for propagated uncertainty in the mass.  You can see (similar to part 1 of this lab) that each partial derivative shows how much mass is thrown off by, multiplied by a value of how uncertain we were with our measurements.  For the forces (spring scale), we chose an uncertainty of 0.02 Newtons, and for the angle we chose an uncertainty of 1 degree, or 0.017 radians.

Calculations for "Weight 3"

• Our formula for mg showed that that the mass was 1.081 kg, or 1081 grams.  The actual mass of the object was 938 grams.  If we subtract the propagated uncertainty of 47.6 grams from our mass, it gives us 1033.4 grams, showing that our result does not fall within our calculated uncertainty of the mass.

Calculations for "Weight 1"

• Our formula for mg showed that that the mass was 0.7353 kg, or 735.27 grams.  The actual mass of the object was 693 grams.  If we subtract the propagated uncertainty of 18.7 grams from our mass, it gives us 716.57 grams, showing this result also does not fall within our calculated uncertainty of the mass.

Errors:

The results of this lab could have been off due to a number of factors including the accuracy of the instruments - while we were doing the lab other groups were coming up to look at measurements of the spring scale which could have adjusted the calibration.  The spring scales themselves did not seem that high tech.  We also may not have been lenient enough when determining how uncertain our measurements were - it may have helped to choose an uncertainty of 2 degrees.  

Conclusion:

This lab allowed us to understand how calculating the partial derivatives in a formula affect measurements.  We took the partial derivatives in each factor of our equations and added them up to allow for error in our experiment.  However, for the second part of the lab our measurements did not fall within range of propagated uncertainty.  This makes sense because the tools we used for part 2 of the experiment did not seem as accurate as the tools we used in part 1.  Regardless, it helped us to understand how to calculate propagated error for each experiment.