Introduction to propagated error calculations

Introduction to propagated error calculations

Purpose:
Through measuring the density of Metal Cylinders and determination of the unknown mass, learn about how to calculate the propagated error of the experiment.

Part One   Measuring the Density of Metal Cylinders
Experiment:
1. Use the caliper to measure the diameter and the height of the metal cylinder.
2. Use the balance to measure the mass of the cylinder.
3. Determine the uncertainty of each measurement.
4. Calculate the density using the data.
5. Take the derivative of the density with respect to each variable.
6. Plug in the derivative and the uncertainty into the formula and calculate the propagated error.
7. Repeat the procedure and calculate the density and propagated error for other two metal cylinder.

Calculation:

Result:

1. density of the first metal: 7620 kg/m^3 

    propagated error of the first calculation: 134 kg/m^3

2. density of the second metal: 1460 kg/m^3 

    propagated error of the second calculation: 25 kg/m^3

3.density of the third metal: 2190 kg/m^3 

    propagated error of the third calculation: 41.26 kg/m^3

Pat Two   Determination of an unknown mass

Experiment:

1) Get some spring scales (5 or 10 N should be right). Note that there are adjusting screws to zero the scales. 

2) Adjust the scale so that it reads zero when nothing hangs on it and that it reads the appropriate weight when a known weight is hung on it (document what you did here.) 

3) Obtain an unknown mass. Record which unknown mass you used. (It should have some sort of label on it). 

4) Set up two clamps onto the edge of a lab table as shown, with a long rod in each. Suspend the two spring scales at asymmetric angles and hang the unknown mass on them. 

In the Lab, steps one to four are omitted since the apparatus has been set up.

5) Measure the angles and record the scale readings. Estimate the uncertainty in your angle readings and scale readings. 

6) Use your measured values to determine the mass of the unknown mass. 

7) Take the derivative of the mass with respect to each variable

8)  Plug in the uncertainties and derivative to determine the propagated uncertainty in the calculated value of the mass. 

9) Use a different set of angles and a different hanging mass. Do steps 5-8 again. 

Calculation:

Result:

1. weight of the first set-up: 5.3 N

    propagated error of the first set-up: 0.23 N

2. weight of the first set-up: 5.19 N

    propagated error of the first set-up: 0.24 N

Conclusion:

Through this experiment, we learn about the use of caliper and mass balance and how to calculate the propagated error to examine the reliability of the result of a experiment.