Sep-7-2016: LAB 6 - Propagated Uncertainty in Measurements

LAB 6 - Propagated Uncertainty in Measurements

Tony Wu, Leslie Zhao, Isaiah Hernandez

September 7, 2016

Lab Goal: To calculate and understand the propagated error in each of our measurements. Determine whether or not our measurements are within the experimental uncertainty of the accepted values.

Theory/Introduction: Accuracy in measurements and calculations is an absolute must in the real world.  Structural engineers must work with a level of accuracy that is acceptable and safe.  With that said, 100% accuracy and precision is not possible.  Every measurement will have a certain degree of uncertainty.  These small degrees of uncertainty can propagate and drastically alter the final results.  In this lab we are learning to calculate the levels of uncertainty in order to explicitly state the range of possible values.  We learned how to use calipers to measure to an accuracy of 0.01 cm meters.

Apparatus: This lab only uses calipers for measuring length and a scale for measuring mass.

Experimental Procedure: We used calipers to measure the diameter and length of two metal cylinders. The diameter and length measurements have an uncertainty of ±0.01 cm.  We then measured the mass of each cylinder using a scale.  The mass measurements have an uncertainty of ±0.1 g.   Now that we have the mass, diameter, and length of each cylinder, we can derive the density of each. .

Data and Calculations:

 

Shown above are the calculations to derive the density of each cylinder. We use calculus in order to calculate the values for the propagated uncertainty. We take the formula for the density and take the partial derivatives of each variable. We then take the magnitude of the sum of all the variables.

Conclusions: Accuracy in measurements can never be 100% accurate. In order to take this into account, measurements must be given with an acceptable degree of uncertainty. Each little bit of uncertainty propagates to affect the final calculations. The important thing is to keep track of the limitations that every measurement tool has in order to calculate the propagated uncertainty.