March 6, 2017 Lab 6: Propagated Uncertainty in Measurements

Lab 6: Propagated Uncertainty in Measurements

Chris Ceron, Amy Chung, John Choi

Purpose

The purpose of this lab is to calculate the density of metal cylinders and the propagated error in our measurements. 

Vernier Caliper with micrometer measurements

The two cylinders used, Copper and Zinc respectively 

Items Used:

• Metal (Copper and Zinc) cylinders
• Vernier Caliper with micrometer measurements
• Electronic Scale

Theory/Introduction

This lab applies the use of propagated uncertainties to accurately calculate the range of the densities for our metal cylinders. Propagated errors are found in every measurement taken. To find the total propagated uncertainty, we first found the individual uncertainty with each measurement of height (cm), diameter (cm), and mass (g).

Individual Uncertainties

Our Vernier Caliper used micrometer measurements, which divides each millimeter with 10 division markers. This allowed us to measure the height and diameter of each cylinder to the nearest hundredth, Our electronic scale was accurate to the nearest hundredth's place. The result of the individual certainties were:

Height(cm): 0.01 cm
Diameter(cm): 0.01 cm
Mass(g): 0.01 g

Method Used to Calculate Total Propagated Error

To calculate density, we used the following formula:

Because we measured the diameter of the cylinder and not the radius, we expressed r in respect to the diameter and simplified to end up with:

To find the total propagated error, we took the natural log on both sides, and used a natural log identity to separate each variable:

Taking the derivative of both sides resulted in:

Using the square root form, the final result of the formula resulted in:

Data

	Mass (g)	Height (cm)	Diameter (cm)
Cylinder 1	28.93	3.28	1.29
Cylinder 2	81.48	4.89	1.59

Calculations

Calculation of density and propagated error for both cylinders

Conclusion

We were able to calculate the propagated error of the density using the uncertainty of each measurement as shown above. To determine our greatest source of error, we went back and calculated which measurement gave us the greatest uncertainty. The uncertainty of each of our measurements were by a unit of .01; however, the uncertainty within the measurement of the diameter gets doubled by a coefficient of two, producing a greater number within the delta rho calculation which causes greater uncertainty in our calculation.

Our value of density for zinc:


Our value of density for copper: