Physics 1050Experiment 1

Introduction to Measurement andUncertainty

Physics 1050Experiment 1

Prelab Questions

These questions need to be completed (on scrap paper) before entering thelab. Once you begin the lab, write out the answers from your workings. Youwill be penalized if you are not prepared to answer these questions. Pleaseshow all workings.

Prelab 1:A rectangular object has length L = 25.3 +/- 0.2 cm and widthW = 18.6 +/- 0.5 cm.What are the respective absolute uncertainties in length and width?Find the respective relative uncertainties in length and width.Prelab 2:Given a set of length measurements: 60.42, 60.38, 60.51, 60.47, 60.66 cm.Find: the average (mean) length

the standard error (refer to pages VII - X of your Lab Workbook).

Physics 1050Experiment 1

Contents

Introduction

Part I: Measuring a Wooden BoardLength MeasurementsWidth MeasurementsPerimeter and Area Calculations

Part II: Measuring Metal CylindersVernier CalipersMeasurementsCalculationsData AnalysisResults

Part III: Summary

Physics 1050Experiment 1

Objective

In this experiment, you will study measurement anduncertainty by measuring the physical parameters of arectangular piece of board and various metal cylinders.You will then use uncertainty rules to estimate theuncertainty in the perimeter and area of the board.You will make a graph to determine the density of themetal cylinders and the associated uncertainty.

Physics 1050Experiment 1

Introduction

When a physical quantity is measured, a reading is made on ascale. The number of significant figures in the reading is limited bythe device itself and the size of the smallest division on the scale.The uncertainty is a reflection of the confidence in themeasurement.Examples are given in the following two slides.

Physics 1050Experiment 1

Introduction

Example 1:A new metre stick measuring a board with a well defined edge:The position can reliably be measured to a fraction of a mm.The uncertainty will be a fraction of a mm (between 0.2 and 0.5 mm).

The position of theedge may be readin this image as60.46 ± 0.03 cm.

Physics 1050Experiment 1

Introduction

Example 2:An old metre stick measuring a board with a rough edge:The position can only reliably be measured to a single mm.The uncertainty will be a single mm.

The position of theedge may be readin this image as60.3 ± 0.1 cm.

Physics 1050Experiment 1

Definitions

Absolute and Relative UncertaintyIf δx represents the absolute uncertainty in a measured quantity, then relative uncertainty is given by δx/x.The uncertainty is usually rounded to 1 significant figure. A result is generally reported in the form x ± δx.

For the example, if we make a reading of the length of an object with ametre stick of 32.45 cm ± 0.05 cm, the absolute uncertainty δx is 0.05 cmand the relative uncertainty is 0.05/32.45 = 0.002.

Standard ErrorThe standard error in a measurement, x, that is one of a series ofmeasurements is an indication of how any individual measurement isdifferent from the mean, . The method of calculation is given on pages VII - X of your laboratoryworkbook.

!

x

Physics 1050Experiment 1

Rules for the propagation of uncertainties

Rule 1:When adding or subtracting measured quantities, the absolute uncertainty inthe result is the sum of the absolute uncertainties in the measured quantities.

Example 1: z = x – y, Example 2: z = x + y,δz = δx + δy δz = δx +δy

Rule 2:When multiplying or dividing measured quantities, the relative uncertaintyin the result is the sum of the relative uncertainties in the measured quantities. Example 1: z = x/y Example 2: z = xy

δz/z = δx/x + δy/y δz/z = δx/x + δy/y

These and the other rules are further discussed in the introductory material.

Physics 1050Experiment 1

Part I: Measuring a Wooden BoardLength Measurements

All measurements must be made to either 3 or 4 significant digits. Using a metre stick, measure the length, L, of the board provided.Take three measurements of the length - one near each side of theboard, and one near the centre.Record your data in your Activity Log Table 1.Complete Table 1 by calculating: the mean deviation for each value deviation squared sum of the deviation squared average deviation standard errorSee pages VII - X for help in filling out this table

Physics 1050Experiment 1

Part I: Measuring a Wooden BoardLength Measurements

Reminder: Your uncertainty is an estimate of the reliability of yourmeasurement.

QUESTION 1: Estimate the absolute uncertainty associated withyour metre stick. In your estimate, take into account the size of the smallest scale division, andthe condition of your measuring device as described previously.

Physics 1050Experiment 1

Length Uncertainty

The absolute uncertainty in the average value of length L iswritten as ± δL where δL is either the standard error (recordedin Table 1) or the estimated uncertainty in an individualmeasurement (Question 1), whichever value is greater.QUESTION 2: Write down the average length of the board

in the form L ± δL, where δL is either the standard error or the uncertainty of the metre stick (whichever is greater).Find the relative uncertainty in L.(Refer to Definitions slide.) Be sure to use the correct number of significant figures.

Reminder: relative uncertainty = δL/L

Physics 1050Experiment 1

Width Measurements

Using a metre stick, measure the width, W, of the board provided. Takethree measurements of the width - one near each side of the board, and onenear the centre.Record your data in your Activity Log Table 2.Use Graphical Analysis to determine the standard error in width:

Click to open Graphical Analysis.

Enter your width values in the table.Click Analyze then Statistics.Record the mean, standard deviation, and point count in Table 2.Calculate the standard error and record your result in Table 2.QUESTION 3: Write down the average width of the board in the form

W ± δW where δW is either the standard error or the uncertainty in an individual measurement (whichever isgreater). Find the relative uncertainty in W.

Physics 1050Experiment 1

Perimeter and Area Calculations

QUESTION 4: Calculate the perimeter P of the board, the absolute uncertainty in P, and the relative uncertainty in P.

Reminder: Rule 1:When adding or subtracting measured quantities, the absolute uncertainty inthe result is the sum of the absolute uncertainties in the measuredquantities.

Example 1: z = x – y, Example 2: z = x + y,δz = δx + δy δz = δx +δy

QUESTION 5: Write down the perimeter of the board in the formP ± δP to the correct number of significant figures.

Physics 1050Experiment 1

Perimeter and Area Calculations

QUESTION 6: Calculate the area A of the board, the relative uncertainty, and the absolute uncertainty in A.

Reminder: Rule 2:When multiplying or dividing measured quantities, the relative uncertainty inthe result is the sum of the relative uncertainties in the measured quantities. Example 1: z = x/y Example 2: z = xy

δz/z = δx/x + δy/y δz/z = δx/x + δy/y

QUESTION 7: Write down the area of the board in the form A ± δA to the correct number of significant figures.

Physics 1050Experiment 1

Part II: Measuring Metal CylindersVernier Calipers

Vernier calipers are commonly used inlaboratories for the precise measurementsof dimensions.

Turn on the calipers with the OFF/ON switch.Make sure the calipers are set to measure inmillimetres by pressing the mm/inch buttonuntil the display shows mm.Close the jaws of the calipers completely andpress ZERO to zero the calipers.

Be sure to convert any measurementsto cm.

The uncertaintyassociated with thecalipers is 0.01mm.

Physics 1050Experiment 1

Use of Vernier Calipers

To measure outer dimensions of an object, the object isplaced between the jaws, which are then moved together untilthey secure the object. The screw clamp may then betightened to ensure that the reading does not change while thescale is being read.

Physics 1050Experiment 1

Measurements of Metal Cylinders

Using a triple beam balance, weigh each of the four metalcylinders and record the masses in Table 3.Using the Vernier calipers, measure the length of each cylinderand enter it in Table 3.Using the Vernier calipers, measure the diameter of eachcylinder and record it in Table 3.Calculate the volume of each cylinder and record it in Table 3.

Careful with units when entering your results in Table 3!

Physics 1050Experiment 1

Relationship between mass andvolume

We will now analyze the data graphically by plotting the data anddetermining the slope and associated uncertainty.But first, we examine what we are plotting and why.We have a relationship between physical quantities in anequation:Density equals mass divided by volume or ρ=m/V. (the Greekletter ρ (rho) is the symbol for density).We will plot the measured data of mass on the y-axis and volumeon the x-axis.

QUESTION 8: To what physical quantity does the slope of themass versus volume graph equal correspond?

Physics 1050Experiment 1

Graphical Analysis

Open Graphical Analysis by clicking on the icon belowCLICK HERE CLICK HERE

Enter the values of volume in the x column of graphical analysisand enter the values of mass in the y column. Label your axeswith include proper units and title your graph (see Page XI of theintroductory material for instructions).Draw a line of best fit by clicking Analyze then Linear Fit. Double-click on the pop-up box labelled “Linear Fit” and check the options“Standard Deviations” for both Slope and Intercept. Click OK.

Record the slope and its uncertainty in Table 4.

CHECKPOINT: Have an instructor check the value of your slope.

Enter your names in the “Notes” window and select File, Print.

Physics 1050Experiment 1

Part II: Results

QUESTION 9: Using the slope and uncertainty from your graph,what are the maximum and minimum values for the density of the cylinders?

QUESTION 10: By comparing your value of density with those inthe table, what kind of metal are your cylinders made of?

7.85steel

10.5silver

11.3lead

7.75Iron

2.7aluminum

Density (g/cm3)Substance

Physics 1050Experiment 1

Part III: Summary

QUESTION 11: List at least three sources of the experimental uncertainties involved in Parts I and II of this experiment. Classify them as random or systematic and justify your classification.

QUESTION 12: As an example of the propagation of errors, consider the perimeter of the board: Was the absolute uncertainty of perimeter larger or smallerthan the absolute uncertainties of length and widthof the board? Is this an expected result? Explain.

Physics 1050Experiment 1

Wrap it up!

Check that you have completed all the Tables in your ActivityLogMake sure that you have answered all Questions completely.Attached to your Activity Log should be your graph of massversus volume.