Physics lab report

Error Propagation: Volumes Introduction

In this lab you will practice error analysis by measuring volumes of a few regular shapes.

You need to understand the Data Analysis document to successfully complete this

exercise. The aim is to understand how errors propagate through formulas such as the

volume of a box V = LWH. If we measure the width W, height H, and length L many

times to establish errors for each quantity, what will the error in the volume be when we

multiply the average values of the three quantities together? The need to answer this type

of question arises in all areas of the physical sciences. (See Philip Bevington and D.

Keith Robinson “Data Reduction and Error Analysis for the Physical Science” for more

details).

Procedure

Determine the volume of various objects using Vernier calipers to measure relevant

dimensions. The Vernier is accurate to 0.05 mm. Take several independent readings for

each dimension. Make sure everyone learns to use the calipers and that you understand

how a Vernier works (ten divisions on one scale equals nine on the other, etc.). Compute

the mean and standard deviation of these readings for all partners for each dimension

using Excel or Graphical Analysis.

Measure the relevant dimensions for each of the following objects:

1. A rectangular prism. 2. A triangular prism. 3. A hollow cylinder (find the volume of solid material). Be sure to also

measure the “wall thickness” directly with the calipers. 4. A bullet shaped object, approximated as a cylinder plus hemisphere of

matching diameter.

5. A small plastic cup. Estimate the volume of the cup by finding the volume of water that fills a rectangular box.

Analysis

Find the volume of each of the above objects and give the error in the volume. Express

your answer in cm3. To do this you need the mean and standard deviation of each of the

dimensions on which the volume depends (this is your raw data). Note that the error in

the calipers is only 0.05 mm, so many of your measurements may give the same result.

Take five measurements of each dimension to get the mean and standard deviation.

Do the error analysis using the three cases outlined in the Data Analysis document. They

are reproduced below:

Case I: pure product function   xyyxf ,

For product (and division) functions, the square of the fractional error of

the result is the quadratic sum of the fractional errors of the variables. Or,

you can use partial derivatives approach:

Case II: power function   mxyxf ,

For power functions, the fractional error of the result is m times the

fractional error of the variable. Or, you can use partial derivatives

approach:

This result holds for m negative or fractional as well, corresponding to division

(x-1) and roots (x1/2). Please note the above equation corrects a typo in the original

document. The absolute value of the power m is needed so the relative error is a positive

quantity. Cases I and II can be combined as   nm yCxyxf , where C is a constant:

Case III: pure addition   ByAxyxf , where A and B are constants. Note that A or B may be negative, corresponding to subtraction.

For sum/difference functions, absolute (not fractional) errors add quadratically.

Or, you can use partial derivatives approach:

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In the above equations x and y represents the standard errors of the quantity x and y

accordingly. The standard error is also called the standard deviation of the mean. It is

obtained by calculating the standard deviation x from a series of measurements of the

quantity x first. The program graphical analysis will calculate standard deviation x for

you, or you can use the STDEV function in EXCEL, or the std dev button on many

calculators. Then use the equation for std. dev. of the mean N

x 

where N the total number

of measurements made of quantity x to calculate the standard error.

The values of x and y

are average/mean values.

Hints and Notes

What is below is not meant to be an exhaustive list of what should be put in your

reports, but should be thought of as reminders of things to think about during your lab

period. Please see the course syllabus for more details about what should go into

your report.

1. For each shape state clearly the results: mean value of the dimensions and its error; its volume and the error in the volume.

2. In Data Analysis include a sample calculation of error in the dimensions; value of volume and its error propagation for either hollow cylinder or bullet-shaped object

and one of the other objects.

3. You do not need replicate the original data/statistics in the report. Be sure your data sheets are attached to your report!

4. Your Results section need to have results for all cases.