Microsoft Word - Exercise_E

or Propagation.docx

1

E

or Propagation: Volumes

Introduction

Physics is an empirical science. Every theoretical concept has to be supported with the

experiment to be fully accepted. Experience has shown that no measurement is exact. All

measurements have some degree of uncertainty due to the limits of instruments and the

people using them. In science it is critically important to specify that uncertainty and

include its reliable estimate in the final statement of the result of the measurement. The

concept of the e

or propagation becomes a vital part of each physics experiment.

There are two concepts associated with measurements: accuracy and precision. The

accuracy of the measurement refers to how close the measured value is to the true or

accepted value. For example, if you used a balance to find the mass of a known standard

100.00 g mass, and you got a reading of 78.55 g, your measurement would not be very

accurate because the percent discrepancy between the measured and “true value” is 21%.

Precision refers to how close together a group of measurements actually are to each other.

In the example above, if the three measurements of mass that obtained are 78.55g,

78.60 g and 78.50 g the measurements can be considered precise. Precision has nothing

to do with the true or accepted value of a measurement, so it is quite possible to be very

precise and totally inaccurate as in the example with an object’s mass.

In many cases, when precision is high and accuracy is low, the fault can lie with the

instrument. If a balance or a thermometer is not working co

ectly, they might

consistently give inaccurate answers, resulting in high precision and low accuracy. One

important distinction between accuracy and precision is that accuracy can be determined

y only one measurement, while precision can only be determined with multiple

measurements. A measurement can be accurate but not precise, precise but not accurate,

neither, or both. For example, if an experiment contains a systematic e

or*, then

increasing the sample size generally increases precision but does not improve accuracy.

Eliminating the systematic e

or improves accuracy but does not change precision.

Commonly the best estimate of the true value of the measured quantity x can be obtained

y calculating the mean (average) value ( x ) of some significant number of measurements

(N) of that quantity repeated using the same equipment and procedures:

N

x

x

N

i

i∑

== XXXXXXXXXX)

* Systematic e

ors: Commonly caused by a flaw in the experimental apparatus. For

example bad cali

ation in the instrumentation will give systematic e

or.

2

In such case the measure of the uncertainty (random e

or*) in the mean value will be the

standard deviation of the mean:

Nx

σ

σ = (2)

where

xσ is the standard deviation of the mean, σ – is the standard deviation and N – is

the number of measurements. The standard form for reporting a measurement of a

physical quantity x is:

xxx σ±= (3)

This statement expresses our confidence that the co

ect value of x probably lies in (or

close to) the range from

xxx σ−= to xxx σ+=

There are two rules that will be followed in this course for stating the final result: a)

uncertainty should be rounded to one sig.fig. unless that sig. fig. equals one; if the very

first significant figure is one than the following digit should be reported; b) The last

sig.fig. in the mean value should be in the same decimal place as the first significant

figure in uncertainty. For example: l = XXXXXXXXXX ± .006) mm

To estimate the accuracy of the final result the percent discrepancy or percent difference

is calculated.

%100*%

exp

theo

ertheo

x

xx

disc

−

= XXXXXXXXXX)

%100*%

2exp1exp

ave

ere

x

xx

diffe

−

= (5)

A measure of the accuracy can only be determined if some prior knowledge of the true

value is available.

The precision of the measurements shows the quality of the measurements and can be

expressed as the relative uncertainty (the standard deviation of the mean divided by the

mean value):

elative uncertainty %100*

x

xσ= (6)

* Random or statistical e

or: Class of e

ors produced by unpredictable or unknown

variations in the measuring process. The effect of the random e

or may be reduced by

epetition of the experiment.

3

The mean value of the physical quantity, as well as the standard deviation of the mean,

can be evaluated after a relatively large number of independent similar measurements

have been ca

ied out. These measured quantities in many cases serve as a basis for the

calculation of other physical quantities of interest. In such situations the uncertainties

associated with the directly measured quantities affect the overall uncertainty in the final

quantity of interest. To estimate the end uncertainty we use the rules of e

or propagation.

For example, the volume of a box is V = LWH. If we measure the width W, height H, and

length L many times to establish e

ors for each quantity: ∆ W,∆ H,∆ L, what will the

e

or in the volume ∆ V be when we multiply the average values of the three quantities

together? Or, in the other words, how the e

or of volume propagates.

In this lab e

or analysis will be practiced by measuring volumes of a few regular shapes

objects. The partial derivatives approach will be used through the course to propagate the

e

or:

XXXXXXXXXX)

where x and y are the measured quantities with the standard deviation of the mean (also

called standard e

ors)Δx and Δy.

Objectives:

• to practice e

or propagation concept and reporting experimental result in the

co

ect format.

Equipment: hollow cylinder, vernier caliper.

Procedure:

A student was asked to use Vernier caliper to measure the necessary dimensions of

hollow cylinder, bullet-shapes object, 3D rectangular wood object to calculate the volume

of each object and its e

or. The Vernier caliper is precise to 0.05 mm, so many of the

measurements gave the same results. Each dimension was measured 6 times. The

collected data is provided in table 1, table 2 and table 3.

Table Hollow cylinder

Height, mm Thickness, mm Inner diameter, mm

XXXXXXXXXX

XXXXXXXXXX

XXXXXXXXXX

XXXXXXXXXX

XXXXXXXXXX

XXXXXXXXXX

Based on the acquired data, use Logger Pro to find the mean value of each dimension and

the standard deviation. Using the above measurements to calculate the standard e

or

(standard deviation of the mean) for each dimension1; the mean value of volume for

22

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Δ

∂

∂

+⎟

⎠

⎞

⎜

⎝

⎛ Δ

∂

∂

=Δ y

y

fx

x

ff

4

hollow cylinder and its standard e

or. The final results should be reported in

centimeters, using the co

ect format.

1How to find the mean value of each dimension and the standard deviation of the

mean, using the Logger Pro:

Create columns in Logger Pro and label them appropriately. Copy the acquired data from

the table and paste it to Logger Pro.

Hint: The default file of Logger Pro consists of a two-column table and a graph area.

Double click on the column name to change its name. Enter the co

esponding name and

units for the measured quantity in the appropriate box. Create in this way columns for

each dimension of the object. Name data set by the name of the object.

Use the statistics feature of Logger Pro to find the mean values for each dimension as

well as the standard deviation.

Hint: To get the statistical values (mean, standard deviation etc.) for a set of data you

need to put the same data on both axes of the graph and next press STAT button on

the tool bar. Unless noted otherwise when you present any data in a graph form avoid

lines connecting the data points → in Logger Pro you change graph features by double-

click on the graph window. To add a new graph: INSERT→GRAPH.

To paste the data for the other object add manual columns; paste data and show its

statistics.

Calculate standard deviation of the mean for each of the measured dimensions using

equation (2). Use those values to propagate the e

or in volume with the partial derivative

approach.

Complete the provided Template “E

or Propagation exercise” to submit your work for

grading.

Template_E

or Propagation Exercise

(9 extra points toward final grade)

Template_E

or Propagation Exercise

(9 extra points toward final grade)

At the top of the first page (1 point):

Title of the exercise; Your name (prominent); Date of the Experiment; TA’s name; Lab Section number.

Beginning of the new page:

Experimental Data section (1points)

The raw data that will be used for further calculations in Data Analysis should be presented in the table below.

Hollow cylinde

Dimension

Mean value, unit

Stand. Deviation, unit

E

or (stan.dev.of the mean Logger Pro), unit

(Add/delete rows as needed)

Important: All logger pro graphs need to be attached at the end of the lab report. The graphs can also be inserted as a picture under the tables.

Data analysis section (4 points)

1. Volume calculations and its E

or Propagation

1.1 The equation that was used to calculate volume of the Hollow cylinder is:

Equation:

Calculations:

1.2 The standard deviations

Answered 2 days AfterJan 22, 2022

_E

or Propagation Exercise

E

or Propagation Exercise

Title: To practice e

or progression concept for calculating e

or in the volume of cylinder.

Student name: OLUWASEUN OLOWOYO

Date of Experiment: 22.01.2022

TA’s name:

Lab Section number:

A. Experimental Data section

The raw data used for further calculations is presented in the table below.

Hollow cylinde

Dimension

Mean value, unit

Stand. Deviation, unit

E

or (stan.dev.of the mean Logger Pro), unit

Height (mm)

128.0 mm

0.06055 mm

0.15 mm

Thickness (mm)

4.975 mm

0.1541 mm

0.40 mm

Inner diameter (mm)

76.05 mm

0.1612 mm

0.45 mm

(Add/delete rows as needed)

Important: All logger pro graphs are attached at the end of the lab report.

B. Data analysis section

1. Volume calculations and its E

or Propagation

1.1 The equation that was used to calculate volume of the Hollow cylinder is:

Equation:

Volume of hollow cylinder:

= pie *height * (outer diameter2 – Inner diameter2 )/4

=pie*height *[(Inner diameter + 2* thickness)2 – Inner diameter2]/4

Calculations:

Pie = 22/7

Height (H) = 128.0mm

Inner diameter (D0) = 76.05 mm

Thickness = 4.975 mm

Volume of hollow cylinder:

= pie*height *[(Inner diameter + 2* thickness)2 – Inner diameter2]/4

= (22/7)* 128.0mm *...

or Propagation Exercise

E

or Propagation Exercise

Title: To practice e

or progression concept for calculating e

or in the volume of cylinder.

Student name: OLUWASEUN OLOWOYO

Date of Experiment: 22.01.2022

TA’s name:

Lab Section number:

A. Experimental Data section

The raw data used for further calculations is presented in the table below.

Hollow cylinde

Dimension

Mean value, unit

Stand. Deviation, unit

E

or (stan.dev.of the mean Logger Pro), unit

Height (mm)

128.0 mm

0.06055 mm

0.15 mm

Thickness (mm)

4.975 mm

0.1541 mm

0.40 mm

Inner diameter (mm)

76.05 mm

0.1612 mm

0.45 mm

(Add/delete rows as needed)

Important: All logger pro graphs are attached at the end of the lab report.

B. Data analysis section

1. Volume calculations and its E

or Propagation

1.1 The equation that was used to calculate volume of the Hollow cylinder is:

Equation:

Volume of hollow cylinder:

= pie *height * (outer diameter2 – Inner diameter2 )/4

=pie*height *[(Inner diameter + 2* thickness)2 – Inner diameter2]/4

Calculations:

Pie = 22/7

Height (H) = 128.0mm

Inner diameter (D0) = 76.05 mm

Thickness = 4.975 mm

Volume of hollow cylinder:

= pie*height *[(Inner diameter + 2* thickness)2 – Inner diameter2]/4

= (22/7)* 128.0mm *...

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