Physics 1 Lab -Complete

Physics I experiments

By: Atheer Dawood Mahir

2008© 00971503916861 [email protected]

POBox 346 , Ajman, UAE

Preface II

Preface

Recent welcome changes in practical physics taught in the

University stage have been of two kinds:

(1) the incorporation of new experiments with modern apparatus,

and (2) Computer analysis of experimental data

For the lab course in physics, this text focuses on the useful, hands-

on computer-based skills used day-to-day in implementing an actual

research project. Provides laboratory students with comprehensive

training in data acquisition and analysis by Microsoft excel as

computerized method and by traditional old methods.

Laboratory sessions are designed with a number of outcomes in

mind. We certainly want to investigate many of the concepts and

phenomena that you meet in the lecture part of the course. We also want

you to become proficient in the use of the computer to take and analyze

data and to report the results of your investigations. Learning experimental

techniques and working with each other as you investigate these

phenomena shows you how researchers work together and share ideas.

Thus we will do a variety of things in the lab, which may or may not be

exactly in synch with your class schedule. We want you to act and feel like

researchers - who need to know a variety of skills and information in order

to investigate the world in which they live.

Preface

III

The general requirements for any physics lab

(1) Each student has a specific lab section and should attend that

section.

(2) You will normally work in pairs. You may choose your own laboratory

partners. During the course of the semester you have to change

partners several times in order to experience the different approaches

that different partners bring to the lab.

(3) You need to read the lab before you arrive in lab.

(4) For most experiments, the theory will have already been covered in

class and the methods or procedures will be specified in the

laboratory manual. Thus, the laboratory manual is always considered

to be a part of each report so that this information does not need to

be copied into the lab report.

How to Write a Laboratory Report

The ability to write an effective lab report is essential in all scientific fields.

Therefore, you will be required to adhere to the following guidelines.

Reports may be typed or handwritten in ink. The format of the reports

should resemble that generally used in scientific journals. Use the general

headings shown below. (Other headings may be used when

appropriate.)

a. Cover Page: The cover page should include the title of your report,

your name , ID, email ,course name , course number, your section

number, and date of experiment.

Preface

IV

b. Title (Name of Experiment): The title should be brief and descriptive,

and should appear at the top of your report.

c. Purpose: Describe in general terms, why you are doing this

experiment. What do you hope to learn in the process? What skills

can you develop by doing this experiment?

d. Equipment and Materials (apparatus): List the materials and

equipment that you used.

e. Procedure: Briefly explain how you did the experiment. Provide

enough detail to let the reader know why your steps led you to the

associated conclusion.

f. Results and Calculations: Outline the results of your experiment. Give

all necessary qualitative and quantitative observations. Generally,

the quantitative aspects of the results are best presented in graphs

and tables. These should be accompanied by a verbal description

of the data and the trends that may occur. Remember to include

your units and to use scientific notation. When applicable, you

should include a sample calculation.

g. Discussions and Conclusions: In your discussion, interpret your results

and observations. Keep in mind the purpose of the experiment

while interpreting your results. You should also answer the questions

that were presented to you throughout the experiment in this

section. In your Conclusions, explain the significance of your results

and summarize your thoughts on them.

h. References: Cite any and all reference material(s) that you used in

writing the lab report and/or answering the questions that were

presented to you.

Preface

V

Ten laboratory reports in this text are designed with the same divisions

above with some updated requests like links to web, Java, animations,

and photos related to these reports.

As you will see in each practical report, you need to study carefully the

theory and steps of procedure to fill out the Result and Discussion sections.

You need also to prepare graph papers and excel graphs for different

experiment to attach it with result and discussion sections and submit it to

your instructor. You do not need to submit all sections of your report to

correct; just the last two sections (results and discussions) with related

graphs and excel sheets. After correction return it to suitable place to

have a complete report for that experiment.

Atheer Dawood Mahir

Aug.22, 2008, Ajman , UAE

Preface

VI

The contents Preface.................................................................................II

The general requirements for any physics lab .................. III

How to Write a Laboratory Report ................................... III

The contents ...................................................................... III

Significant Digits ............................................................... 3

Graphing Lab and excel ...................................................... 3

Density using different tools............................................... 3

Vectors (free body diagram) ............................................... 3

Motion Along Straight line and Newton's laws .................. 3

Friction................................................................................ 3

Spiral Spring – Hooks Law................................................. 3

Simple Pendulum................................................................ 3

Angular simple harmonic motion ....................................... 3

Moment of inertia ............................................................... 3

Physics Lab AUST

Significant Digits Page 7/102

Experiment 01:

Significant Digits Purpose:

This experiment will demonstrate how to determine the significant digits of a number like(52, 502, 5020, 0.05020, 1.05020) and perform calculations with the correct significant digits:

• The purpose of significant digits. • Determining significant digits • Addition and subtraction • Multiplication and division.

Apparatus : • A support stand with a string

clamp, • Measuring tape, • a stop Watch, • a small spherical ball, • string and scissor, • Transparent ruler , 2 Pencil

(HB) and Eraser • Scientific calculator.

Web:

Significant Figures:

New Theory:

http://www2.wwnorton.com/college/physics/om/_content/_ind

ex/tutorials.shtml (Click on Significant Digits)

http://homepage.mac.com/dtrapp/experiments/SignificantFigur

es.html

http://phoenix.phys.clemson.edu/tutorials/sf/index.html

http://ostermiller.org/calc/sigfig.html

Old Theory:

http://www.ausetute.com.au/sigfig.html

http://www.hazelwood.k12.mo.us/~grichert/sciweb/phys8.htm http://www.chem4free.info/calculators/signdig.htm

Physics Lab AUST


The Art of Making Measurements ------------ Professor Lewin MIT 1999

A measurement is meaningless without knowledge of its uncertainty. The lengths of an aluminum rod and the length of a student are both measured standing straight up and lying down horizontally to test whether the student's length is larger when he is lying down than when he is standing straight up. Within the uncertainty of the measurements, the difference between standing and lying is substantial for the student (NOT for the aluminum rod).

Theory1: Determine Significant Digits:

Applying principles of significant digits is a way to communicate the precision of

any measured number. When performing calculations keeping track of significant digits

is important. A calculated value can not have more significant digits than the value from

which it was derived.

We Assume that every well-defined measurable quantity has a certain true

value(Fig.1 & 2).

Fig. 1

Fig.2

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However, our equipment will only permit us to measure that true value to some

more or less limited precision. The stopwatch is a fairly precise instrument, but the

measured value obtained from it is still only an approximation of the true value.(fig.3)

Fig. 3

When the measurement is reported as 1.8 s that means that the true value is

believed to be somewhere between 1.75 s and 1.85 s.(fig.4)

Fig.4

This is a less precise stopwatch, fig.5, capable of measuring only to the nearest

second.

Physics Lab AUST


Fig.5

The measured value from it should be reported as 2 s, meaning that we can only

tell the true value is somewhere in the range from 1.5 s to 2.5 s.(fig.6)

Fig.6

If there is a sensor put in place that can detect more precisely when the car crosses

the finish line, then the measured value can be reported as 1.81 s.(fig.7 & 8)

Physics Lab AUST


Fig.7

Fig.8

The more significant digits there are , the more precise the measurement. That

would imply a more precise value is known. A scientist who presents this kind of data

might be accused of unethical conduct. (fig.9)

Fig.9

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When you see a number, it is important to be able to tell how many significant

digits are in it, so that you can tell how much precision is being implied. Numbers can be

written to include non-significant digits as well as significant digits. The following rules

will enable you to tell which digits are the significant ones.

Rule 1: Any nonzero digit is significant.

Rule 2: Any zero to the left of all nonzero digits is not significant.

Rule 3: Any zero between significant digits is significant.

Rule 4: Zeroes at the end of a number and to the right of a decimal point are

significant.

Rule 5: Zeroes at the end of a number without a decimal point are not significant.

Example Significant Digits Rules

52 2 Rule 1

5.03 3 Rule 1,3

5.20 3 Rule 1,4

0.2000 4 Rule 1,2,4

0.0020 2 Rule 1,2,4

52000 2 Rule 1,5

52000.0 6 Rule 1,3,4

Significant Digits for calculated quantities:

Often , we will start with two measured values that each have a certain number of

significant digits. Then we will calculate another quantity based on those measured

values. For example, we may know distance and time and then calculate speed from

them. How many significant digits should the calculated quantity have? (Fig.10)

Fig.10

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Adding and subtracting: When adding or subtracting numbers, there is always

some decimal position at which one or both numbers run out of significant digits. That

decimal position is then a “weakest link” that determines the number of significant digits

in the final answer.

Two examples will illustrate this point

Physics Lab AUST


Multiplying and Dividing: In a calculation involving multiplication or division,

the significant digits of the answer are determined by the number with fewest significant

digits.

An example will illustrate this point:

Scientific Notation: is a system in which numbers are expressed as a number

between 1 and 10 multiplied by power of 10. 110203.103.12 ×=

There is one digit before the decimal point. This makes it easy to determine the

number of significant digits. You simply determine the number of significant digits on

the number before the multiplication sign.

By using the idea that for free falling that htα , where (t) is falling time from

specific height (h): If 12 2hh = which means 41.100.20.750.150

1

2

1

2 ====cmcm

hh

tt , prove

that 1

2

tt within the significant digits rules is equal to 0.2 using digital stopwatch.

Physics Lab AUST


Procedure: 1. Pick small spherical ball suspended by a light string which is attached to a support

stand by a string clamp.

2. Adjust the height of the spherical ball bottom to about 150.0 cm from the ground.

3. Prepare stopwatch and scissor, cut the string and measure falling time using

stopwatch.

4. Repeat previous step for 5 times.

5. Adjust the height of the spherical ball bottom to about 75.0 cm from the ground.

6. Repeat steps 3 & 4.

7. Tabulate your results.

8. Get the significant digits for each measurement.

9. Compare between theoretical result and practical result.

Physics Lab AUST


Students Information (1 Mark):

Name Sec: ID Contact # Email Date of Experiment

Experiment # Experiment Name 01 Significant Digits

Results (6 Marks):

cmh ..........................2 =

1t s 2t s 3t s 4t s 5t s averageht −2 s # of Significant Digits

cmh ..........................1 =

1t s 2t s 3t s 4t s 5t s averageht −1 s # of Significant Digits

..................................................................

1

2 ==hh

................................................................

1

2 ==−

−

averageh

averageh

tt

.......................2

1

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

averageh

averageh

tt

Is 1

2

2

1

2

hh

tt

averageh

averageh =⎟⎟⎠

⎞⎜⎜⎝

⎛

−

− as a result of using significant digits?

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Physics Lab AUST


Discussion (3 Marks): # of significant digits2:

507,320 => 5 significant digits

0.00507320 => 6 significant digits

5.07320 => 6 significant digits

1) Calculate using significant digits rules: (1.5 marks)

=×528010 ………… =+ 9.516.5 …………

=+32000.0

0445.410.6 …………

2) If you measure student length when he is standing straight up and lying down horizontally. Count how many significant digits are in student length.(1.5 Marks) Straight Up= ……..cm, Lying down= ……..cm, Average= ……… cm,

# Significant Figures= …… digits

Physics Lab AUST

Graphing Lab and excel Page 18/102

Experiment 02:

Graphing Lab and excel Purpose:

• Learn how to plot points on graph paper. • Learn how to plot points on excel. • Learn how to find the best straight line (not through

origin) by graphing (hand Drawing) • Learn how to find the best straight line through origin

by graphing (hand Drawing) • Learn how to find the best straight line through origin

by Least square method (Statistical Method) • Learn how to find the best straight line through origin

by excel (Computerized Method) • Learn how to find the best straight line (not through

origin) by Least square method (Statistical Method) • Learn how to find the best straight line (not through

origin) by excel (Computerized Method) Apparatus :

• PC with Printer. • Graph papers (A4 Size). • Transparent ruler (30 Cm). . • Scientific calculator. • 2 Pencil (HB). • Eraser

Web: plotting Plotting Points on a Coordinate System: http://www.wisc-online.com/objects/index_tj.asp?objID=ABM201 Print Free Graph Paper: http://www.printfreegraphpaper.com/ Plotting Data on Linear Graph Paper: http://www.boomer.org/c/php/pk0201a.php Plotting Points in Rectangular Coordinate System: http://www.analyzemath.com/graphing_calculators/rectangular_coordinate.html How To Construct a Line Graph On Paper: http://staff.tuhsd.k12.az.us/gfoster/standard/bgraph.htm

Web: for regression http://www.ece.uwaterloo.ca/~ece204/TheBook/06LeastSquares/linear/theory.html http://www.people.ex.ac.uk/SEGLea/psy2005/simpreg.html http://people.hofstra.edu/stefan_waner/realworld/calctopic1/regression.html http://mathworld.wolfram.com/LeastSquaresFitting.html An interactive, visual flash demonstration of how linear regression works.: http://www.dangoldstein.com/regression.html

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وضع النقاط

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Physics Lab AUST


Theory: To understand graphing, knowledge of key terms is essential:

• Graph: A visual representation of comparative information.

• Graph paper: Paper that has been pre-divided into equal-sized squares which

ensures

o Accurate placement of ordered pairs,

o Correct intersection point of multiple graphs,

o More accurate demonstration of slop.

• Print Free Graph Paper: http://www.printfreegraphpaper.com/

• Ordered pair: A specific point on a graph.

• Relation: Any combination of ordered pairs , written as

{(x1,y1),(x2,y2),(x3,y3),…,(xn,yn)} where subscript numbers denote separate

ordered pairs.

• X-Axis: The horizontal plane.

• Y-Axis: The vertical plane.

• Origin: The (0,0) point.

• Domain: All the x-values within a relation.

o It is very important in dividing the x-axis on graph paper: each cm will

represent some equal quantity of real data.

o Help us to determine the start and end point of x-axis on the graph paper.

o We could exclude zero value for x-axis if our data not through the origin. X-

axis may be all in positive values only or negative values only.

o If our data through the origin, so, It is necessary to include zero value for x-

axis.

• Range: All the y-values within a relation.

o It is very important in dividing the y-axis on graph paper: each cm will

represent some equal quantity of real data.

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تمثيل

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نسبي

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تتضمن

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نقطة التقاطع مع المحور Y

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توضيح

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منظمة

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موحد

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نستبعد

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ضمن

Physics Lab AUST


o Help us to determine the start and end point of y-axis on the graph paper.

o We could exclude zero value for y-axis if our data not through the origin. Y-

axis may be all in positive values only or negative values only.

o If our data through the origin, so, It is necessary to include zero value for y-

axis.

How To Construct a Line Graph On Paper Step What To Do How To Do It

1 Identify the variables

a. Independent Variable - (controlled by the experimenter)

• Goes on the X axis (horizontal) • Should be on the left side of a data

table. b. Dependent Variable -

(changes with the independent variable) • Goes on the Y axis (vertical) • Should be on the right side of a data

table.

2 Determine the variable Domain and range.

a. Subtract the lowest data value from the highest data value.

b. Do each variable separately.

3 Determine the scale of the graph.

a. Determine a scale, (the numerical value for each square), that best fits the range of each variable.

b. Spread the graph to use MOST of the available space.

4 Number and label each axis.

• This tells what data the lines on your graph represent.

5 Plot the data points.

a. Plot each data value on the graph with a dot. b. You can put the data number by the dot, if it

does not clutter your graph.

6 Draw the graph.

a. Draw a curve or a line that best fits the data points.

b. Most graphs of experimental data are not drawn as "connect-the-dots".

7 Title the graph.

a. Your title should clearly tell what the graph is about.

b. If your graph has more than one set of data, provide a "key" to identify the different lines.

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ننشئ

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على انفراد

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عددي

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وضع علامة

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فوضى

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تجريبي

Physics Lab AUST


Linear Regression3 4 5:

Given a set of points (xi, yi) for i = 0, 1, 2, ..., n, we may not be able (or may

not want) to find a function which passes through all points, but rather, we may want

to find a function of a particular form which passes as closely as possible to the

points. For example, in Figure 1, it would make much more sense to try to find the

straight line which passes as closely as possible to each of the points.

Figure 1. Linear regression of a straight line on a set of points.

We will look at three techniques for finding functions which are closest to a

given curve:

• Linear regression using linear polynomials (matching straight lines),

• General linear regression (polynomials, etc.), and

• Transformations to linear regression (for matching exponential functions).

As well, we will discuss how we can use regression curves for extrapolation and an

efficient method (QR decomposition) for calculating least squares curves.

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تراجع

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خصوصي

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تعبر

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معنى أو يشعر

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تبديل , تحويل

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أسي

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تقدير عن طريق القراءة

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Terminology:

This processes is called regression because the y values are regressing (or

moving towards) the value on the curve which we find.

The term linear in linear regression refers to the coefficients of the matching

function. As a special case, we begin by looking at linear regression using linear

polynomials (i.e., y = ax + b).

Simple Linear Regression:

Consider the points (xi,

yi) shown in Figure 2.

It looks like the points

appear to lie in a straight line,

something of the form y(x) = ax +

b where a and b are unknown real

values. The question is, how can

we find the best values for a and

b. For example, Figure 2 shows

the two functions y(x) = 1.2 x +

2.4 and y(x) = 1.3 x + 2.5 in red

and blue, respectively. The blue

line looks better, but how did we

even pick the values of 1.3 and

2.5, and can we do better?

To begin, we must define the term regression. In this case, we are regressing

the values of y to some value on a curve, in this case, y(x) = c1x + c2. Because this is

an expression which is linear in c1 and c2, it is termed linear regression. (This has

nothing to do with the fact that the function is linear.)

The technique we will use to find the best fitting line will be called the method of least

squares.

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الاصطلاح

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لقب

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معامل

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ملائمة , اتففاق

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على التتابع

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Derivation of the Method of Least Squares

Given the n points (x1, y1), ..., (xn, yn), we will find a straight line which

minimizes the sum of the squares of the errors, that is, in Figure 3, we have an

arbitrary curve and the errors are marked in light-blue.

Figure 3. The errors between an arbitrary curve y(xi) = c1xi + c2 and the points yi.

Writing this out as mathematically, we would like to minimize the sum-of-the-

squares-of-the-errors (SSE):

∑=

+−=n

iii cxcySSE

1

221 ))((

Notice that the only unknowns in this expression are c1 and c2. Thus, from

calculus, we know that if we want to minimize this, we must differentiate with-

respect-to these variables and solve (simultaneously) for 0:

0)(21

211

=−−−= ∑=

n

iiii xcxcySSE

dcd , 0)(2

121

2

=−−−= ∑=

n

iii cxcySSE

dcd

Expanding the first equation (and dividing both sides by -2), we get:

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استنتاج

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The Sum of the Squares of the Errors

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01

21

21

1=−− ∑∑∑

===

n

ii

n

ii

n

iii xcxcyx

If we, with some foresight, define the following, the sum of the x's (Sx), the sum of the

y's (Sy), the sum of the squares of the x's (SSy), and the sum of the products of the x's

and y's (SPx, y), that is,

∑∑∑∑====

====n

iiiyx

n

iix

n

iiy

n

iix yxSPxSSySxS

1,

1

2

11

,,,

the we get the linear equation:

yxxx SPcScSS ,21 =+

Expanding the second equation (and dividing both sides by -2), we get:

011

21

11

=−− ∑∑∑===

n

i

n

ii

n

ii cxcy

By calculating the third sum and rearranging, we get the linear equation:

yx SnccS =+ 21

We could solve these the long way (as you probably did in high school), however, we

note that this describes the system of equations:

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

y

yx

x

xx

SSP

cc

nSSSS ,

2

1 , this is a system of linear equations which we can, quite

easily, solve.

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠

⎞⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛

==

⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛

==

⎪⎭

⎪⎬

⎫

+=

==

∑∑

∑∑

∑∑∑

==

==

===

n

xayb

xxn

yxyx

baxy

cbcaFitBest

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

iii

11

2

11

2

111

21

intercept

nslopea

where So

,, :

…..(1)

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حكمة , ذكاء

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يواجه

Physics Lab AUST


Equations 1 are for general linear regression for best straight line. If we put limits to our graph and force it to be through origin, point (0,0), then equations 1 will change by the assumption which b= 2c =0.

∑=

−=n

iii xcySSE

1

21 ))((

0)(21

11

=−−= ∑=

n

iiii xxcySSE

dcd

01

21

1=− ∑∑

==

n

ii

n

iii xcyx

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

===⎪⎭

⎪⎬

⎫

=

=

∑

∑

=

=

x

yxn

ii

n

iii

SSSP

x

yxslopea

axy

ca,

1

2

11

where So

, :originh Fit througBest

…(2)

We may ask: If the given points do not lie on a straight line, is there a way we can tell

how far off they are from lying on a straight line?

There is a way of measuring the "goodness of fit" of the least squares line, called the coefficient of correlation 2R . We measure this by the fraction

∑

∑

∑∑

∑

=

=

=

=

=

−

−−=

−

+−−=−=

−=

n

ii

n

iii

n

i

n

ii

i

n

iii

yy

xyy

n

yy

baxy

SSMSSER

R

1

2

1

2

1

21

1

2

2

2

))(mean(

))((1

)(

))((1 1

mean thefrom deviations squared of sumline thefrom deviations squared of sum1

….(3)

Physics Lab AUST


Procedure: A. These are readings of some experiments, please use your hand to plot these points on two separated graph paper , then , draw the best straight line (not through the origin) for the first graph and best straight line through origin for the next graph.

According to How To Construct a Line Graph On Paper o Independent Variable is X , Dependent Variable is V o Variable Domain=10-2=8 m, Range=23.5-7.1=16.4~17 m/s o Scales :

o X-scale: (1 m) for each (2cm) on the graph paper ~ need at least 8*2cm=16 cm x-axis.

o Y-scale: (1 m/s) for each (1cm) on the graph paper~ need at least 17*1cm=17 cm y-axis.

o Number and label each axis, Plot the data points and Draw a line that best fits the data points.

o Title the graph.

1. For 1st graph paper a. Start X-axis with 2 m and end point with 10 m. b. Start Y-axis with 5.0 m/s and end point with 24.0 m. c. Draw a line that best fits the data points (not through the origin) d. Find two suitable points from the drawn line: ( ) ( )2211 , ,, yxyx . e. Find the slope=rise/run= ( ) ( )1212 xxyya −−= and intercept 11 axyb −= . f. Write the equation of the best fit: baxy +=

Figure 4: Slope and intercept 6

2. For 2nd graph paper a. Start X-axis with 0 m and end point with 10 m. b. Start Y-axis with 0 m/s and end point with 24.0 m. c. Draw a line that best fits the data points through the origin. d. Find one suitable point from the drawn line: ( )11, yx . e. Find the slope=rise/run= 11 xya = . f. Write the equation of the best fit through origin: axy =

3. What the difference between these two graphs?

X m V m/s 2 7.1 4 10.9 6 16 8 18

10 23.5

Physics Lab AUST


B. Make the graph papers, but now use the statistical equations 1, 2, 3 1. For 3rd graph paper

a. Start X-axis with 2.0 m and end point with 10 m. b. Start Y-axis with 5.0 m/s and end point with 24.0 m. c. Use equations 1 to find the equation for the best fitting line. d. Draw a line that best fits the data points by selecting two values of x from x-

axis values. Let 9 ,3 21 == xx and find ? ?, 21 == yy . e. Draw these two points ( ) ( )2211 , ,, yxyx on the graph paper to draw the line

through them which is representing the equation of best fit. f. Compare it with 1st graph of section A. g. Find the coefficient of correlation 2R .

2. For 4th graph paper

a. Start X-axis with 0 m and end point with 10 m. b. Start Y-axis with 0 m/s and end point with 24.0 m. c. Use equation 2 to find the equation for the best fitting line through origin. d. Draw a line that best fits the data points by selecting one value of x from x-

axis values. Let 71 =x and find ?1 =y . e. Draw these two points ( ) ( )11, ,0,0 yx on the graph paper to draw the line

through them which is representing the equation of best fit through origin. f. Compare it with 2nd graph of section A. g. Find the coefficient of correlation 2R .

3. What the difference between these two graphs (3rd and 4th) according to

coefficient of correlation 2R ? C. Excel7:

1. For 5th graph paper (Not through origin) : Follow up these graphical steps: 1. Open Excel:

Physics Lab AUST


You will get like this shape:

2. Input in column A values of X, and column B values of Y. When you finish ,

Select the first cell call A1 in excel sheet as in fig below

3. Now , Select Chart (be sure that you select Cell A1):

You will get this figure:

Physics Lab AUST


4. Select XY (Scatter) as in figure below:

5. Click Next twice to get this figure below:

Fill in the blanks information about your experiment: chart title, X-axis and Y-axis,

like this figure:

6. Click next, and select (As object in:) as in figure:

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After clicking on the Finish bottom, you will get this figure:

Then right click by mouse on the empty area , select from list (clear) function for

white background. As in figure:

You will get then this figure:

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7. Select any point on the graph, right click on it, select (Add Trendline…) function

Then, you will get this figure with linear type:

Physics Lab AUST


8. Now, click another tab (Options), and check (Display equation on chart and

Display R-squared value on chart) as in figure below,

9. Click ok to get following figure with all information , print out this figure , attach

it with your report as excel results:

10. Compare this graph with 1st and 3rd graphs.

Physics Lab AUST


C. Excel: 2. For 6th graph paper (Through Origin): Follow up these graphical steps:

11. (Repeat steps 1st to 7th of part C.1)

12. Now, click another tab (Options), and check (Display equation on chart , Display

R-squared value on chart, and set intercept = 0) as in figure below,

13. Click ok to get following figure with all information , print out this figure , attach

it with your report as excel results:

14. Compare this graph with 2nd and 4th graphs.

Physics Lab AUST


Students Information: Name Sec: ID Contact # Email Date of Experiment Experiment # Experiment Name

02 Graphing Lab and Excel Results (8 Marks): (Attach all your 6 graphs with this section: Write your name and ID on each graph with its title) You have these readings,

X m 2 4 6 8 10 V m/s 7.1 10.9 16 18 23.5

1. (2 Marks) Follow steps in procedure sections A.1 and A.2 to draw 1st and 2nd graphs.

A.1: ( ) ( ) ..).......... , ...(........., ..),.......... , ...(........., 2211 == yxyx

( ) ( ) ( )( ) . yb ,

- -

111212 =−===−−= axxxyya

. ............ ............ +=+= xbaxy

A.2: ( ) ..).......... , ...(........., 11 =yx , ( )( )

1

1 ===xya , . ............ xaxy ==

What the difference between these two graphs? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2. (4 Marks) B.1 Complete the table below Using equations 1 & 3:

2))((line thefrom deviations squaredSE baxy ii +−== , 2)(mean thefrom deviations squaredSM MeanYyi −==

Reading x y xy x^2 Y^2 SE SM 1 2 7.1 2 4 10.9 3 6 16 4 8 18 5 10 23.5

N Sx Sum X

Sy Sum Y

SPx,y Sum Xy

SSx Sum X^2

SS y Sum Y^2

Sum SE SSE

Sum SM SSM

5 30 75.5 Mean X Mean Y N a b SSM

SSE=2R

6 15.1 5 Draw points: ( ) ( ) ..).......... , 0.9(, ..),.......... , 0.3(, 2211 == yxyx

Compare the resulting line with that of 1st graph in section A: ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Physics Lab AUST


B.2 Complete the table below Using equations 2 & 3: 2)(line thefrom deviations squaredSE ii axy −== ,

2)(mean thefrom deviations squaredSM MeanYyi −== Reading x y xy x^2 Y^2 SE SM

1 2 7.1 2 4 10.9 3 6 16 4 8 18 5 10 23.5

N Sx Sum X

Sy Sum Y

SPx,y Sum Xy

SSx Sum X^2

SS y Sum Y^2

Sum SE SSE

Sum SM SSM

5 30 75.5 Mean X Mean Y N a b SSM

SSE=2R

6 15.1 5 0 Draw points: ( ) ( ) ..).......... , 0.7(, ,0,0 11 =yx

Compare the resulting line with that of 2nd graph in section A: __________________________________________________________________________________________________________________________________________________________________________________________

What the difference between these two graphs (3rd and 4th) according to coefficient of correlation 2R ? __________________________________________________________________________________________________________________________________________________________________________________________

3. Excel (2 Marks)

C.1: Print out excel graph for this section (5th graph) and compare it with 1st

and 3rd graphs:

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

C.2: Print out excel graph for this section (6th graph) and compare it with 2nd

and 4th graphs:

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Physics Lab AUST


Discussion (2 Marks):

1. What you learn from this lab:

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2. What are you prefer, excel or statistical or hand drawing? why?

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Physics Lab AUST

Density using different tools Page 37/102

Experiment 03:

Density using different tools Purpose:

Using Measuring tools to calculate density depending on significant digits rules.

Apparatus : • Transparent ruler , 2 Pencil

(HB) and Eraser • Scientific calculator. • Vernier, Micrometer ,

measuring tape and digital balance.

• Metal piece to find its density.

Web: Vernier: Theory: http://www.rit.edu/~uphysics/VernierCaliper/caliper.html

Virtual: http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=52

Micrometer : Java: reading : http://members.shaw.ca/ron.blond/Micrometer.APPLET/ http://www.upscale.utoronto.ca/PVB/Harrison/Micrometer/Flash/MicSimulation.html Animations: take measurement http://www.upscale.utoronto.ca/PVB/Harrison/Micrometer/Flash/FullAnimation.html

Other precise devices : http://www.microscopyu.com/tutorials/java/reticlecalibration/

Physics Lab AUST


Theory8:

Accuracies of measuring tools are 0.1 cm , 0.01 cm, 0.001 cm for ruler , vernier

and micrometer, respectively.

In a engineering laboratory it is often necessary to determine the length and

masses of objects. Sometime it is necessary to do this with some degree of precision.

Various measuring tools exist for performing such measurements, such as vernier

callipers, micrometers, dial gauges. These instruments are capable of giving very precise

answers, provided the instruments are used with some degree of care.

In this experiment, you will be given a metal sample. You will then take readings

of its spatial dimensions. From these reading you will then determine a value of the mass

of the sample which you will confirm using an electronic balance.

We will describe the operation of a vernier calliper, a micrometer and a dial gauge

(Optional Device). Once you have read the section, the specific experiments that you

perform will be described.

1. Vernier calipers

The diagram below shows a picture of a vernier caliper. This instrument has two

sets of jaws and a stem probe.

One set of Jaws is used for measuring the dimensions of an object by simply

closing the jaws around the outside of the body. The second set of jaws is used for

measuring the interior diameter of a cavity. The stem probe is used for measuring the

depth of a cavity. The screw clamp is used to lock the jaws of the vernier so it can be read

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without modifying the position of the sliding scale. The precision of length measurements

may be increased by using a device that uses a sliding vernier scale. This instrument has a

main scale (in millimetres) and a sliding vernier scale. In figure 1 below, the vernier scale

(below) is divided into 10 equal divisions and thus the highest precision of the instrument

is 0.1 mm (note, some calipers have a scales giving a precision of 0.02 mm). Both the

main scale and the vernier scale readings are taken into account while making a

measurement. The main scale reading is given by the position of the zero mark on the

sliding vernier.

The zero is between the 3 mm and 4 mm on the main scale, this tells us that the

reading should be somewhere between 3 and 4 mm. The 7 mark on the vernier coincided

exactly with one of the divisions (i.e. the 17 mm mark) on the main scale.

Therefore the reading on the vernier is 3.0 + 0.7 = 3.7mm.

The reading on the second vernier below is 15.8 mm.

In is not unusual to find vernier scales with a nominal precision of 0.02 mm as

shown in the picture below. Note that the inset in the top right of the photograph shows

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an enlargement of the vernier near the position where vernier and main scale marks

coincide.

The zero of the vernier is between 37 and 38mm on the normal scale. So the reading is somewhere between 37 and 38 mm. When we read the vernier scale we note that the 46 mark on the vernier coincides with one of the marks on the upper main scale.

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2. The micrometer screw gauge

The micrometer screw gauge is used to measure even smaller dimensions than the vernier

calipers.

The micrometer screw gauge also uses an auxiliary scale (measuring hundredths of a

millimetre) which is marked on a rotary thimble. Basically it is a screw with an

accurately constant pitch (the amount by which the thimble moves forward or backward

for one complete revolution). The micrometers in our laboratory have a pitch of 0.50 mm

(two full turns are required to close the jaws by 1.00 mm). The rotating thimble is

subdivided into 50 equal divisions. The thimble passes through a frame that carries a

millimetre scale graduated to 0.5 mm. The jaws can be adjusted by rotating the thimble

using the small ratchet knob. This includes a friction clutch which prevents too much

tension being applied. The thimble must be rotated through two revolutions to open the

jaws by 1 mm.

In order to measure an object, the object is placed between the jaws and the thimble is

rotated using the ratchet knob until the object is secured. Only the ratchet knob should be

used to secure the object firmly between the jaws. This ensures that the instrument will

gives consistent readings as well as preventing damage to the screw mechanism. The

manufacturer recommends 3 clicks of the ratchet before taking the reading. The lock may

be used to ensure that the thimble does not rotate while you take the reading

The first significant figure is taken from the last graduation showing on the sleeve

directly to the left of the revolving thimble. Note that an additional half scale division

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(0.5 mm) must be included if the mark below the main scale is visible between the

thimble and the main scale division on the sleeve. The remaining two significant figures

(hundredths of a millimetre) are taken directly from the thimble opposite the main scale.

The micrometer reading below is 7.38 mm. In figure 11 the last graduation visible to the left of the thimble is 7 mm and the thimble lines up with the main scale at 38 hundredths of a millimetre (0.38 mm); therefore the reading is 7.38 mm.

In the figure below, the last graduation visible to the left of the thimble is 7.5 mm. Hence the reading is 7.5 mm plus the rotary scale reading of 0.22 mm, giving a total of 7.72 mm.

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Physics Lab AUST


Procedure:

To train on the internet about using these devices , please, work with these websites:

http://members.shaw.ca/ron.blond/Vern.APPLET/index.html

http://members.shaw.ca/ron.blond/Micrometer.APPLET/

After training, Follow these steps:

10. Pick metal piece (Cylindrical Piece like Dirham).

11. Get the diameter and thickness by ruler, vernier, and micrometer, be careful about

significant digits.

12. Get mass using digital Balance.

13. Get volume and density.

14. Tabulate your results.

Physics Lab AUST


Students Information (1 Mark): Name Sec: ID Contact # Email Date of Experiment

Experiment # Experiment Name

03 Density using different tools Results (6 Marks): First: Tabulate your reading with significant figures: Digital Balance Value Unit Mass …………………………….. ……………………………. Diameter ; d Value Unit Vernier …………………………. …………………………. Micrometer …………………………. …………………………. Thickness; h Value Unit Vernier …………………………. …………………………. Micrometer …………………………. ………………………….

Device 2, drraduis =

( )cm

hThickness,

( )cm hrV

VVolume..

,2π=

( )3cm

mMass,

( )gm VmDensity =ρ,

( )3cmgm

Vernier ………… ………… ………… ………… …………

Micrometer ………… ………… ………… ………… …………

Physics Lab AUST


Discussion (3 Marks): 1. What is the reading of

the vernier scale show in the picture1?

…………………….

2. What is the reading of the micrometer shown in the picture1?

………………………..

3. Using rules of significant digits ,and find (review experiment 01): 1.2345 x 52.36 = ………… 1.2345 + 52.36 = ………… 4253/5421 = ………… 4253/5421358 = ………… 23Km+500m = ………… 23Km+50m = …………

Physics Lab AUST

Vectors (free body diagram) Page 46/102

Experiment 04: Vectors (free body diagram)

Purpose: • Study Vectors Depending upon Forces Acting on Body. • Gain skills for finding the resultant using graph paper. • Acquire a capability to analysis vectors. • Learn how to get the length and angle of vectors from its

components. • Understand the equilibrium case and how it is useful to find

practically the resultant of some vectors. • Extend the vectors concepts by doing dot and cross products.

Apparatus: • Inclined plane with

trolley and screw model • 2 Precision

dynamometers , 1.0 N • Protractor. • Transparent ruler. • 2 HB pencils. • Graph Papers. • Scientific Calculator.

Source9

Web Sites: Vector Arithmetic Java Visualization:

http://www.pa.uky.edu/~phy211/VecArith/ Vector Calculator:

http://comp.uark.edu/~jgeabana/java/VectorCalc.html Vector Addition:

http://www.walter-fendt.de/ph11e/equilibrium.htm http://home.a-city.de/walter.fendt/phe/resultant.htm http://www.phy.ntnu.edu.tw/java/vector/vector.html http://www.math.sfu.ca/~hebron/archive/2000-

1/math251/jsp/vectoraddition.html http://www.phys.hawaii.edu/~teb/java/ntnujava/vector/vector

.html http://physics.bu.edu/~duffy/java/VectorAdd.html

Graphing Vector Calculator: http://www.frontiernet.net/~imaging/vector_calculator.html

Vectors and force table: http://explorer.scrtec.org/explorer/explorer-

db/rsrc/783751800-447DED81.2.PDF http://www.phy.olemiss.edu/~thomas/weblab/221_Lab_Man

ual_sum2002/221Vectors_update_sum2002.pdf http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l

3a.html http://www.utm.edu/~cerkal/forcet.htm

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Theory: Resultant is the vector sum of individual vectors. There are some methods to get resultant Graphical (Geometrical) Method: There are two famous methods, Polygon Method http://www.walter-fendt.de/ph11e/resultant.htm and Parallelograms Method http://www.phy.ntnu.edu.tw/java/vector/vector.html ,

Polygon Method10 Parallelograms Method

Two vectors A and B are added by

drawing the arrows which represent

the vectors in such a way that the

initial point of B is on the terminal

point of A. The resultant C = A + B, is

the vector from the initial point of A to

the terminal point of B.

Many vectors can be added together in

this way by drawing the successive

vectors in a head-to-tail fashion:

In the parallelogram method for vector

addition, the vectors are translated, (i.e.,

moved) to a common origin and the

parallelogram constructed as follows:

The resultant R is the diagonal of the

parallelogram drawn from the common

origin.

Determine the scales by using as large as possible of graph area to get more accurate

values. The most accurate method is analytical (component) method.

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Physics Lab AUST


Analytical (component) method:

In this method, analyze each vector to its components, as in the right area of this java applet web sit http://www.pa.uky.edu/~phy211/VecArith/ . If you get the length and angle (magnitude and direction) of any vector, you can easily translate it to its components.

Suppose that Av

in the XY plane has an angle α with x-axis, so, x-component is αcos.AAx = and y-component is αsin.AAy = . If B

v in the XY plane has an angle β

with x-axis, so, x-component is βcos.BBx = and y-component is βsin.BBy = . To find the resultant (vector sum) as length and angle, just apply the following equations:

tan ,

where, ˆˆ

sin.cos.

where, ˆˆ

sin.cos.

where, ˆˆ

where,

22⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎩⎨⎧

+=+=

+=

⎩⎨⎧

==

+=

⎩⎨⎧

==

+=

+=

−

x

yyx

yyy

xxxyx

y

xyx

y

xyx

RR

RRR

BARBAR

jRiRR

BBBB

jBiBB

AAAA

jAiAA

BAR

θ

ββαα

r

r

r

rrr

……(1)

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Physics Lab AUST


Practical Method:

Free body diagram is the vector diagram studies the forces acting on some body. When two or more forces are applied to the object, their vector sum, or resultant, can be found by finding the additional force needed to exactly balance the applied force. For example, if two forces ( A

rand B

r) are applied, the resultant R

r, or vector sum, is

RBArrr

=+ ….(2)

the magnitude and direction of Rr

may be found by finding a third force, ERr

, such that

0=++ ERBArrr

…(3)

When the net force on the object is zero it will remain in equilibrium. The sum of Ar

and Br

must then be equal in magnitude, but opposite in direction, to ERr

, i.e.,

ERBArrr

−=+ and ERRrr

−= …(4)

Current experiment, reading of the inclined balance will represent equilibrium force ERr

. The resultant R

rwill be the same magnitude and opposite direction,

i.e., angle of resultantθ = Eθ ± 180° as in fig.

θE

θE

RE

A

R

B

A

B

Y

X

Fig.1

Free Body Diagram

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Physics Lab AUST


Procedure: 1. Reset Newton's balance to zero reading by displace section a in fig.2

2. Find the weight of trolley, this will represent vector Ar

.

3. Setup your equipments as in fig 2.

4. Fix support at 3 cm for S (in fig.2). Record the width S and height h in table 1.

5. Vector Br

represent the reading of the normal force to the inclined surface by Newton's

balance (always reset balance in the direction of reading).

6. Vector ERr

represent the reading of the parallel force to the inclined surface by

Newton's balance (always reset balance in the direction of reading).

7. Get the resultant Rr

and denote it as methodpracticalR .

rpractically by using equation 4.

(Table1)

Fig. 2

8. Use a graph paper to perform Parallelograms Method to get Rr

and denote it as

methodParallR .

r in the table 2, http://www.phy.ntnu.edu.tw/java/vector/vector.html .

9. Use a graph paper to perform Polygon Method to get Rr

and denote it as methodpolygonR .

r in

the table 2, http://www.walter-fendt.de/ph11e/resultant.htm.

10. Use Analytical (component) method, the most accurate value using equations 1, to

get Rr

and denote it as methodAnalyticalR .

r. (Tables 3 and 4)

Physics Lab AUST


Students Information (1 Mark): Name ........................................................... Sec# ......................... ID ............................................................ Contact # ......................... Email ........................................@................ Experiment Date ......................... Experiment # Experiment Name

04 Vectors (free body diagram) Calculations & Results (6 Marks):

Table 1

cmS ..........= cmh ..........= ( )0.......tan =⎟⎠⎞

⎜⎝⎛= −

Sh

Eθ

Force Vector Magnitude Angle to +x-axis Ar

α = Br

β =

ERr

Eθ =

Resultant: methodpracticalR .

r methodpractical.θ =

Table 2: Graphical (Geometrical) methods

Resultant Magnitude Angle to +x-axis

methodParallR .

r methodParall .θ =

methodpolygonR .

r methodpolygon.θ =

Table 3: Analytical (components) Method- Analysis vectors Ar

& Br

Vector (use Table 1) X-component Y-component

Ar

.............cos. == αAAx .............sin. == αAAy

Br

............cos. == βBBx ............sin. == βBBy

Table 4: Analytical (components) Method- Getting Resultant Components Resultant

..............=+= xxx BAR ......................22. =+== yxmethodAnalytical RRRR

r

..............=+= yyy BAR ( )00. ......180tan =+⎟⎟

⎠

⎞⎜⎜⎝

⎛= −

x

ymethodAnalytical R

Rθ

Physics Lab AUST


Discussion (3 Marks): 1. We found In this experiment: methodParallR .

r, methodpolygonR .

r, methodpracticalR .

r and methodAnalyticalR .

r.

Discuss the magnitudes of these vectors:

______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

2. What skills you acquired from this lab? ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 3. In this experiment, we just use addition procedure, you know about dot and cross product, so, if you have these two vectors:

kjiB

kjiAˆ3ˆˆ2

ˆˆ3ˆ

−−=

−+=r

r

Get

( ) ( ) ( )

( ) ( ) ( )

...............................................................

............

............

ˆˆˆ

........................

==×

=•

kji

BA

BA

rr

rr

Physics Lab AUST

Motion Along Straight line and Newton's laws Page 53/102

Motion Along Straight line and Newton's laws

Experiment 05

• Study motion along a straight line through inclined surface. • Find the dragging force acting for motion and acceleration by

Newton's Second Law. • Understand the effect of rotational dynamics on the motion of

rigid body.

Purpose

• Inclined

Plane with support .

• Cylinders (hollow and solid)

• Stopwatch. • Measuring

Tape.

Apparatus:

• http://occawlonline.pearsoned.com/bookbind/pubbooks/young

_awl/chapter2/objectives/deluxe-content.html: 2.1.4 Sliding on

an incline.

• http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l3e.html

Web

Physics Lab AUST


Theory11: Part A: Neglecting the rotational concepts

When the body moves with acceleration, it means that displacement and velocity

is accelerated by the net force depended on Newton's Second Law as follows:

)1.....(/

/

lim

lim

0

0

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=∆∆

==

∆∆

==

→∆

→∆

MaFtvdtdva

txdtdxv

t

t

Equations (1) are for motion along a straight line, where X is a displacement, V is

the velocity, a represent the body acceleration, and t is the motion time. Acceleration

depends totally on the net force acting on the body, so, constant net force will produce

constant acceleration:

[ ]

)2.(...........

21)(

.,

,

200

00

00

00

0

00

onacceleraticonst

attvxxdtatvdx

atvdtdxatvv

dvdtaconstadvadt

dtdva

dtdxv

tx

x

v

v

tv

v

t

4444444 34444444 21

+=−⇒+=

+=⇔+=⇒

=⇔==

==

∫∫

∫∫∫∫

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If the initial velocity 0v equal zero and S= 0xx − is in figure, so:

)3.......(..........221

22

tSaaatS Calculated ==⇒=

To get the expected (theoretical) value of a, apply Newton's second law:

)4( ............ Sh

Shsin ,sinsin

expected gaa

gamamgmaFX

==⇒

==⇒=⇒=∑ θθθ

Eq.4 represents constant acceleration, if θ kept constant, and eq.3 can written as

)5.......(..........22 Sa

tCalculated

=

Eq.5 reveal the linear relation between S and 2t as long as θ is constant. If we draw

between S on horizontal x-axis and 2t on y-axis, we'll get linear relation through origin

with a slope equal toCalculateda

2 , so:

η

θmgcosθ θ

mgsinθ a

X+ mg

θ

S hمسند

Free body diagram of the body

Initial time

Final time

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(6) ........ 2slope

aCalculated = . And,

...(7) %100%expected

expected ×−

=a

aaError Calculated

Part B: Considering rotational concepts into calculations.

As you see here, we have a rigid body rotates about its axis. We have to include

the rotational concepts to minimize the error above because some of the kinetic energy

will convert to rotational energy. That’s why expected acceleration is substantially

greater than calculated in part A.

Now, we are considering the rotational concepts, the kinetic energy K and

potential energyU :

22 .

21.

21 wIvMK CMCM += , MghU = where

o M : Mass of rigid body,

o CMv : Center of Mass velocity,

o ∑= 2iiCM rmI : Moment of inertia and 2cMRICM = for symmetric bodies of

radius R , c is numerical constant depends on the distribution of mass around axis

of rotation.

• Moment of inertia for hollow cylinder is 2MRICM = gave me 1=c for it.

• Moment of inertia for solid cylinder is 2

21 MRICM = gave me 5.0=c for it.

• Moment of inertia for solid sphere is 2

52 MRICM = gave me 4.0=c for it.

o w : Angular velocity, R

vw CM=

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Here we have two equations of motions, one for translation (Linear or center of

mass - cm) motion and another for rotational motion:

(8) ........ )(sin xcmx mafmgF −=−+=∑ θ

(9) ........ .c.m.R2zzcmz IfR αατ ===∑

Solve 8 and 9 to get xcma − , where zxcm Ra α=− , cylinder rolls without slipping. Eliminating

zα and f :

)10........(sin1

1sin

θ

θ

gc

a

acag

xcm

xcmxcm

⎟⎠⎞

⎜⎝⎛+

=

=−

−

−−

Equation 10 represents the expected value of acceleration. It is depends on the factor

c that is coming from the moment of inertia of rigid body.

)11......(sin 500.0cexpected_h θga =, for hollow cylinder (hc).

)12......(sin667.0cexpected_s θga =, for solid cylinder (sc).

)13......(sin714.0sexpected_s θga =, for solid sphere (ss).

Equation 6 represents the Calculateda value.

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Procedure:

1. Put the support (5 cm height) under the point of 100 cm on S scale.(as in fig. below for hollow cylinder)

2. Put the Cylinder on inclined plane on the position S=100 cm. 3. Let the body move from rest by activating the stopwatch and stopping it when the

body reaches the end of S. Put time as t. 4. Take other values of S (90 cm, 80 cm, 70 cm, and 60 cm) and repeat step 3. 5. Draw values of S on horizontal x-axis and 2t on y-axis. Get the slope of the line

through origin. 6. Get Calculateda using eq.6. 7. Repeat the same experiment with solid cylinder. (repeat steps 1-6)

Part A: Neglecting the rotational concepts

8. Get expecteda using eq.4 and get percentage error using eq.7. 9. Discuss the error.


10. Get expecteda using equations 11 and 12 and get percentage error using eq.7. 11. Compare the errors here with the errors in part A. give me your opinion.

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Student Details(1 Mark): Name ........................................................... Sec# ......................... ID ............................................................ Contact # ......................... Email ........................................@................ Experiment Date ......................... Experiment # Experiment Name

05 Motion Along Straight line Calculations & Results (6 Marks):

cm 100S = cm .... h = .......sin == Shθ 1. Write down your data:

Hollow cylinder )( st

# (m) S )( 1 st )( 2 st )( 3 st )( 3)( 321 stttt ++= )( 22 st

1 1.00 2 0.90 3 0.80 4 0.70 5 0.60

2. Plot on graph paper and on excel the )( 22 st versus (m) S , and find the slope, Graph Paper Excel equation: ………………

Slope hcCalculateda − Slope hcCalculateda − Value Unit Value Unit Value Unit Value Unit

Solid cylinder )( st

# (m) S )( 1 st )( 2 st )( 3 st )( 3)( 321 stttt ++= )( 22 st

1 1.00 2 0.90 3 0.80 4 0.70 5 0.60

3. Plot on graph paper and on excel the )( 22 st versus (m) S , and find the slope, Graph Paper Excel equation: ………………

Slope scCalculateda _ Slope scCalculateda _ Value Unit Value Unit Value Unit Value Unit

Physics Lab AUST


Part A: Neglecting the rotational concepts

4. Get expecteda using eq.4 and obtain percentage error using eq.7 for hollow cylinder-

hc and solid cylinder-sc.

Hollow Cylinder - hc expecteda Graph Paper Excel

Value Unit hcerror% hcerror%

Solid Cylinder - sc expecteda Graph Paper Excel

Value Unit scerror% scerror%

5. Discuss the error.


6. Get cexpected_ha using eq.11 and obtain percentage error using eq.7 for hollow

cylinder-hc.

Hollow Cylinder - hc cexpected_ha Graph Paper Excel

Value Unit hcerror% hcerror%

7. Get cexpected_sa using eq.12 and obtain percentage error using eq.7 for solid

cylinder-sc.

Solid Cylinder - sc cexpected_sa Graph Paper Excel

Value Unit scerror% scerror%

8. Discuss Error and compare it with part A.

Physics Lab AUST



1. Discuss the percentage errors in parts A and B:

______________________________________________________________

______________________________________________________________

______________________________________________________________

2. If S=10.0 m and angle of inclined surface is 30°,

• Get the acceleration for:

slipping object (no friction): _______________

Solid sphere without slipping: _______________

slipping Solid cylinder (no rotation): _______________

• Is there any change to the acceleration if S=20.0 m? Why?

______________________________________________________________

3. Which body rolls down in incline fastest, and why? (Hint use ffii UKUK +=+

to find cmv for each body OR use equation 10)12

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

Physics Lab AUST

Friction Page 62/102

Experiment 06: Friction

Purpose13: • Understanding the friction principle and find the static

friction for Horizontal and Inclined Surface. • Investigating the friction as a function of the area, the weight

and the material • Comparison of friction as a function of the weight and

determining the coefficient of friction Apparatus:

• Pair of wooden

blocks for friction

experiments

• Some Weights, 0.1-

0.5 kg

• 2 Dynamometers

(1.1 N)

• 5 stand rods, 25 cm

Web Sites:

http://www.leybold-didactic.com/literatur/hb/e/p1/p1261_e.pdf

http://www.fearofphysics.com/Friction/frintro.html

http://www.physics.capcollege.bc.ca/lab104/Forces%20friction/i

ndex.htm

http://www.aspire.cs.uah.edu/textbook/experiment.html

http://zebu.uoregon.edu/1999/ph161/friction.html

Java

http://www.phy.ntnu.edu.tw/java/Reaction/reactionTime.html

http://www.fearofphysics.com/Friction/friction.html

Physics Lab AUST


Theory14:

Frictional Forces: We have seen several problems where a body rests or slides

on a surface that exerts forces on the body. Whenever two bodies interact by direct

contact (touching) of their surfaces, we describe the interaction in terms of contact forces.

The normal force is one example of a contact force: we’ll look in detail at another contact

force, the force of friction.

Kinetic and Static Friction When you try to slide a heavy box of books across

the floor, the box doesn’t move at all unless you push with a certain minimum force.

Then the box starts moving, and you can usually keep it moving with less force than you

needed to get it started. If you take some of the books out, you need less force than before

to get it started or keep it moving. What general statements can we make about this

behavior? First, when a body rests or slides on a

surface, we can think of the surface as exerting a

single contact force on the body, with force

components perpendicular and parallel to the

surface (Fig. 1) 15. The perpendicular component

vector is the normal force, denoted by nr . The

component vector parallel to the surface (and

perpendicular to nr ) is the friction force, denoted

by fr

. If the surface is frictionless, then fr

is zero

but there is still a normal force. (Frictionless

surfaces are an unattainable idealization, like a massless rope. But we can approximate a

surface as frictionless if the effects of friction are negligibly small.) The direction of the

friction force is always such as to oppose relative motion of the two surfaces.

The kind of friction that acts when a body slides over a surface is called a kinetic

friction force Kfr

. The adjective “kinetic” and the subscript “k” remind us that the two

surfaces are moving relative to each other. The magnitude of the kinetic friction force

usually increases when the normal force increases. This is why it takes more force to

slide a box full of books across the floor than to slide the same box when it is empty. This

principle is also used in automotive braking systems: The harder the brake pads are

squeezed against the rotating brake disks, the greater the braking effect. In many cases

Physics Lab AUST


the magnitude of the kinetic friction force Kfr

is found experimentally to be

approximately proportional to the magnitude n of the normal force. In such cases we

represent the relationship by the equation

force)friction kinetic of (Magnitude nf KK µ= (1) where Kµ (pronounced “mu-sub-k”) is a constant called the coefficient of kinetic

friction. The more slippery the surface, the smaller the coefficient of friction. Because it

is a quotient of two force magnitudes, Kµ is a pure number, without units.

CAUTION Friction and normal forces are always perpendicular Remember that Eq. (1) is not a vector equation because Kfr

and n are always perpendicular. Rather it is a scalar relationship between the magnitudes of the two forces.

Friction forces may also act when there is no relative motion. If you try to slide a

box across the floor, the box may not move at all because the floor exerts an equal and

opposite friction force on the box. This is called a static friction force Sf . In Fig.216a, the

box is at rest, in equilibrium, under the action of its weight wr and the upward normal

force nr . The normal force is equal in magnitude to the weight ( wn = ) and is exerted on

the box by the floor. Now we tie a rope to the box (Fig.2b) and gradually increase the

Physics Lab AUST


tension T in the rope. At first the box remains at rest because, as T increases, the force of

static friction Sf also increases (staying equal in magnitude toT ).

At some point T becomes greater than the maximum static friction force max−Sf

the surface can exert. Then the box “breaks loose” and starts to slide. Figure 2c shows the

forces when T is at this critical value. If T exceeds this value, the box is no longer in

equilibrium. For a given pair of surfaces the maximum value of Sf depends on the

normal force. Experiment shows that in many cases this maximum value, called max−Sf , is

approximately proportional to n ; we call the proportionality factor Sµ the coefficient of

static friction. Table 117 lists some representative values of Sµ .

friction) static ofnt (Coefficie max_ nf SS µ= (2) In a particular situation, the actual force of static friction can have any magnitude

between zero (when there is no other force parallel to the surface) and a maximum value

max−Sf . In symbols,

force)friction static of (Magnitude nf SS µ≤ (3)

Physics Lab AUST


Like Eq. (1), this is a relationship between magnitudes, not a vector relationship. The

equality sign holds only when the applied force T has reached the critical value at which

motion is about to start (Fig.2c). When T is less than this value (Fig.2b), the inequality

sign holds. In that case we have to use the equilibrium conditions (∑ = 0Fr

) to find Sf .

If there is no applied force ( 0=T ) as in Fig.2a, then there is no static friction force either

( 0=Sf ).

As soon as the box starts to slide (Fig.2d), the friction force usually decreases; it’s

easier to keep the box moving than to start it moving. Hence the coefficient of kinetic

friction is usually less than the coefficient of static friction for any given pair of surfaces,

as Table 1 shows. If we start with no applied force ( 0=T ) and gradually increase the

force, the friction force varies somewhat, as shown in Fig.2e.

Rolling Friction: It’s a lot easier to move a loaded filing cabinet across a

horizontal floor using a cart with wheels than to slide it. How much easier? We can

define a coefficient of rolling friction rµ , which is the horizontal force needed for

constant speed on a flat surface divided by the upward normal force exerted by the

surface. Transportation engineers call rµ the tractive resistance. Typical values of rµ

are 0.002 to 0.003 for steel wheels on steel rails and 0.01 to 0.02 for rubber tires on

concrete. These values show one reason railroad trains are generally much more fuel

efficient than highway trucks.

force)friction rolling of (Magnitude nf rr µ= (4)

This experiment verifies that the static friction force and the kinetic friction force

are independent of the size of the contact surface and proportional to the normal force.

The coefficients of friction depend on the material of the contact surfaces. As the static

friction force is always greater than the kinetic friction force, we can always say

kS µµ > (5) To distinguish between Kinetic and rolling friction, the friction block is placed on

top of multiple stand rods laid parallel to each other. The rolling friction force

Physics Lab AUST


Procedure18: 1. Prepare clean, dry and smooth experiment surfaces (e.g. laboratory bench) for the

friction experiments.

2. If the resulting frictional forces are too slight, use different base surfaces.

3. Using the dynamometer, determine the weight (force of gravity) 1W of the large

wooden block and 2W for the small block.

Part A. Static and kinetic friction as a function of the area, the weight and the material:

4. Place the small block on the experiment surface with the plastic side down.

5. Using the dynamometer, measure the maximum horizontal pulling force at which

the body remains stationary on the experiment surface as the static friction

force max−Sf . (Fig.3)

6. Measure the horizontal pulling force which maintains a uniform motion on the

experiment surface as the kinetic friction force Kf .

7. Place the wooden block on the base surface with the wide wooden side and then

the narrow wooden side down and repeat your measurements for max−Sf and Kf .

Physics Lab AUST


8. Repeat the measurements with the large block for friction experiments.

9. Repeat the measurement on other surfaces as desired.

10. Tabulate your results in Table A and answer the questions related to the obtained

data.

Part B. Static and kinetic friction as a function of the force of gravity:

11. Place the small block on the experiment surface

with the plastic side up and measure the static and

kinetic friction force. (fig. 4)

12. Increase the weight of the block by adding in turn

the weights 0.1 kg, 0.2 kg, 0.5 kg and 1.0 kg and repeat the measurements.

13. Plot on graph paper and on excel the ( NfS max− ) and NfK on normal axis and

( Nn ) on horizontal axis, find slopes then Sµ and kµ for Plastic and wooden sides.(4 straight lines on the same graph OR make 4 separate graphs for each case). [All lines should be through origin]

14. Slope of any straight line represent the coefficient of friction corresponding to friction kind.(According to equations 1 and 2)

Physics Lab AUST


Part C. Rolling and kinetic friction as a function of the area, the weight and the material:

15. Lay the stand rods next to each other and place the

large block on the rods with the plastic side down.

16. Measure the horizontal pulling force which maintains

a uniform motion on the rolling rods as the rolling friction force rf .(fig.5)

17. Increase the weight of the block by adding in turn the weights 0.1 kg, 0.2 kg, 0.5

kg and 1.0 kg and repeat the measurements.

18. Align the block parallel to the rod axes and measure the sliding friction force.

19. Plot on graph paper and on excel the ( NfK ) and Nfr on normal axis and

( Nn ) on horizontal axis, find slopes then kµ and rµ .(2 straight lines on the same graph OR make 2 separate graphs for each case). [All lines should be through origin]

20. Slope of any straight line represent the coefficient of friction corresponding to friction kind.(According to equations 2 and 3).

21. Discuss your results.

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Student Details(1 Mark):

Name ........................................................... Sec# ......................... ID ............................................................ Contact # ......................... Email ........................................@................ Experiment Date ......................... Experiment # Experiment Name

06 Friction Calculations & Results (6 Marks):

9. Write down your data and answer the questions:

1W = _________ N

2W = _________ N

Part A. Static and kinetic friction as a function of the area, the weight and the material: Table A

NceNormal for Material 2rea/cmA NfS max− NfK Sµ kµ

1W = Plastic 12x6

1W = Wood 12x6

1W = Wood 12x3

2W = Plastic 12x6

2W = Wood 12x6 As the measuring results:

• Both the static friction force max−Sf and the kinetic friction force Kf depend on the material properties of the friction surfaces and on the weight of the blocks. Is this true?

______________________________________________________________ • Are the friction forces dependent of the size of the friction area?

______________________________________________________________ • Are the coefficients of friction depending on the contact area?

______________________________________________________________

Physics Lab AUST


Part B. Static and kinetic friction as a function of the force of gravity:

Table B1- Experiment surface: Wood-coated benchtop Plastic Side Wooden Side

Nn NfS max− NfK NfS max− NfK

10. Plot on graph paper and on excel the ( NfS max− ) and NfK on normal axis and ( Nn ) on horizontal axis, find slopes then Sµ and kµ for Plastic and wooden sides.(4 straight lines on the same graph OR make 4 separate graphs for each case).

Table B2- Coefficient of friction represents the slope of line Graph Paper Excel

Plastic Side Wooden Side Plastic Side Wooden Side Sµ kµ Sµ kµ Sµ kµ Sµ kµ

As the measuring results:

• Did you satisfy the equation 5? ______________________________________________________________

Part C. Rolling and kinetic friction as a function of the area, the weight and the material: Table C

Plastic Side Graph Paper Nn Nfr NfK rµ kµ

Excel rµ kµ

As the measuring results:

• Did you satisfy that the rolling coefficient is much less than kinetic coefficient? ______________________________________________________________

Physics Lab AUST



4. Is the increasing of the contact area of rubber tires (for race cars) with concrete is to increase the coefficient of friction? Why?

______________________________________________________________

______________________________________________________________

______________________________________________________________

5. Theoretically, do you expect that Sµ is the same in Table A and Table B2 for the same material? Why?

______________________________________________________________

______________________________________________________________

______________________________________________________________

6. Put the right sentence in the blanks:

Static Friction Force Kinetic Friction Force Symbol Direction

Magnitude 1. Fs 2. Fk 3. Opposite direction to the resultant of horizontal forces acting on static body. 4. Parallel direction to the resultant of horizontal forces acting on static body. 5. Opposite direction to the resultant of horizontal forces acting on moving body. 6. Parallel direction to the resultant of horizontal forces acting on moving body. 7. Opposite direction to the moving body. 8. Parallel direction to the moving body. 9. snµ≥

10. snµ≤

11. snµ=

12. knµ≥

13. knµ=

14. knµ≤

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Spiral Spring – Hooks Law Page 73/102

Experiment 07: Spiral Spring – Hooks Law

Purpose:

(1) to study simple harmonic motion,

(2) to determine the spring constant of a spiral spring.

(3) to determine the effective mass of spring.

Apparatus :

(1) A support stand,

(2) a spiral spring,

(3) a set of weights,

(4) a weight hanger,

(5) a meter stick,

(6) and a stop watch.

Web:

http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html

http://oneweb.utc.edu/~Harold-Climer/Sconstantlab.pdf

http://www2.dsu.nodak.edu/users/edkluk/UPWeb/UPLab_HT

ML/UPM7/UPM7_w.html

http://cougar.slvhs.slv.k12.ca.us/~pboomer/labsphys/physlabo

ok/lab13.html

http://www.walter-fendt.de/ph11e/springpendulum.htm

http://hyperion.cc.uregina.ca/~bergbusp/uglabs/p112/Experime

nts/Expt01SHM07.pdf

http://www.phys.utk.edu/labs/HookesLaw.pdf

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Theory:

The restoring force of a stretched spring is proportional to its elongation, if the

deformation is not too great. This relationship for elastic behavior is known as Hooke's

law and is described by19

xkF vv−= (1a)

where Fv

is the force, k is the spring constant, and xv is the elongation of the spring. The

force is in the opposite direction to its elongation, as shown by the minus sign. For a

system such as in figure 1, the spring's elongation, xv , is determined by the spring

constant k and the mass m, and is independent of the spring's own mass. Thus applying

equation 1 to this system results in

kxmg = (2)

then the motion of the spiral spring will be simple harmonic motion.

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject

to the linear elastic restoring force given by Hooke’s Law. The motion is sinusoidal in

time and demonstrates a single resonance frequency20:

X

Fig 1 F=W

1 2 Spiral Spring

F=W

Fsp=KX

X

-X

Free-body diagram

Physics Lab AUST


.

period1

T1f f, 2 and ,

mk where

sin sin

2 ====

==

πωω

ω tmkAtAy

(1b)

and its period T can be calculated using the equation for the period of simple harmonic

motion.

kmT π2= (3)

where m is the equivalent mass of the system. The equivalent mass of the system is the

sum of the mass, M , which hangs from the spring and the spring's equivalent mass

(effective mass) , om , or

omMm += (4) Note that om is not the actual mass of the spring but the equivalent (effective) mass.

Substituting this into equation 3 and the result is

( )omMk

T +=2

2 4π (5)

From the graph of 2T vs. mass, M , k and om can be found.

Physics Lab AUST


Procedure21:

1. Hang a 0.100 kg mass to the spiral spring, start the oscillation in a vertical

direction, and measure the time required for twenty complete oscillations. The

time for twenty oscillations should be repeated at least 3 times. [The small

amplitude is required in the oscillation. (2 or 3 cm will provide better result.)]

2. Calculate T, the system's period of oscillation.

3. Repeat step 2 for the masses 0.002 kg, 0.300 kg, 0.400 kg and 0.500 kg.

4. On Graph paper: Plot a graph of 2T vs. mass, M , and determine the spring

constant k from the relationship of slopek /4 2π= of this line.

5. On Graph paper: Determine om the equivalent (effective) mass of the spring by

extending the line down to cross the x-axis and figure the intercept on the x-axis.

6. On Excel: Graph the period T as a function of mass, M , using the chart feature

of Excel. The mass is the independent variable and should be plotted on the

horizontal axis or abscissa (x axis). The period is the dependent variable and

should be plotted on the vertical axis or ordinate (y axis).

7. On Excel: Use the trendline feature to draw a smooth curve that best fits your

data. To do this, from the main menu, choose Chart and then Add Trendline . . .

from the dropdown menu. This will bring up a Add Trendline dialog window.

From the Trend tab, choose Power from the Trend/Regression type selections.

Then click on the Options tab and select Display equations on chart and Display

R-squared value on chart options.

8. Examine the power function equation that is associated with the trendline. Does it

suggest the relationship between period and mass given by Equation (3)?

9. Examine your graph and notice that the change in the period per unit mass, the

slope of the curve, decreases as the mass increases. This indicates that the period

increases with the length at a rate less than a linear rate. The theory and Equation

(3) predict that the period depends on the square root of the mass. A graph of 2T versus M should result in a straight line.

10. Square the values of the period measured for each mass and record your results in

the table.

Physics Lab AUST


11. On Excel: Use the chart feature again to graph the period squared, T2, as a

function of the mass. The period squared is the dependent variable and should be

plotted on the y axis. The mass is the independent variable and should be plotted

on the x axis.

12. Examine your graph of 2T versus M and check to see if there is a linear

relationship between 2T and M so that the data points lie along a line.

13. On Excel: Use the trendline feature to perform a linear regression to find a

straight line that best fits your data points. This time from the Add Trendline

dialog window. Choose Linear from the Trend/Regression type selections. Click

on the Options tab and once again select the Display equations on chart and

Display R-squared value on chart options. This should draw a straight line (not

through origin) that best fits the data and should display the equation for this

straight line.

14. On Excel: Equation (5) is of the form baxy += where 2Ty = and ka /4 2π= ,

oamb = , Mx = . A graph of 2T versus M should therefore result in a straight line

whose slope, a, is equal to k/4 2π . From the equation for the trendline, record the

value for the slope, a, the interception, b. From the equation ka /4 2π= find k , the

spring constant, and from the equation oamb = find om , the spring equivalent

(effective) mass.

Physics Lab AUST


Students Information (1 Mark): Name Sec ID Contact # Email Date of Experiment


07 Spiral Spring – Hooks Law Results (6 Marks):

20 Oscillations time : t (s) Mass M (Kg) t1

(S) t2 (S)

Mean Value (S)

T=t/20 (S)

2T (S2)

Plot ( 2T ) on normal axis against (M) on horizontal axis. Get the slope then Find K and

om Y= …. X + …… Excel Graph Paper

y-Intercept Slope x-Intercept Slope Unit Value Unit Value Unit Value Unit Value

om SlopeK 24π= om SlopeK 24π= Unit Value Unit Value Unit Value Unit Value

Conclusion: ______________________________________________________________

______________________________________________________________

______________________________________________________________

Physics Lab AUST


Discussion (3 Marks): Q1: If the amplitude of the vibration for a spring is doubled, what happens to the period?

Answer this question by measuring the period of an oscillating 250 gram mass whose

amplitudes of vibration are 2 cm, 4 cm, and 8 cm.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

Q2: Choose End points or Equilibrium point for each question.

Question End points

Equilibrium point

At which point(s) does the mass on a vibrating spring have the greatest acceleration?

At which point(s) does it have the least acceleration? At which point(s) does the mass have the largest force exerted on it?

At which point(s) does the mass have the smallest force exerted on it?

At which point(s) does the mass on a vibrating spring have its largest velocity?

At which point(s) does the mass have its smallest velocity?

Q3: You will do this experiment on the moon. What is the result of spring constant K in

this case? Why?

______________________________________________________________

______________________________________________________________

______________________________________________________________

Physics Lab AUST

Simple Pendulum Page 80/102

Experiment 08:

Simple Pendulum Purpose:

(1) to study the motion of a simple pendulum, (2) to study simple harmonic motion, (3) to learn the definitions of period, frequency, and amplitude, (4) to learn the relationships between the period, frequency, amplitude and length of a simple pendulum and (5) to determine the acceleration due to gravity using the theory, results, and analysis of this experiment.

Apparatus : (1) A support stand with a string clamp, (2) a small spherical ball, (3) a 125 cm length of light string, (4) a meter stick, (5) a vernier caliper, (6) and a timer.

Web:

http://lectureonline.cl.msu.edu/~mmp/labs/labpend/lab.htm http://phoenix.phys.clemson.edu/labs/223/sample/ http://www.phy.ntnu.edu.tw/java/Pendulum/Pendulum.html http://www2.dsu.nodak.edu/users/edkluk/UPWeb/UPLab_HTML/UPMC6/UPMC6_w.html http://www.phys.utk.edu/labs/SimplePendulum.pdf

Theory:

A simple pendulum may be described ideally as a point mass suspended by a massless

string from some point about which it is allowed to swing back and forth in a place. A

simple pendulum can be approximated by a small metal sphere which has a small radius

and a large mass when compared relatively to the length and mass of the light string from

which it is suspended. If a pendulum is set in motion so that is swings back and forth, its

motion will be periodic. The time that it takes to make one complete oscillation is defined

as the period T. Another useful quantity used to describe periodic motion is the frequency

of oscillation. The frequency f of the oscillations is the number of oscillations that occur

Physics Lab AUST


per unit time and is the inverse of the period, f = 1/T. Similarly, the period is the inverse

of the frequency, T = l/f. The maximum distance that the mass is displaced from its

equilibrium position is defined as the amplitude of the oscillation.

When a simple pendulum is displaced from its equilibrium position, there will be a

restoring force that moves the pendulum back towards its equilibrium position. As the

motion of the pendulum carries it past the equilibrium position, the restoring force

changes its direction so that it is still directed towards the equilibrium position. If the

restoring force Fv

is opposite and directly proportional to the displacement x from the

equilibrium position, so that it satisfies the relationship22

xkF vv−= (1)

then the motion of the pendulum will be simple harmonic motion and its period can be

calculated using the equation for the period of simple harmonic motion.

kmT π2= (2)

It can be shown that if the amplitude of the motion is kept small, Equation (2) will be

satisfied and the motion of a simple pendulum will be simple harmonic motion, and

Equation (2) can be used.

A

l

S

F=mgsinθ mgcosθ

mg

Simple Pendulum

T

Free body Diagram at A

θ

Y

F=mgsinθ mgcosθ

mg

X

T

Physics Lab AUST


The restoring force for a simple pendulum is supplied by the vector sum of the

gravitational force on the mass. mg, and the tension in the string, T. The magnitude of the

restoring force depends on the gravitational force and the displacement of the mass from

the equilibrium position. Where a mass m is suspended by a string of length l and is

displaced from its equilibrium position O by an angle θ and a distance x along the arc OA

through which the mass moves. The gravitational force can be resolved into two

components, one along the radial direction, away from the point of suspension, and one

along the arc in the direction that the mass moves. The component of the gravitational

force along the arc AO provides the restoring force F and is given by

θsinmgF −=v

(3) where g is the acceleration of gravity, θ is the angle the pendulum is displaced, and the

minus sign indicates that the force is opposite to the displacement. For small amplitudes

where θ is small, sinθ can be approximated by θ measured in radians so that Equation (3)

can be written as

θmgF −=v

(4)

The angle θ in radians is lx the arc length divided by the length of the pendulum or the

radius of the circle in which the mass moves. The restoring force is then given by

lxmgF −=

v (5)

and is directly proportional to the displacement x and is in the form of Equation (1) where

k =l

mg . Substituting this value of k into Equation (2), the period of a simple pendulum

can be found by

⎟⎠⎞

⎜⎝⎛

=

lmgmT π2 (6)

and

glT π2= (7)

Therefore, for small amplitudes the period of a simple pendulum depends only on its

length and the value of the acceleration due to gravity.

Physics Lab AUST


Procedure23:

15. The simple pendulum is composed of a small spherical ball suspended by a long,

light string which is attached to a support stand by a string clamp. The string

should be approximately 125 cm long and should be clamped by the string clamp

between the two flat pieces of metal so that the string always pivots about the

same point.

16. Use a vernier caliper to measure the diameter d of the spherical ball and from this

calculate its radius r. Record the values of the diameter and radius in meters.

17. Adjust the length of the pendulum to about 50 cm. The length of the simple

pendulum is the distance from the point of suspension to the center of the ball.

Measure the length of the string ls from the point of suspension to the top of the

ball using a meter stick. Add the radius of the ball to the string length ls and

record that value as the length of the pendulum L=ls +r.

18. Displace the pendulum about 5º from its equilibrium position and let it swing

back and forth. Measure twice the total time that it takes to make 20 complete

oscillations. Record that time in your table.

19. Increase the length of the pendulum by about 0.20 m and repeat the measurements

made in the previous steps until the length increases to approximately 1.5 m.

20. Calculate the period of the oscillations for each length by dividing the total mean

time value by the number of oscillations, 20. Record the values in the appropriate

column of your data table.

21. On Excel: Graph the period of the pendulum as a function of its length using the

chart feature of Excel. The length of the pendulum is the independent variable and

should be plotted on the horizontal axis or abscissa (x axis). The period is the

dependent variable and should be plotted on the vertical axis or ordinate (y axis).

Physics Lab AUST


22. On Excel: Use the trendline feature to draw a smooth curve that best fits your

data. To do this, from the main menu, choose Chart and then Add Trendline . . .

from the dropdown menu. This will bring up a Add Trendline dialog window.

From the Trend tab, choose Power from the Trend/Regression type selections.

Then click on the Options tab and select the Display equations on chart and

Display R-squared value on chart options.

23. Examine the power function equation that is associated with the trendline. Does it

suggest the relationship between period and length given by Equation (7)?

24. Examine your graph and notice that the change in the period per unit length, the

slope of the curve, decreases as the length increases. This indicates that the period

increases with the length at a rate less than a linear rate. The theory and Equation

(7) predict that the period depends on the square root of the length. If both sides

of Equation 7 are squared then lg

T2

2 4π= If the theory is correct, a graph of T2

versus l should result in a straight line.

lg

T2

2 4π= (8)

25. Square the values of the period measured for each length of the pendulum and

record your results in the table.

26. On Excel: Use the chart feature again to graph the period squared, T2, as a

function of the length of the pendulum l . The period squared is the dependent

variable and should be plotted on the y axis. The length is the independent

variable and should be plotted on the x axis.

Physics Lab AUST


27. Examine your graph of T2 versus l and check to see if there is a linear relationship

between T2 and l so that the data points lie along a line.

28. On Excel: Use the trendline feature to perform a linear regression to find a

straight line that best fits your data points. This time from the Add Trendline

dialog window. Choose Linear from the Trend/Regression type selections. Click

on the Options tab and once again select the Display equations on chart , Display

R-squared value on chart and set intercept =0 options. This should draw a

straight line through origin that best fits the data and should display the equation

for this straight line.

29. On Excel: Equation (8) , lg

T2

2 4π= , is of the form y=ax where y= T2 and a=

g

24π , x= l . A graph of T2 versus l should therefore result in a straight line whose

slope, a, is equal to g

24π . From the equation for the trendline, record the value

for the slope, a, and from the equation a= g

24π find g, the acceleration due to

gravity, which represent calculated value practicalg .

30. On Graph Paper: Plot ( 2T )on normal axis against (L) on horizontal axis. Get the

slope then Find practicalg .

31. Compare your result with the expected value of the acceleration due to gravity 9.8

m/ s2. Calculate the Percent Error in your result and the accepted result.

Physics Lab AUST


Students Information (1 Mark): Name Contact # ID Day and Time Email Date of Experiment


08 Simple Pendulum Results (6 Marks):

20 Oscillations time : t (s) Pendulum Length

L (m) t1 (S)

t2 (S)

Mean Value (S)

T=t/20 (S)

2T (S2)

On Excel: Graph the period of the pendulum as a function of its length using the chart feature of Excel. Examine the power function equation that is associated with the trendline. Does it suggest the relationship between period and length given by Equation (7)?

______________________________________________________________ ______________________________________________________________ ______________________________________________________________

On Graph Paper: Plot ( 2T )on normal axis against (L) on horizontal axis. Get the

slope then Find practicalg .

Excel Y=…….X Graph Paper Slope Slope

Unit Value Unit Value

practicalg practicalg Unit Value Unit Value

%Error %Error

Physics Lab AUST


Discussion (3 Marks): Q1: Discuss your errors: ______________________________________________________________

______________________________________________________________

______________________________________________________________

Q2: At °20 angle, I can not apply the same formula. Why? ______________________________________________________________

______________________________________________________________

______________________________________________________________

Q3: Use three different weights of pendulum, and measure the oscillation time. From your results, is oscillation time increase or decrease (support your answer with expected result and calculated result). A. Discuss theoretical (Expected) Result: ______________________________________________________________

______________________________________________________________

______________________________________________________________

B. Discuss practical results: ______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

Physics Lab AUST

Angular simple harmonic motion Page 88/102

Experiment 09 : Angular simple harmonic motion

Purpose24:

o Measuring the period of oscillation of a thin transverse rod with

weights on a torsion axle as a function of the distance of the

weights from the torsion axle.

o Confirming the proportionality between the moment of inertia of

the weights and the square of the distance.

o Determining the restoring torque of the torsion axle.

Apparatus:

• torsion axle

• stand base, V-

shape, 20 cm

• stopwatch

Web Sites:

http://www.colorado.edu/physics/phys2010/phys2010LabMan2000/

2010labhtml/Lab4/EXP4LAB99.html

http://www2.dsu.nodak.edu/users/edkluk/UPWeb/UPLab_HTML/U

PMC11/CPMC11_w.html

virtual:

http://www.explorelearning.com/index.cfm?method=cResource.dsp

View&ResourceID=34

Physics Lab AUST


Theory:

A mechanical watch keeps time based on the oscillations of a balance wheel (Fig.

1) 25. The wheel has a moment of inertia I about its axis. A coil spring exerts a restoring

torque zτ that is proportional to the angular

displacement θ from the equilibrium position. We

write κθτ =z , where κ (the Greek letter kappa) is

a constant called the torsion constant. Using the

rotational analog of Newton’s second law for a

rigid body, 22 dtdIIz θατ ==∑ , we can find

the equation of motion:

θIκ

dtθdI −==− 2

2

or ακθ

The form of this equation is exactly the same for the acceleration for simple harmonic

motion, with x replaced by θ and mk replaced by Iκ . So we are dealing with a form

of angular simple harmonic motion. The angular frequency ω and frequency f are given

by equation 1:

SHM)(angular 1period , 21 and

fT

If

I====

κπ

κω (1)

The motion is described by the function )cos( φωθ +Θ= t where Θ (the Greek letter

theta) plays the role of angular amplitude.26

The moment of inertia27 is a measure of the inertia that a body exhibits when a

torque acts on it causing a change of its rotational motion. It corresponds to the inertial

mass in the case of translational motions. In rotational oscillations, for example, the

period of oscillation T is the greater, the greater the moment of inertia J of the

oscillating system is. From equation 1:

SHM)(angular 2 κ

π IT = (2)

The moment of inertia of a pointlike mass m moving on a circular path with radius r is

Physics Lab AUST


21 rmI ⋅= (3a)

The moment of inertia of two equal masses m that are rigidly connected and have the same distance r from the axis of rotation is:

22 2 rmI ⋅⋅= (3b)

In both cases, the moment of inertia is proportional to the square of the distance r .

In the experiment, the rigid connection between the two masses is established by

means of a thin rod whose middle is fixed to the torsion axle (see Fig. 2). After deflection

from the equilibrium position, the system oscillates with the period of oscillationT .

From Equation 2 it follows that:

SHM)(angular 2

2

⎟⎠⎞

⎜⎝⎛⋅=π

κ TI (4)

Physics Lab AUST


However, the moment of inertia is composed of the moment of inertia 2I of the two

weights and the moment of inertia 0I of the rod:

022 IrmI +⋅⋅= (5)

Therefore the period of oscillation 0T of the rod without weights is measured in another

measurement, which leads to

π

T κrm π

T κ2

022

22

2⎟⎠⎞

⎜⎝⎛⋅+⋅⋅=⎟

⎠⎞

⎜⎝⎛⋅

Or

20

22

2 8 TrmT +⋅⋅⋅

=κπ (6)

Thus a linear relation between the square of the period of oscillation T and the square of

the distance r is obtained. From the slope of the straight line,

8slope 2

κπ⋅⋅

==ma (7)

the restoring torque κ can be calculated if the mass m is known.

Physics Lab AUST


Procedure28:

The experimental setup is illustrated in Fig 3.

1. Fix the middle of the transverse rod to the torsion axle and arrange the weights

symmetrically at a distance of 30 cm from the torsion axle.

2. Mark the equilibrium position on the table.

3. Rotate the transverse rod to the right by 180° and release it.

4. Start the time measurement as soon as the transverse rod passes through the

equilibrium position and stop the measurement after five oscillations.

5. Repeat the measurement four times, alternately deflecting the rod to the left and to

the right.

6. Calculate the period of oscillation T from the mean value of the five measured

values.

Physics Lab AUST


7. One after another reduce the distance to 25 cm, 20 cm, 15 cm, 10 cm and 5 cm,

each time repeating the measurement.

8. Remove the weights, and repeat the measurement.

9. Calculate 2r and 2T for each r . Record the values in the appropriate column of

your data table.

10. On Graph paper: Plot a graph of 2T vs. 2r , and determine the torsion constant κ

from the relationship of ( ) slope8 2πκ ⋅⋅= m of this line.

11. On Excel: Use the chart feature to graph the period squared, 2T , as a function of

the square of the distance 2r . The period squared is the dependent variable and

should be plotted on the y axis. The square of the distance is the independent

variable and should be plotted on the x axis.

12. On Excel: Examine your graph of 2T versus 2r and check to see if there is a linear

relationship between 2T and 2r so that the data points lie along a line. Use the

trendline feature to perform a linear regression to find a straight line that best fits

your data points. From the Add Trendline dialog window. Choose Linear from

the Trend/Regression type selections. Click on the Options tab and once again

select the Display equations on chart and Display R-squared value on chart

options. This should draw a straight line (not through origin) that best fits the data

and should display the equation for this straight line.

13. On Excel: Equation (6) is of the form baxy += where 2Ty = and

( ) 8 2 κπ⋅⋅= ma , 20aTb = , 2rx = . A graph of 2T versus 2r should therefore

result in a straight line whose slope, a, is equal to ( ) 8 2 κπ⋅⋅m . From the

equation for the trendline, record the value for the slope, a, the interception, b.

From the equation ( ) 8 2 κπ⋅⋅= ma findκ , the torsion constant.

14. Get the percentage errors for each shape, if you know that Nm023.0expected =κ

where ( )[ ] %100% ⋅−= ExpectedCalcultedExpectederror .

Physics Lab AUST



Name ........................................................... Sec# ......................... ID ............................................................ Time & Date ......................... Email ........................................@................ Experiment Date ......................... Experiment # Experiment Name

10 Moment of inertia and body shape Calculations & Results (6 Marks):

distances Time for 5 oscillations S Oscillation time

mr 1t 2t 3t 4t 5t S

55

5

1

×=∑ it

T

22 mr 22 ST

0.30 0.25 0.20 0.10 0.05

Without weights

Plot ( 22 ST ) on normal axis against ( 22 mr ) on horizontal axis. Get the slope then Find κ where Kg 0.240m = .

Graph Paper Excel Y= …. X + …… Slope Slope

Value Unit Value Unit

( ) Kg 0.240m ,slope8 2 =⋅⋅= πκ m ( ) Kg 0.240m ,slope8 2 =⋅⋅= πκ m

Value Unit Value Unit

%error = %error= Conclusion: ______________________________________________________________

______________________________________________________________

______________________________________________________________

Physics Lab AUST



1. Discuss the %error of each item:

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

2. Using same setup: Can I use equation 4 to find the mass of solid sphere on some

spaceship where there is no gravity? How?( mRRmI esolidspher 145.02 ,..52 2 == )

______________________________________________________________

______________________________________________________________

=m __________________________________________________________

3. Did A hollow body and a solid body with same mass make the same T. Why?

______________________________________________________________

______________________________________________________________

______________________________________________________________

Moment of inertia Page 96/102

Experiment 10 : Moment of inertia

Purpose29:

• Determining the moments of inertia of rotationally symmetric

bodies from their period of oscillation on a torsion axle.

• Comparing the periods of oscillation of two bodies having different

masses, but the same moment of inertia.

• Comparing the periods of oscillation of hollow bodies and solid

bodies having the same mass and the same dimensions.

• Comparing the periods of oscillation of two bodies having the

same mass and the same body shape, but different dimensions.

Apparatus:

• torsion axle

• set of cylinders

for torsion axle

• sphere for the

torsion axle

• stand base, V-

shape, 20 cm

• stopwatch

Web Sites:

http://www.colorado.edu/physics/phys2010/phys2010LabMan2000/

2010labhtml/Lab4/EXP4LAB99.html

http://www2.dsu.nodak.edu/users/edkluk/UPWeb/UPLab_HTML/U

PMC11/CPMC11_w.html

virtual:

http://www.explorelearning.com/index.cfm?method=cResource.dsp

View&ResourceID=34


Theory30:

The moment of inertia is a measure of the resistance of a body against a change of

its rotational motion and it depends on the distribution of its mass relative to the axis of

rotation. For a calculation of the moment of inertia I, the body is subdivided into

sufficiently small mass elements im∆ with distances ir from the axis of rotation and a

sum is taken over all mass elements:

(1) . 2∑∆=i

ii rmI

For bodies with a continuous mass distribution, the sum can be converted into an integral.

If, in addition, the mass distribution is homogeneous, the integral reads

iii

iiiii

ii vmVMrvrvrmI ∆=∆=∆=∆=∆= ∑∑∑ ρρρρ , , ... 2

i

22

(2) .1. 2∫=V

dVrV

MI

M: total mass, V: total volume, r: distance of a volume element dV from the axis of

rotation.

The calculation of the integral is simplified when rotationally symmetric bodies

are considered which rotate around their axis of symmetry. The simplest case is that of a

hollow cylinder with radius R. As all mass elements have the distance R from the axis of

rotation, the moment of inertia of the hollow cylinder is:

(3) . 2RMI nderhollowcyli =


In the case of a solid cylinder with equal mass M and equal radius R, Eq. (2) leads

to the formula:

.Hπ.RVdrHrrV

MIR

dersolidcylin2

0

2 with ...2.1. == ∫ π and the result is:

(4) ..21 2RMI dersolidcylin =

That means, the moment of inertia of a solid cylinder is smaller than that of the

hollow cylinder as the distances of the mass elements from the axis of rotation are

between 0 and R.

An even smaller value is expected for the moment of inertia of a solid sphere with

radius R (see Fig. 1). In this case, Eq. (2) leads to the formula:

3

0

222

34 with ...2.1. π.RVdrrRrr

VMI

R

esolidspher =−= ∫ π

(5) ..52 2RMI esolidspher =

Thus, apart form the mass M and the radius R of the bodies under consideration a

dimensionless factor enters the calculation of the moment of inertia, which depends on

the shape of the respective body.

The moment of inertia is determined from the period of oscillation of a torsion

axle, on which the test body is fixed and which is connected elastically to the stand via a

helical spring. The system is excited to perform harmonic oscillations. If the restoring

torque κ is known, the moment of inertia of the test body is calculated from the period

of oscillation T according to:

(6) 2

.2

⎟⎠⎞

⎜⎝⎛=π

κ TI


Procedure31:

15. The experimental setup is illustrated in Fig 2.

16. Put the sphere on the torsion axle, and mark the equilibrium position on the table.

17. Rotate the sphere to the right by 180° and release it.

18. Start the time measurement as soon as the sphere passes through the equilibrium

position and stop the measurement after five oscillations. (repeat this 3 times)

19. Calculate the average period of oscillation T.

20. Replace the sphere with the disk, and repeat the measurement.

21. Replace the disk with the supporting plate.

22. Repeat the measurement with the solid cylinder and then with the hollow

cylinder.

23. Finally carry out the measurement with the empty supporting plate.

24. Get calculated Moment of Inertia calculated I using Eq.(6) as in table.

25. Get expected Moment of Inertia expectedI using Eqs.(3-5).

26. Get the percentage errors for each shape.%100% ×

−=

ExpectedCalculatedExpected

error



Name ........................................................... Sec# ......................... ID ............................................................ Time & Date ......................... Email ........................................@................ Experiment Date ......................... Experiment # Experiment Name

10 Moment of inertia and body shape Calculations & Results (6 Marks):

The restoring torque κ of the torsion axle required for the calculation was determined by instructor radNm /023.0=κ or use the practical value of κ from previous experiment.

Time for 5 oscillations

Oscillation time calculatedI

S 1t S 2t S 3t S

53

3

1

×=∑ it

T 2

2

.2

.

mKg

TI ⎟⎠⎞

⎜⎝⎛=π

κ Shape

=sphI Solid Sphere =sdI Solid Disc

=spI Supporting Plate (S.P.) & =spscI Solid Cylinder + (S.P.) & =sphcI Hollow Cylinder + (S.P.)

M 2.R expectedI calculatedI

Kg m Formula Value 2.mKg Formula Value

2.mKg

% error

Shape

0.930 0.145 5/..2 2RM sphI Solid Sphere

0.340 0.220 2/. 2RM sdI Solid Disc

0.330 0.090 2/. 2RM spspscsc III −= & Solid

Cylinder

0.360 0.090 2.RM spsphchc III −= & Hollow

Cylinder



1. Discuss the %error of each item:

______________________________________________________________

______________________________________________________________

______________________________________________________________

2. Evaluation, from your results, Answer these questions:

Solid Sphere Different

Masses & Same I Solid Disc

Select Answer: … (a) T depends on mass of shape. (b) T depends on I (Moment of inertia) of shape.

Solid

Cylinder Approximately Same M

& R Hollow Cylinder

Select Answer: … (a) I depends on mass & radius only. (b) I depends on shape of a rotationally symmetric body.

Solid Disc Approximately same M &

Shape Solid Cylinder

Select Answer: … (a) If M & shape equal , I is proportional to

2R only. (b) If M & shape equal , I is proportional to R only.

From Evaluation, moment of inertia I depends on three significant parameters:

1. ______________________________________________________________

2. ______________________________________________________________

3. ______________________________________________________________

3. A hollow body has a greater moment of inertia than a solid body with same mass and

dimensions. Why?

______________________________________________________________

______________________________________________________________

______________________________________________________________

References page 102/102

1 According the web site of the book: H.C. Ohanian and J.T.Markert, “Physics for engineers and scientists” , 3rd edition, W. W. NORTON & COMPANY STUDYSPACE. Website is : http://www.wwnorton.com/college/physics/om/_content/_index/tutorials.shtml 2 According the web site of the book: H.C. Ohanian and J.T.Markert, “Physics for engineers and scientists” , 3rd edition, W. W. NORTON & COMPANY STUDYSPACE. Website is : http://www.wwnorton.com/college/physics/om/_content/_index/tutorials.shtml 3 http://www.ece.uwaterloo.ca/~ece204/TheBook/06LeastSquares/ 4 http://www.people.ex.ac.uk/SEGLea/psy2005/simpreg.html 5 http://people.hofstra.edu/stefan_waner/realworld/calctopic1/regression.html 6 http://online.redwoods.cc.ca.us/instruct/mbutler/BUTLER/math99/ppt/graphs.pdf 7 Photos here (graphing Lab) are a screen snapshot from Microsoft office excel 2003 8 All photos here (Graphing Lab) From website http://www.phy.uct.ac.za/courses/c1lab/vernier1.html 9 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1251_e.pdf 10 These figures (vectors exp.) are from http://www.physchem.co.za/Vectors/Addition.htm 11 The colored 3d-inclined plane (Motion Along Straight line exp.) was designed by A.D.Mahir using Google sketch up software. 12 Young & Freedman , “University physics with modern physics”, 12th ed., Pearson/Addison Wesley, 2008, p326 13 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1261_e.pdf 14 Young, and Freedman, "University physics with modern physics", 12ed. Pearson/Addison Wesley, 2008, p 149-155. 15 Young, and Freedman, "University physics with modern physics", 12ed. Pearson/Addison Wesley, 2008, p 149. 16 Young, and Freedman, "University physics with modern physics", 12ed. Pearson/Addison Wesley, 2008, p 151. 17 Young, and Freedman, "University physics with modern physics", 12ed. Pearson/Addison Wesley, 2008, p 150. 18 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1261_e.pdf , http://www.leybold-didactic.com/literatur/s1/e/d1/d1241_e.pdf, http://www.leybold-didactic.com/literatur/s1/e/d1/d1242_e.pdf 19 Young & Freedman, “University physics with modern physics”, 12th ed, Pearson AW, 2008, P421-427 20 From http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html 21 Some of procedure steps are coming from http://www.phys.utk.edu/labs/SimplePendulum.pdf 22 From http://www.phys.utk.edu/labs/SimplePendulum.pdf and Young & Freedman, “University physics with modern physics”, 12th ed, Pearson/Addison Wesley, 2008, P421-427 23 Some of procedure steps are coming from http://www.phys.utk.edu/labs/SimplePendulum.pdf 24 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1451_e.pdf 25 From Young and Freedman, “University physics with modern physics”, 12th ed, Pearson/Addison Wesley, 2008.pages: 433 26 Young and Freedman, “University physics with modern physics”, 12th ed, Pearson/Addison Wesley, 2008.pages: 433-434 27 http://www.ld-online.de/literatur/hb/e/p1/p1451_e.pdf page 1 28 http://www.ld-online.de/literatur/hb/e/p1/p1451_e.pdf Page 2 29 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1452_e.pdf Page 1 30 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1452_e.pdf Page 1 31 From http://www.leybold-didactic.com/literatur/hb/e/p1/p1452_e.pdf page 2

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