Physics 103 Lab Manual
Prepared By:
Dr. Imad Ladadwa & Mr. Mohammad Al-Zaatreh
Table
of Contents
A.
Introduction ................................................................................................................................... 2
B.
Errors............................................................................................................................................. 9
C.
Graphs ......................................................................................................................................... 26
Experiment #1 Measurements with Micrometer and Vernier Caliper ........................................................ 30
Experiment #2 Free Fall Motion ................................................................................................................. 32
Experiment #3 Addition and Resolution of Vectors ................................................................................... 35
Experiment 4 Measurement of g using simple pendulum ........................................................................... 41
Experiment #5 Hooke’s Law ...................................................................................................................... 45
Experiment 6:DC Circuits: Series and Parallel Circuits ............................................................................. 50
Experiment 7: Kirchhoff’s laws .................................................................................................................. 54
Experiment 8: Oscilloscope in AC Measurements ..................................................................................... 58
Experiment 9: Charging and Discharging Of a Capacitor .......................................................................... 62
Experiment 10: Measurement of magnetic fields ....................................................................................... 66
Lab Reports Templates ............................................................................................................................... 72
1
A. Introduction
Physics 103L is an introductory laboratory course. In doing the experiments of
the course, you will meet some physics theory, and some of it will be unfamiliar to
you. Don’t worry about that, we will try to make the theoretical part easy
understandable, but that is not the main of the course. The main purpose is to help
you learn how to do an experiment: how to plan an experiment, how to setup the
apparatus, how to take measurements, how to analyze your measurements, how to
draw conclusions from your results, and how to write a lab report.
Here in more detail are some of the things that you will learn:
 How to use some common measuring instruments, such as Vernier
Calipers, micrometers, voltmeters, ammeters, stop-watches, digital Multi-meters
and the Oscilloscopes.
 How to use some common electrical components, such as resistors, rheostats and
capacitors.
 How to find the uncertainty in your measurements and your calculated result.
 How to plot and draw good graphs on linear graph paper and on logarithmic graph
paper, and how to use the graphs to obtain useful information.
 How to present your work in a clear, clean and concise way in a lab report.
Experimental physics and theoretical physics are the two paths by which we
discover how the physical world operates, Both are necessary; without experiment,
theoretical physics loses its relation with the real world, without theory,
experimental physics produces only lists off numbers.
In this course the emphasis is on experimental physics. Some scientists like to
imagine that nature is a clever opponent with many secrets and that the job of the
experimenter is to persuade or trick nature into revealing her secrets. We hope that
in this course you will enjoy learning some of the tricks.
2
A-II Lab Reports
A lab report is intended to provide information to someone who did not perform
the experiment. It should tell him what you set out to do, how you did it and what
results you obtain. You may find it helpful to pretend that you are writing the report
so that it can be understood by who was absent from the experiment. When writing
the report, you have to fill a lab report form.
A lab report (form) should include the following sections:
1. Purpose: This is a short statement which tells the aim of the experiment and
mentions, without details, the method used. In addition, it contains the main
result of the experiment.
2. Theory: This section includes any relevant theory together with
needed mathematical formulae and error equations. Do not copy
from manual books, just write basic things which clarify the
background of the experiment to the reader.
3. Procedure: This section includes a description of the apparatus used (with
diagrams needed), and an account of the method used. Write procedure as you
have performed it, do not copy from manual.
4. Data: The data are the measurements; usually the data are presented in a table.
The following points should be taken into consideration when writing data.
 Data should be written directly in the table. Never write data on scrap paper
“to be copied later into the table later “.
 Data are written in ink.
 Data should be accompanied by units. In experimental physics, pure
numbers do not tell much about themselves. Usually units are written with
the headings of the table.
 If you make a mistake in recording an entry in the data table, do not
obliterate it and do not write on top of the incorrect entry. Simply cross out
lightly on the incorrect entry and write near it the correct on as shown:
Not this: 16.9 and not this: 16.9 But this: 16.9 16.4
3
5. Calculations: This section includes any graphs that you plot together with the
measurements you made from the graph. It also includes sample calculations
leading to the final result of the experiment. It should include also error
calculation. Here you explain how you obtained the uncertainties in your
measurements and how you combine these uncertainties to obtain the uncertainty
in the final result.
6. Conclusion: Here you present your final result with its uncertainty, and you
make final remarks about it. For example if the final remark is a known constant
like acceleration of gravity, you state whether your result agrees with the known
value within the experimental error. You may also include short discussion of
any systematic errors and how they affect your result. Random errors are usually
not discussed her since these are always present in the experiment.
Finally, the questions at the end of the experiment may help you in writing your
conclusion.
Remember that:
a. Long reports do not necessarily mean good reports, (you are restricted to space
given in the form).
b. Clarity and readability of the report is an important factor in report grading.
The sample reports follows illustrates how a report should look. (Do not worry
if at the beginning of the course, there are some parts of the report which you do
not understand.)
4
A-III Sample Lab Report
Determination of g
Abstract:
The aim of the experiment is to determine the acceleration of a falling body g.
The method used is by measuring the time of fall of a metal weight over a measured
distance.
The main result is:
g  730  30 cm / sec 2
Theory:
An object acted on only by its weight falls with constant acceleration (g). If it starts
from rest at time t=0, then at time t, it has fallen a distance y
y
If y 
1 2
2y
gt , then g  2
2
t
Dividing the last equation by g:
Hence;
1 2
gt
2
Hence g 
2y 2 y
 3  2t
t2
t
g 2y t 2 2 y
t2
 2   3 2t 
g
2y
t 2y t
g y
t

2
g
y
t
Procedure: A solenoid was
connected to an alternating voltage
of 50 Hz, so the core frequency is 50
Hz. A falling weight was connected
to a narrow strip of paper, when the
object is released, the core – which
pushes periodically on the strip leaves
marks on the paper strip every 1/50.0
=0.02 sec.
5
Figure A.1
Initially the core is moved to the left, making the zero point on the paper
strip, and the weight is hung on the core by means of a Hooke. When the
solenoid is turned on, the core begins to oscillate, allowing the weight
to fall.
Data:
Frequency of the solenoid f  50.0  0.1 Hz
Time between marks t=1/50=0.02 sec
Estimated uncertainty in y, y  0.2 cm
Mark
No.
0
1
2
3
4
5
6
7
8
9
Table A.1
Time
t(sec)
0.000
0.020
0.040
0.060
0.080
0.100
0.120
0.140
0.160
0.180
Distance
y(cm)
0.0  0.15
0.4
0.7
1.9
3.1
4.9
6.9
19.5 9.5
12.6
15.6
t2
(sec2)
0.0000
0.0004
0.0016
0.0036
0.0064
0.0100
0.0144
0.0196
0.0256
0.0324
Calculations:
1. f=50 Hz. Time between marks is T=1/50.0 =0.02 sec. Time for the
nth mark is nT=0.02n.
2. Centroid (for the graph): ( y, t )  (7.5cm,0.0154 sec )
y
2
3. Slope of the graph m= 2  487cm / sec (see the graph paper)
t
1 2
The equation y  gt is similar to the straight line equation y=mx
2
2
2
6
with slope equal to m , which means that m 
1
g , so g=2m=974 cm/sec2.
2
4. Calculation of g :
Using the equation derived in the theory;
g y
t

2
g
y
t
T
but
1
f
Figure A.2
Determination of g ; Y vs. t
2
20.0
Best Line
Centroid
Data points
17.5
15.0
Slope 
Y
 487.2 cm / sec 2
2
t
Y (cm)
12.5
Y
10.0
7.5
t
5.0
2
2.5
0.0
0.0
0.5
1.0
1.5
2.0
2
2.5
2
t sec x 10
3.0
3.5
-2
7
then
T f
0.1


 0.002
T
f
50.0
Note also that t  nT so
t T

 0.002
t
T
y  0.2 cm, Clearly y / y =will be different for different values of y, so we use
the average value of y, y  7.5cm .
y / y  0.2 / 7.5  0.027
Then, g / g  0.027  2(0.002)  0.031
g  0.031( g )  0.031(974)  30.2cm / sec2 .
Results and Conclusion:
g  970  30cm / sec 2
The accepted value of g is 980 cm/sec2, but at the altitude of Tabuk,
We expect g to be less than 980 cm/sec 2. Our results of g agree with the accepted
value within the experimental error.
There was a systemic error from the fact that the weight was not allowed to fall
freely, because of friction and because of the blows by the solenoid core on the paper
strip. These effects would result in a lower value of g than the actual one.
8
B.
Errors
Introduction:
It is known that the experimental conditions in the lab. are not ideal, so we don’t
expect that when we measure the value of a physical quantity we will obtain its real
value which represents this quantity; this is because there are many sources of error
like: errors which are caused by instruments, the way we do the experiment, errors
which can occur by the person who is doing the experiment and the way we measure
the physical quantity. Each time we do an experiment we try our best to make the
error as small as possible, but we never obtain a zero error, this means that we can
never measure the real value of a physical quantity, but we can get a value which is
very close to it when we make the error very small.
B-I Random Errors:
Uncertainty in a measurement :
Figure B.1 shows part of a voltmeter scale. As you measure the position of the
voltmeter,
you may say “that’s 5.4
volts and a little bit more”.
but How much more?
the pointer is between
almost half-way between
5.4 and 5.5, but less than
half-way. Is it closer to
5.44 or 5.45 or 5.43 ? Most
probably it is 5.44 . You are
uncertain about the last
figure by about  0.01 volt in this example.
To inform others about your uncertainty, you report your measurement as:
V  5.44  0.01 volt. This means that if the measurement is repeated, the new
measurement probably will not differ from the first one by more than  0.01 volt.
Now suppose that you use the voltmeter shown in figure B.2, which has a more
finely divided scale. The reading is clearly 5.43 and a little bit more. How much
more? A gain you must estimate the position of the pointer in the space between 5.43
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and 5.44. Is it 5.436, 5.437, or 5.438? again the last figure is uncertain. You would
therefore report your measurement, as V  5.437  0.001 volt.
The estimated uncertainty depends on how finally the scale is divided. It also
depends on how carefully you read the instrument.
But always there will be some uncertainty in the last figure. Hence, all measured
quantities are uncertain and the true values of all measured quantities are unknown.
The best one can do it is to use more precise instruments and be more careful to
reduce the magnitude of these uncertainties.
Estimating Uncertainties:
In the example above, the uncertainty
was found by estimation, taking into
consideration the instrument used and the
care taken using in using it. If only one
measurement of a quantity is made, then
the only way to find the uncertainty is to
estimate it.
If several measurements of the same quantity are made, however we can find the
uncertainty in another way.
Suppose that five students measured the voltage drop (V) across a resistance R;
their measurements in volts were as shown below:
No.
Voltage
(Volts)
1
5.45
Table B.1
2
3
5.44
5.43
4
5.46
5
5.43
10
The best estimate of the true value is the average value defined as:
1 N
Average value x   xi
N i1
B.1
5
In this example N=5 and V  Vi =5.442. The best estimate of the uncertainty is
i 1
the standard deviation of the sample defined as:
Sample standard deviation:
s 
1 N
( xi  x ) 2

N  1 i 1
B.2
You can show that the sample standard for the measurements shown in table
B.1 is :  s  0.013 volts .
The advantage of using  s as the uncertainty in the measurement is that it
allows us to give a more precise meaning to what we mean by “uncertainty “.
Earlier we said that any one measurement “probably“ does not differ from another
one by more than the uncertainty. But we did not say what we meant by
“probably”. When we use  s as the uncertainty, the word “probably has a clear
meaning. The probability is about 2/3 that any one measurement does not differ
from another by more than  s .
 s is called the sample standard deviation because it is the uncertainty in only one
measurement, namely a sample measurement.
Uncertainty in the mean:
The best estimate we can make of the true “true” value is the average of the
mean of our measurements. But we need to know how confident the average value
is close to the true value. Clearly the more measurements we make, the more we
approach the true value.
11
Statistical theory recommends that we define the uncertainty in the mean or the
“standard deviation of the mean“ as:
Standard Deviation of the Mean

m  s
B.3
N
We note here that the probability that the mean value does not differ from the
true value by more than σm is about 2/3.
Importance of knowing the uncertainty:
Suppose you want to find out by experiment whether the length of a metal rod
depends on its temperature. You can measure the length at two different
temperatures, you measurements are:
Length L
(cm)
Temperature
Table B.2
98.025 98.034
10
20
C
Now, we would like the question “Does L depend on the temperature ?“. We cannot
answer this question unless we know the uncertainty in L. If the uncertainty is  0.01
cm then you cannot answer the question, since the difference in the two
measurements is smaller than the uncertainty.
So, we note that the result of an experiment must include the value of the
measured quantity and its uncertainty. Both are necessary.
Note that the uncertainties we have treated above are plus or minus sign
quantities, this is because measured quantities are spread on both sides of the true
value. If we take any other measurement we can not predict whether it is above or
below the true value. Consequently, these uncertainties are called random.
12
We have consistently used the word “uncertainty”. In statistical theory the word
“error” is usually to mean the same thing. In this sense, “error” does not mean a
mistake.
Exercise 1:
Five students measured the length of a rod to be as shown below:
No.1
1
L(length) 87.5
(cm)
2
87.4
Table B.3
3
87.6
4
87.8
5
87.2
1) What is the best estimate for the length of the rod?
2) What will be the uncertainty in L.?
B-II Systematic Errors:
Random errors are always present in an experiment. They are equally likely
to be positive or negative. They cause the several measurements to spread out
around the true value.
In the other hand, systematic errors are different; the measured values spread
on one side of the true value, they are either below or above the true value. If there
is a systematic error (in addition to random errors), the several measurements
spread out around a mean which is different from the true value.
Let’s introduce to you three examples that can cause systematic errors:
1. Suppose that you want to measure the voltage drop across a resistance R in
your circuit , and suppose that you used a scale voltmeter to take this
measurement , if such a scale voltmeter reads a value of –0.2 volts when the
voltmeter is not connected to any thing . Then if you measure the voltage
across the resistance, this voltage will be smaller by –0.2 volts. So in this
example the systematic error is –0.2 volts.
We note that it is easy to eliminate this systematic error by checking the zero
setting before doing the experiment. You can adjust the voltmeter to read zero
volts when it is not connected, this process of adjusting is called calibration of
13
the instrument, note that if you cannot calibrate the instrument, you have to add
(or subtract) the systematic error from all measurements.
Always check for systematic errors in any used instrument and try to eliminate
them in advance.
2. Suppose you measured the length of a rod by using the meter stick shown
below in figure B.3:
If the meter stick in figure B.3 is an old one or one that has been roughly used,
it may be ½ mm of the end worn off. Then all your measurements of the length of
the rod will be ½ mm big, there is a systematic error of + ½ mm in this case.
There is no easy way to know the size of the systematic error, but you can
eliminate it by start measuring from a mark far from the end of the meter stick.
3. Systematic errors are not caused only by the non-calibration of the instruments
, this will be obvious from this third example . Consider the example when a
student tries to measure the length of a rod, suppose that the rod has a width (w)
as shown in figure B.4, and he tries to measure
14
The length of the rod, suppose that the student is not looking perpendicularly to
the scale, then he will make an incorrect reading of the position of the edge of the
rod, this is case (a) of looking in the figure B.4. The correct way of looking is
shown as case (b) in the figure; it is straight down perpendicular to the scale.
In principle, systematic errors can be eliminated. But there is no simple way to be
sure that you have eliminated all of them in the experiment. One of the qualities of
a good experimenter is that he is able to imagine all possible sources of systematic
error and to do whatever necessary to eliminate them.
B-III Precision and Accuracy:
Suppose that a measurement has a small random error, then we say that this
measurement has high precision; it is a precise measurement. To make this clear to
you we give an example; suppose that two students measured the voltage across a
resistance R using a scale voltmeter, their measurements were as shown below:
student1 V1  5.443  0.002 volts
student 2 V1  5.44  0.03 volts
Since the measurement of student 1 has less random error compared to the
random error of student 2, the measurement of student 1 is more precise than the
measurement of student 2.
Small Random Errors means
High Precision
If the systematic errors are neglected, so that the measured value is close to the
true value, we say it has high accuracy, it is an accurate measurement.
Negligible Systematic Errors
means High Accuracy
A measurement can be precise but not accurate. This is due to the presence of
systematic errors which shift the measured value from the true value. For example,
15
suppose that we measured the diameter of a wire using a micrometer, the
measurement is:
D  2.13  0.01 mm
Suppose that we forgot to calibrate the micrometer when doing the above
measurement, (when the micrometer is closed, it reads 0.02 mm), then our
measurement will be big by 0.02 mm.
We also note that a measurement can be accurate but not precise. For example,
a junior physics student measured the acceleration of gravity and obtained as a result
g  9.9  0.2m / s 2 . This measurement of g is not very accurate, but it agrees with,
within the reported uncertainty, with the accepted value of 9.82 m/sec2.
In most student experiments, the true value is known, so that it is possible to
know the accuracy of your measurement. Note that the true value is the value
accepted by the community of scientists, because it is the value obtained by
experienced, skillful, and trust worthy experimenters.
A comparison between measured and accepted values:
There is very little point in performing an experiment if one does not draw some
calculations. Almost all experiments have quantitative conclusions, involving the
statement of a numerical result. An interesting conclusion must involve the
comparison of two or more numbers, mainly the comparison of a measurement with
the accepted value. It is this comparison of numbers that error analysis is so
important.
Suppose that a student made a measurement of a physical quantity x, so he
reported his results as:
Re sult  x  x
The meaning of the uncertainty x is that the correct value of x “probably” lies
between xbest  x and xbest  x ; It is certainly possible that the correct value lies
slightly outside this range.
Therefore a measurement can be regarded as satisfactory even if the accepted value
lies slightly outside the estimated range of the measured value.
16
On the other hand, if the accepted value is well outside the measured range
(discrepancy much more than twice the uncertainty say ) then there is good reason
to think that something has gone wrong.
Example:
Two students A and B measured the speed of sound in air (at standard temperature
and pressure). The measurements are as shown below:
VA  338  2m / sec
VB  325  5m / sec
Since the known accepted value at standard temperature and pressure is 331
m/sec2, then for student A the discrepancy D=338-331=7 m/sec2,
Note that 2  VA  2  2  4m / sec , hence D  2VA .
For student B the discrepancy between measured and accepted values D=3312
325=6 m/sec2, and 2  VB  2  5  10m / sec , hence D  2VB .
The result of student A is not accepted, while the result of student B is well
accepted. Student A has to check his measurements and calculations to find out what
has gone wrong. Unfortunately tracing his mistake can be a tedious job since there
are so many possibilities; he may have mistake in his measurement or calculations,
he may have estimated his uncertainty incorrectly, he also be comparing his
measurement with the wrong accepted value.
Finally, and perhaps most likely, a discrepancy may indicate undetected source
of systematic error, The detection of such systematic error will require careful
checking of the calibration of all instruments and detailed review of all procedures,
so always check the following when you got a strange results (strange result mean
that the range of the measured value does not agree with the range of the accepted
value):
 Check for the calibration of the instruments you use.
 Check to find out if the data you wrote down is the same as the data you
obtained from the instruments you use.
 Check to find out if you made possible mistakes during the calculation.
 Check to find out if you are comparing your result with the wrong accepted
value.
17
B-IV Significant Figures:
Suppose that you measure the length (L) of a rod using a meter stick with an
uncertainty of  0.1cm , then it would be silly to report the length L as L =2.435
cm, the “3” and “5” are meaningless because the “4” is uncertain by  1 , only the
“2” and “4” have meaning; they are said to significant figures . The length of the rod
should be reported as:
L  2.4  0.1cm
which is with two significant figures.
Now suppose that you made the measurement with a micrometer with an
uncertainty of 0.01mm  0.001cm , then the “2”, the “4”, the “3” and the “5” are
all four significant figures and you should report the length of the rod as:
L  2.435  0.001cm
The significant figures are all the figures up to and including the
figure which is uncertain
Experimental results must always be rounded so that only significant figures are
included. Here are some examples of the correct use of significant figures:
Table B.4
No.
Measurement
# of
Measurement
# of
significant
significant
figures
figures
1
2
2
4.7  0.3
0.47  0.03
2
3
4
473  2
4728  3
2
3
4
3
(4.72  0.03)  10
472.8  0.3
4
5
4
472.84  0.03
4.728  0.003
Here are some examples of the incorrect use of significant figures:
0.47  0.3
4.7  0.03 , 472.8  0.03
, 472.82  0.3 , 473  30
18
When significant figures are correctly used, then even if we are not told the
uncertainty, we can have a rough idea of how much it is. If we are told that L=5.43
cm, we know that the uncertainty is a few hundredth of a centimeter, whereas
L=5.434 cm tells that the uncertainty is a few thousands of a centimeter.
Special attention must be given to zeroes. For example, let R  4.7  0.03 but
R  4.70  0.03 is correct, the zero is significant and must be written. On the other
hand, R  4.70  0.3 is incorrect, the zero is not significant and should be removed.
Significant figures in calculated values:
1) Rounding:
A. If the first non-significant figure is larger than 5, then round the last
significant figure up; that is increase it, as an example :
4.37 is rounded to 4.4
4.368 is rounded to 4.37
B. If the first non-significant figure is less than 5, then fix the last
significant figure, as an example :
4.43 is rounded to 4.4
4.363 is rounded to 4.36
C. If the first non-significant figure equals 5, then it is not clear
whether to round up or round down. The usual custom is to round
up if the last significant figure is odd and round down if it is even.
Thus, as an example:
4.35 is rounded to 4.4
4.45 is rounded to 4.4
In this course, the uncertainty in a result should be rounded to one significant
figure. An exception to this rule is that if the leading digit in the uncertainty is a “1”
then it may be better to keep two significant figures in the uncertainty.
19
For example, if the uncertainty is  0.014 , then to round this to 0.01 would be a 40%
reduction in the uncertainty, thus it would be better if the uncertainty is left with
two significant figures as 0.014 .
Experimental uncertainty should be rounded to one significant
figure unless the leading digit in the uncertainty is 1, then it is
left with two significant figures.
2) Addition and subtraction:
The number with the fewest decimal places limits the number of decimal places
in the result. As an example:
Let R= 10.3 +108.76+0.0349=119.0949, now this result must be rounded, because
according to the rule above, the result must contain one decimal place, so R  119.1
.
When performing in long calculations, keep one extra decimal
place in the result. This is to avoid round-off error, which might
occur if the numbers are rounded off at each step of calculation.
3) Multiplication and division:
Suppose that wants to measure the value of an unknown resistance, so he
measured the voltage across the resistance V=1.7 volts and the current I =11 mA
(note that 1mA  1 10
3
A ), now the resistance R is:
R
V
1.7V

 154.545...
I 0.0011A
If we only know the voltage (V) and the current (I) to two significant
figures, we cannot know their ratio to six significant figures as shown
above, the ratio should be rounded to two significant figures, that is
R=150ΩNote that results, as well as measurements should always be
reported with the correct number of significant figures.
20
Let us give another example here:
Let x1  1.8
x2  2.346 x3  7.86 , we would like to find R where
x  x 1.8  2.346
R 1 2 
 0.537  0.54
x3
7.86
Notice how we rounded the result and we kept only two significant figures.
Note also that if you have a result as some function of a measured quantity such
as:
R  Sin( )
The number of significant figures that you must keep in the result R must equal the
number of significant figures in θ. The same applies for the Cosine function.
For the square root function of a multiplication and division, the number of
significant figures in the result will be given by the fewest number of significant
figures, see the examples below :

a) Sin (12 )  0.21

b) Cos(35 )  0.82
c)
2.3  4.57
 8.8
1.2
4) Significant figures in the uncertainty:
Suppose that the mean value of a set of measurements is L=6.3234 cm and the
standard deviation of the mean is 0.0237 cm, from what has been said above you
should report your result as :
L  6.32  0.02cm
As a second example, a student measured the value of g (acceleration of gravity)
he found that:
g  9.861m / sec 2 g  0.0412m / sec 2
Then he can write his result R as:
21
R  g  g  9.86  0.04m / sec 2
Exercise 3 :
Rewrite the following measurements in their clearest forms with a suitable number
of significant figures:
a) Measured height 4.02  0.02315meter
b) Measured time 17.5143  1Sec.
19
19
 2.41 10 coulombs.
c) Measured charge  9.51 10
d) measured wavelength 0.000,000,432  0.000,000,07meters
e) Measured momentum 2.362  10  21 gm cm/sec.
3
B-V Combining Uncertainties :
In most experiments several quantities are measured and then combined is some
way to find the desired result. Each measurement in uncertain and therefore the final
result will be uncertain. In this we show you how one can find the uncertainty in the
result using the uncertainty in the measured quantity.
1) Addition and subtraction:
Sometimes the desired result is obtained by adding or subtracting
measured quantities. For example, the weight a liquid in a bottle may be found by
subtracting the measured weight of the empty bottle from the total measured weight
of the bottle of liquid.
Let’s consider an example, suppose that the weight of the empty bottle is
M empty  25.7  0.2 gm , and the bottle of liquid is M full  72.3  0.3gm , this means
that the first reading is somewhere between 25.5 gm and 25.9 gm and the second
reading is somewhere between 72.0 gm and 72.6 gm.
The weight of the liquid W=72.3-25.7=46.6 gm, but it might be as big as 72.625.5=47.1 gm , and it might be as small as 72.0-25.9=46.1 gm, thus
M liquid  46.6  0.5gm . We see that the error in measuring the weight of the liquid is
0.5 gm=0.3+0.2 gm, so that the uncertainty in Mliquid is the sum of the uncertainties
in the measured quantities Mbottle and Mempty , so in general if x and y are two
measured quantities , with uncertainties Δx , Δy . Then if the result R  x  y , the
uncertainty in R (ΔR) is:
22
R  x  y
B.4
Notice the that when R=x-y , ΔR=Δx+Δy , since errors always add .
2) Constant multipliers:
Suppose that R=ax+ by , where a and b are constants , x and y are measured
quantities, then we can find ΔR by differentiation with respect to x and y;
B.5
R=a x +b y
3) Multiplication and division :
Consider a rectangular plate of width x and y, you would like to measure its area
(A) and also the error in measuring it, you can do that by measuring x and y, then
A=xy, we can also find the error in A by direct differentiation with respect to x and
y, suppose that the uncertainty in measuring x is Δx, and the error in measuring y is
Δy, then we can find ΔA;
A  yx  xy
B.6
dividing equation B.6 by A we get ;
A
yx xy


A
A
A
B.7
substituting for A=xy in equation B.7, we get ;
A yx xy x y




A
xy
xy
x
y
B.8
4) Raising to a power :
Let R=x2 , and you would like to find ΔR ;
R
2 xx
x
 2 2
R  2 xx and
R
x
x
23
In general if you have a result R=xnymzl, where x, y and z are quantities which
you can measure, n, m and l are integers that is :
n, l , m  1,2,3......  , then;
R  nx n1xy m z l  x n my m1yz l  x n y m lz l 1z
B.9
Dividing equation B.9 by R , we obtain ;
R nx n1xy m z l x n my m1yz l x n y m lz l 1z



n m l
n m l
R
x y z
x y z
xn ym zl
Simplifying, we get;
R
x
y z
 n m l
R
x
y
z
B.10
5) Other functions:
Let us consider how we can find the uncertainty in the result, if the results
depends on other functions such as:
a) Sine and Cosine functions:
R  Sin(x) then
R  Cos( x)x
R  Cos(x) then
R  Sin( x)x
B.11
B.12
b) Natural logarithm:
R  Ln(x) then
R 
1
x
x
B.13
c) Exponential functions:
R  x0e x then
R  x0 e x x
B.14
Note that x0 is treated as a constant in equation B.14 .
24
6) General rule :
If a results R is written as a function of measured quantities x ,y and z that is
R=R(x,y,z) , then one can find the uncertainty in the result as :
R 
R
R
R
x 
y 
z
x
y
z
B.15
Exercise 3
I) A student obtained the following measurements;
a  4  1 cm
b  15  2 cm
c  14  1 cm
t  4.1  0.5 sec.
m  37  1 gm
II) Compute the following quantities and their uncertainties:
1) a+b-c
2) ct2
3)
mb
t
4) Ln(c)
25
C. Graphs
A. Uses of Graphs
In experimental physics, data are often displayed in graphs; there are several
ways in which graphs will be useful;
1. A graph may show more clearly than a table of data the relationship between
variables, for example the relation may be linear, or non-linear. When you
look at the data table, this relation will not be clear to you.
2. A graph may be used to test how well a theory corresponds to the experimental
facts; this is because when you draw the experimental points and the
theoretical curve, you can make a comparison easily between experiment and
theory.
3. Measurements which are represented by a graph can provide the value of a
desired quantity. For example, in the sample lab report the value of the
acceleration of gravity (g) was found by measuring the slope of the graph.
B. Plotting Graphs
The results of our experiments will depend very strongly on how well
the graphs are plotted and drawn. The following rules will help you produce good
graphs (whether it is a linear graph paper or semi-log graph paper)
1. Draw the x-axis line and the y-axis line one at the bottom edge and the other
at the left edge of the graph paper.
2. Label the x-axis and the y-axis with units, for example if you plot the
displacement (D) of a moving object versus time; that is the displacement (D)
on the y-axis and the time (t) on the x-axis , you have to label the x-axis as
“Time (sec)” and the y-axis as “Displacement (D) (meter)”.
3. Choose and use easy scales for the axis. Choose a minimum and a maximum
number of the scale such that it is easily divided.
26
4. Write numbers only on the main divisions of the scale. If the numbers on the
scale are very large or very small , the graph will look neater if a power-of-10
is used , for example if the x-axis measurements are : 121, 133, 152, 164, 171
and the y-axis measurements are : 0.012, 0.021, 0.033, 0.041, 0.052 then you
can draw the points (1.21,1.2), (1.33,2.1), (1.52,3.3), (1.64,4.1), (1.71,5.2) on
the graph paper and multiply the x-axis by 102 and the y-axis by 10-2.
5. Make graphs as large as possible. Fill the entire page.
6. Plot your points on the graph with using crossed circles ⊕.
7. Show the centroid {the point ( x , y ) } on the graph paper by the symbol ●.
8. Give a title to the graph, since the reader needs to know what you are plotting.
C. Linear graphs:
Suppose that you have the following data table (table C.1) for the speed of a
moving object in one dimension; the speed (v) in cm/sec. and the time (t) is in
seconds, suppose you plotted the points along with best straight line as shown in
figure C.1
Time 1.0
(sec)
V
7.0
(cm/s)
2.0
3.0
Table C.1
4.0
5.0
9.5
10.5
13
15.5
6.0
7.0
8.0
16.5
19
21
The best straight line is the average one, the line which passes as close as possible
to as many as of the points. Drawing the best line is a matter of judgment, (note that
you can find equation of the best straight line also using the least square fit method
which you will learn in experiment 7). The best line should have about the same
number of points above it as below it. To determine the best line, it is necessary to
see all the points at one time, you can use a plastic ruler to draw the line in order to
see all points at one time.
27
Figure C.1
Data points
Best Line
Centroid
Speed vs. Time
22
20
18
Centroid
16
V (cm/s)
14
V
12
10
t
8
6
4
2
0
0
2
4
6
8
Time (sec)
In addition to the points which represent the data, there is an additional point to
help you draw the best line. It is the centroid. The cetroid coordinates are the average
values of the measured points, that is:
Centroid  ( x , y )
Now the best straight line should pass through the centroid . ( Notice that in
figure D.1 the centroid is represented by a solid circle ● to differentiate it from
experimental points).
You can obtain the information you need from the graph, in the example of figure
C.1, you can find the following:
You know that if an object is moving under the influence of constant
acceleration is represented by the equation:
28
V  V0  at
C.1
where a is the acceleration of the object V0 is its initial speed, t is the time.
Equation C.1 is similar to the straight line equation y=mx + b, where y=V and
x=t , the slope=m=acceleration (a), the y-intercept is V0=b , (the initial speed), so
from the graph we can find the initial speed of the object and its acceleration.
slope 
y y2  y1 V


 2.0cm / sec 2
x x2  x1
t
From the graph also , the y-intercept =V0=b=5.1 cm/sec2 .
29
Experiment #1 Measurements with Micrometer and Vernier
Caliper
Procedure:
a. Obtain one of the small metal blocks which are provided, caliper and
micrometer.
b. Measure the length L and the
Width W of the block using the
Vernier caliper. Repeat the
measurements five times from
different places.
c. Measure the thickness T of
block with a micrometer.
Repeat the measurements from
different places five times.
Note: your measurements may be in mm units, divide by 10, so that the units of all
measurements are in cm.
d. Use a balance scale to measure the mass of the block, write down your
measurements in table 1.1
Table 1.1
Average
Length L
(cm)
Width
W(cm)
Thickness
T (cm)
30
Calculations:
1. Calculate the mean values L ,W , T and their standard deviation of the mean
 m ( L),  m (W ),  m(T )
Note that L   m ( L), W   m (W ), T   m (T ) .
2. Calculate the best volume V  L  W  T in units of cm3.
3. Calculate the uncertainty in V ;
V L W T



V
L
W
T
Express your results as V  V
4. Calculate the density of the block from  
M
.
V
5. Calculate the uncertainty in the density ρ,



M V

M
V
Write your result as    .
6. Identify the material of the block (which material has this density).
Questions
1. Why did you repeat the measurements in L, W and T in different locations?
2. Do you expect systematic errors to affect standard deviation?
31
Experiment #2 Free Fall Motion
Objective:
To study the motion of a body in one dimension under the influence of the
gravitational force and to determine the magnitude of the acceleration due to
gravity
Theory:
The arrangement in this experiment is designed to study the displacement-time
relationship and so, to determine the gravitational acceleration g. the equation of
displacement as a function of time for a falling object is described by the equation
y(t )  y 0  v0 t 
1 2
gt
2
(1)
In this experiment, the object is dropped
from rest and the distance that it falls is
Figure
measured from the release point. Hence y0 1
v0 are both zeroes, and the above equation
becomes
y (t ) 
1 2
gt
2
and
(2)
The minus sign has been dropped because
are interested in magnitude of the height of the object and not direction.
we
32
Figure 2
As seen from eq.2, if distance versus time is plotted, the graph is a parabola (fig 3).
But if the distance is plotted versus time squared, the graph is a straight line(fig 2).
From eq.2, the acceleration due to gravity can be calculated as:
g
2y
t2
(3)
Procedure:
1. Set up the free fall equipment where the lower cursor indicates to the
position of the impact switch and the upper one indicates to the release
position of the ball. The difference between the two positions represents the
distance y. The several values of y will be given by the class instructor.
2. Be sure that the impact switch is aligned with the steel ball inlet, in other
words; the surface of the switch is on the same line with the inlet (release
point).
3. Turn the electronic timer on and set it up (your instructor will help you in
setting up the timer)
4. Insert the ball release unit and then press the RESET button on the electronic
timer to reset the timer
5. Push the impact switch up.
6. Make the unit ready for the measurement of time (in millisecond) by
pressing the key START
7. Release the ball; it should g\hit the impact switch. If not reset the times and
try again through steps 5 and 6. Record the fall time.
33
8. Repeat step 4-7 three times and calculate the average time.
y(m)
Time t(sec)
t(sec)
t2(sec2) g  2 y (m / s 2 )
t2
(average)
t1
t2
t3
g avg 
 g  ________________  ........................m / s
2
n
Calculations:
1. Use eq.3 to calculate the magnitude of acceleration due to gravity
2. Calculate the average value of the acceleration due to gravity, gavg
3. Plot a graph between y0 on the y-axis and t2 on the x-axis and calculate
the slope from the best fit straight line.
slope 
y
t 2
4. Calculate the experimental acceleration due to gravity from the slope,
where g exp  2  slope
5. Determine the percentage error in the experiment by using
% ERROR 
g th  g exp
g th
 100
where gth=9.8m/s2
Questions:
1. Does the acceleration due to gravity depend on the mass of the falling body? Explain
2. What are the possible experimental errors which make difference between gexp and gth?
3. Calculate the acceleration due to gravity corresponding to the longest and shortest heights
used in the experiment by using eq.3 and compare them. What can you say about these
two readings? Which is better? And what are the reasons for that?
34
Experiment #3 Addition and Resolution of Vectors
Introduction
Many of the concepts used in physics must be described by both a magnitude, or
size, and a direction. Some of these quantities are displacement, velocity, force, and
acceleration. We use a vector to represent these quantities. A vector can be
represented graphically by an arrow whose length is proportional to its magnitude
and which points in the desired direction. It can also be represented mathematically
by giving the components of the vector along three perpendicular directions. In this
experiment we will investigate methods of adding the vectors that represent forces.
We will practice resolving the vectors into components. We will use a force table to
experimentally observe the addition of different force vectors.
Apparatus
Force table (including three pulleys, centering pin, and a ring with strings
attached), weight hangers, weight box, and bubble level
Before the Lab
Read the sections in your text describing vectors and how to add them. You should
learn how to add vectors both graphically (by constructing a triangle or
parallelogram) and by resolving them into perpendicular components and then
adding the components.
Theory
Vector Components:
When we wish to describe a quantity, which has both a
magnitude (size) and direction, we can represent it with a vector.
The vector can be described in terms of components. In Figure 1,
vector A can be broken into components Ax and Ay. We describe
Figure 1
this vector in terms of a horizontal direction, x, and a vertical
direction, y (as shown in the figure1). The angle θ measures the angle from the xaxis to the vector, A. The relationship between the magnitude (or length) of A and
its components are then
35
Ax= A cos(θ) where Ax is the side adjacent (or closest) to the angle
Ay= A sin(θ) where Ay is the side opposite to the angle
We can also find the magnitude and angle associated with A if we know its
components, since
A  ( Ax2  Ay2 )
and
  tan 1 (
Ay
Ax
)
Adding Vectors:
To add two vectors, A and B, to form a resultant vector, C, we can simply add their
components to find the components of C, as shown on the right. Alternatively, the
vector C can be found graphically by positioning A and B head to tail, as in the left
side of Figure 2. C will be the vector that goes from the tail of A to the head of B
and will be the diagonal of the parallelogram formed by A and B. The components
of the vector C can also be found algebraically from the components of A and B as
follows:
If
  
C  A B
then
C x  Ax  B x

C y  Ay  B y
Figure 2
36
Experimental Procedure:
The Force Table is an apparatus used to determine the resultant (or vector sum) of
different forces. Forces are applied radially to a central ring by means of attached
strings, which run over pulleys on the edge of the table, with masses hanging on their
ends. The pull of gravity on the masses (i.e. their weights: mg) gives rise to tension
in the strings that is proportional to the amount of mass hanging. Therefore, the
magnitude of a force may be varied by adding or removing mass. The direction of a
force can be varied by moving a pulley along the circumference of the table. When
two or more forces are applied to the ring, 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 are applied, the resultant, or vector sum, is
 

F1  F2  Ftotal


The magnitude and direction of Ftotal may be found by finding a third force, F3
such that:
 

F1  F2  F3  0
When the net force on the ring is zero it will remain centered around the center pin,
in equilibrium. The sum of rand F1rF2must then be equal in magnitude, but

opposite in direction, to F3 , i.e.,
 

F1  F2   F3
In the following experiments you will practice different methods of adding vectors
and then use the force table to experimentally check your calculations.
Begin by leveling the force table, if necessary. Then, practice balancing forces
until you are able to determine when there is zero net force on the ring. Note that it
is important that the strings tied to the ring slide easily from side to side, so that no
sideways force is applied to the ring. The strings should pull straight outwards
toward positions of the pulleys on the edges of the table. There will be some
uncertainty in your experimental method. Remember that the weight of the hangers
must be included in your total weights.
37
1) Given two force vectors, for example F1 corresponding to the weight of a 100 g
mass at 30o (above the positive x-axis) and F2 corresponding to the weight of a 150
g mass at 1400, find their resultant, or vector sum, by the following three methods.
Record your results in your lab notebook, drawing appropriate diagrams to
describe both your calculations and your experimental method.
Graphically: draw a diagram to scale and construct a parallelogram to find the sum.
Addition of components: on your diagram, define an x- and y-axis. Resolve the
vectors into components along these axis (with your calculator) and find the sum.
Does your answer make sense when you compare the result with the graphical
method?
Experimentally: Use the force table to determine the third force that would be
required to balance the two force vectors previously defined. How does the result
compare to you calculations. What is the uncertainty in your experimental result?
2) Repeat all of the above with two different forces in different directions, e.g. 100
g at 20o and 75 g at -80o. This time try the experimental method first. Set up your
two forces, then pull on the string to find the direction that a third force must be
applied to balance the first two. Add weights to determine the third force. Then
check your result by graphical methods and by addition of components. How do
your results compare? Is this within the expected uncertainty of the experimental
method?
3) Repeat using two perpendicular forces. You can choose the x- and y-axis to be
in the perpendicular directions: for example, F1 = Fx corresponding to 75 g at 0o,
and F2 = Fy corresponding to 100 g at 90o.
4) You can also find the components of a vector experimentally. Place three
pulleys at 240o, 90o and 0o. Hang a total of about 150 g from the string through the
pulley at 240o. You should now be able to find its components along the x-axis (at
0 degrees) and the y-axis (at 90 degrees) by finding the weights that you must hang
from these pulleys to balance the weights. Check your results graphically and
mathematically.
38
Questions
1. Compare the graphical and analytical (addition of components) methods for adding
vectors. Which is more accurate? Give possible sources of error for both methods.
2. What are the possible sources of error in the experimental method? (Why is it necessary
to allow the strings to slip loosely about the ring?)
3. If the weights of all the mass hangers were the same, could their weights have been
neglected? Explain.
4. What is the effect of the weight of the ring? What difference would it make if the ring
were considerably more massive?
5. Do the theoretical values of M3 and θ3 depend upon the acceleration due to gravity g?
Table 1. Experimental data
String 1
String2
M1 (kg)
Ѳ1(deg) M2 (kg)
30
F1= M1g (N)
String3
Ѳ2(deg) M3 (kg)
Ѳ3(deg)(
(Experimentally) Experimentally)
140
F2= M2g (N)
F3= M3g (N) ( Experimentally)
Table 2. Mathematical data
String 1
F1x=F1cos(Ѳ1)
String2
F2x=F2cos(Ѳ2)
String3
F3x=F1x+F2x
F1x=……………(N)
F2x=……………(N)
F3x=……………(N)
F1y=F1sin(Ѳ1)
F2y=F2sin(Ѳ2)
F3y=F1y+F2y
F1y=……………(N)
F2y=……………(N)
F3y=……………(N)
F3=
F
2
3x
 F 3 y  .............( N )
2
 3  tan 1
F3 y
F3 x
 ...............
39
Table 3. Graphical data
The scale used in the graph is …………………………..cm/N

The length of the vector representing F 3 is ………………cm

F 3 = Length of the vector x scale =
Ѳ3 =……………………….(deg)
………….(N)
40
Experiment 4 Measurement of g using simple pendulum
The motion of simple pendulum is an example of Simple Harmonic Motion. Simple
Harmonic motion is the back and forth motion of an object about a point called the
equilibrium point in equal interval called period. The period of the simple harmonic
motion is defined as the time needed to complete one cycle. In our experiment, we
shall study the relation between the length and the period of a simple pendulum.
Figure 3
Using Newton’s second law; we can resolve the total force acting on the mass m
into two parts: radial part and tangential part,
Radial:
Tension  mg cos( )  0
(1)
Tangential:
Ft  mg sin( )
(2)
When the angle θ is small sin(θ) can be approximately written as:
sin( )   (for small θ in radians)
41
Equation (2) becomes,
Ft  m
d 2S
 mg
dt 2
(3)
But S=Lθ at any time, substituting in equation (3) we obtain,
d 2
g
 
2
L
dt
(4)
The solution of equation (4) is,
 (t )   0 sin(t   )
(5)
Where θ0 and δ are constants which can be found from initial conditions, and ω can
be found by substituting of equation (5) into equation (4), we get,
2 
g
L
(6)
From equation (6) we can find the period of the pendulum since
Period  T 
2


2
g/L
 2
L
4 2
T2 
L
g
g
A graph of T2 vs. L is a straight line with a slope s  4 2 / g and y-intercept equals
zero, thus a measurement of the slope and its uncertainty will allow us to determine
the value of g and its uncertainty (
g s

).
g
s
Note: the length of the pendulum L is measured from the point of suspension
to the center of the mass of the bob.
42
Procedure:
I. The relation between the mass of the bob and the period:
1. Use a fixed length of the pendulum, say 70 cm and attach the first bob of mass m1 at the
free end.
2. Pull the string up for a small angular displacement and then leave it. Measure the time
required by the pendulum to complete 10 cycles.
3. Repeat the previous step three times and calculate the average time for 10 cycles.
Calculate the period by dividing the average time by 10.
4. Replace the bob of mass m1 by another bob of mass m2 and repeat steps 2-3
5. Compare the periods. What can you conclude?
II. The relationship between the length of string and the period
1. Measure the length of the string from the support to the center of mass of the bob.
2. Start the pendulum to swing through a small angle (less than 15 degrees). Measure the
time required to swing through 10 cycles.
3. Record time for 10 cycles for various lengths of the pendulum. Calculate period and
period squared.
No.
L(cm)
t1(sec)
10 periods
t2(sec)
10 periods
t3(sec)
10 periods
t average
(sec)
One
Period
T(sec)
T2
(sec2)
43
Calculations:
6. Using a linear graph paper, make a scatter plot of T2 vs. L. and draw the best line
through the points.
7. Find the slope s from the graph and then find g as explained to you by your instructor.
(Theoretical part).
8. Calculate the uncertainty in g using the uncertainty in the slope s.
44
Experiment #5 Hooke’s Law
This laboratory session has the following objective:
Determine the force constant of the spring, k by
1. Measuring the distance the spring is stretched when different weights are added
to it (Hooke’s law)
2. Measuring the period of the motion for several masses attached to the spring
(studying the harmonic motion)
Figure 4
The motion of the block attached at one end of a spring is an example of Simple
Harmonic Motion. Simple Harmonic motion is the back and forth motion of an
object about a point called the equilibrium point in equal interval called period. The
period of the simple harmonic motion is defined as the time needed to complete one
cycle. When we displace the spring by a distance y from its equilibrium position,
then a force F tries to bring it back to the position y0. This force F is proportional to
the displacement Δy= y1-y0 and is called the restoring force. Mathematically, we
describe this motion as
45


F  ky
(1)
Where k is the proportionality constant, called Hooke’s constant or the spring
constant. This constant depends on the quality of the material that the spring is made
of. Eq.(1) is called Hooke’s law and the minus sign indicates that the restoring force
opposes the displacement.
When a weight m is attached to a vertical spring, it will be in equilibrium under the
influence of its weight and the elastic force of the spring,



F  ky  mg
(2)
Or
k
mg
y
(3)
Figure 5
46
In the second part of the experiment we focus on the harmonic motion of the
spring. If the spring is pulled away from its equilibrium and then released, it will
start oscillating about its equilibrium point with its equation of motion is,
𝑚
𝑑2𝑦
𝑑𝑡 2
= −𝑘𝑦 (4)
The solution of this equation is the simple harmonic motion. After solving the
above differential equation, the period is,
T  2
m
k
(5)
But one has to include an effective mass for the spring in the above equation, so
the period for small oscillations would look like the following,
T  2
(m  meff )
k
(6)
Squaring the above equation, we get,
T 2  4 2
m  meff
k

2
4 2 m 4 meff

k
k
(7)
Procedure: Investigation of Hooke’s law,
1. Hang the spring as shown in Fig. 1. Attach a 50 g mass hanger on the spring
and then record the equilibrium position at the bottom of the hanger. This is
the reference point from which we measure the stretch of the spring.
2. Add a series of additional masses to the spring in 50 g increments. Record
the added masses and the corresponding positions of the hanger.
3. Start removing the masses from the spring in 50 g decrements. Record the
corresponding positions of the hanger.
4. Calculate the average displacements.
5. Plot displacement on y-axis with the corresponding masses on x-axis and
measure the slope of the best fit line.
6. Use equation (3) to calculate the value of k, where k 
g
slope
47
1.
2.
3.
4.
5.
Studying the harmonic motion:
Attach a 150 g mass (including the hanger) to the spring.
Pull a little down and then release the mass to oscillate. Calculate the period
for 10 cycles and then determine the period.
Add a series of additional masses and repeat step 2. Work up the largest
mass you can (at least 350 g). Calculate the period T for each step.
Plot T2 (y-axis) vs. m (x-axis) and find the slope of the best fit line.
Use equation (7) to calculate k,
k
4 2
slope
6. Compute g
7. Calculate the uncertainties.
8. Find the effective mass by connecting the best line to the y-axis in T2 vs. m
graph .
9. Calculate the uncertainty
Results and analysis: Investigation of Hooke’s law,
y0=……………………………. (m)
Trial
No.
m (kg)
y(m)
Adding mass
L
Removing mass
R
y=(L+R)/2
(m)
Δy=yy0
1
2
3
4
5
6
7
48
Studying the harmonic motion:
m (kg)
Time of 10 cycles (sec)
t1
t2
t3
t average
T=t/10
T2
(sec)
(sec)
(sec2)
Questions
1. How can you determine the effective mass of the spring meff ?
2. Compare between the two values of k that you obtained by the two methods,
are they equal?
49
Experiment 6: DC Circuits: Series and Parallel Circuits
Introduction:
One can define the resistance R of a metallic conductor by:
R
Voltage V

Current i
(1)
Where V is the potential difference applied between the endpoints of the
conductor and i is the current flowing through the conductor.
Consider two resistors R1 and R2 connected is series, then these resistors can
be replaced by a single equivalent resistor (Rs) where,
Rs  R1  R2
(2)
If the two resistors are connected in parallel, then they are equivalent to a single
resistor of magnitude RP where;
RR
1 1 1
 
 RP  1 2
RP R1 R2
R1  R2
(3)
50
Figure (1) shows how two resistors are connected in series and in parallel.
When they are connected is series, the current passing through them is the same,
when connected in parallel, the potential difference (voltage) across their end
points is the same.
From equation (1), one can find the uncertainty in R (ΔR):
R V i


R
V
i
(4)
Apparatus:
DC power supply, two resistors, Ammeter, Voltmeter, or two Multimeters.
Procedure:
A. Two resistors in series:
1. Connect the following circuit with the resistors R1 and R2 in series as shown
in the following circuit,
51
2. Estimate Δi and ΔV from the scales of the ammeter and the voltmeter.
3. Write down the readings of the ammeter (is ) and the voltmeter (Vs) only
once.
B. Two resistors in parallel :
1. Connect the resistors R1 and R2 as shown in figure (3) in parallel.
2. Estimate Δi and ΔV from the scales of the ammeter and the voltmeter.
3. Write down the readings of the ammeter (ip ) and the voltmeter (Vp) only
once .
4. Write down the values of the two resistors using the color code.
52
Calculations:
Part A of the data :
1. Find the experimental value of the equivalent resistance for the two resistors
connected in series ( Rs(exp)=Vs/is).
Rs Vs is


2. Find the uncertainty in Rs(exp). (note :
)
Rs
Vs
is
3. Compare your result with the values you get from the color codes
(Rs=R1+R2).
Part B of the data:
1. Find the experimental value of the equivalent resistance for the two resistors
connected in parallel ( Rp(exp)=Vp/ip).
R p
2.
Find the uncertainty in Rp(exp). (note :
Rp

V p
Vp

i p
ip
)
3. Compare your result with the values you get from the color codes (
RR
RP  1 2
R1  R2 ).
Questions: Discuss if your measured values of R (R1 or R2 ) , Rs and Rp are
consistent with the values you obtained from color codes , if not discuss why?
53
Experiment 7: Kirchhoff’s laws
Theory
Electric networks are circuits that include many elements such as resistors, voltage
sources and current sources that are connected together in rather complicated way.
In such cases, applying Ohm’s law and the simple parallel and series connection
rules is of no practical help.
Kirchhoff’s Laws
1. Loop theorem: this theorem is just the principle of conservation of energy as
applied to electric circuits. It states that: the algebraic sum of the voltage
drops and electromotive forces (emf’s) in closed electric circuits is always
zero. In other words, the power generated by sources in a closed circuit it
totally consumed by the circuit components,
 Vi  0
i
 k  i j R j
k
j
Where we have accounted for the opposite signs of voltage drops and emf’s.
2. Junction theorem: This theorem is just the principle of conservation of
charge applied to electric circuits. It states that: the algebraic sum of the
currents passing through any circuit junction is always zero,
ij  0
j
 Where the currents entering a junction have opposite signs to those leaving
it.
 One way of finding the values of the currents passing through the different
resistors in a circuit similar to the one shown in figure 1 proceeds as follows:
 Assign a current of arbitrary direction to each of the resistors in the circuit
 Apply Kirchhoff’s junction theorem to all
54
 Apply Kirchhoff’s loop theorem to all independent circuits’ loops.
 You should be able to produce as many independent equations as there are
unknown currents.
 Applying the rules above to the circuit of fig 1 gives the following,
1. Two junctions exist, but both give the same equation
i1+i2-i3=0
2. Three circuit loops exist, but only two of them independent equations could
be formed (Note that the third large loop will result in equation that is the
sum of the two small loop equations).
 1  i1 R1  i3 R3
 2  i2 R2  i3 R3
Figure 6
Solving these linear equations with three unknowns yields the values of the
currents passing through the three resistors. If any current is found to be
negative, its assigned direction must be reversed.
The Superposition Principle (SPP)
If circuit equations are linear, then the mathematical superposition principle is
applicable, it states that: The response at any point in a linear circuit having
more than one source can be obtained as the sum of the responses caused by
each of the independent sources acting alone. Therefore, a circuit that contains
55
independent linear sources and linear circuit components such as resistors,
capacitors and inductors can be analyzed as in the following example
1- Keep ε1 and replace ε2 with a short circuit(Figure 8)
2- Find the current passing through R3 as a result of the presence of ε1 alone
call it i31
3- Keep ε2 and replace ε1 by a short circuit.(Figure 9)
4- Find the current passing through R3 as a result of the presence of ε2 alone
5- Add both currents to find the total current passing through R3
i3=i31+i32
Figure 8
Figure 7
56
Apparatus
Two power supplies, three carbon resistors (1KΩ, 2.2 KΩ, and 4.7KΩ) and a
digital multimeter.
Procedure
a. Connect the circuit as shown in figure (1)
b. Measure the voltage differences across the three carbon resistors and the
current passing through each of them
c. Replace ε2 by a short and repeat part b.
d. Connect ε2 back replace ε1 by a short and repeat part b.
Analysis of results
1. Using SPP, analyze the circuit to find the value of the current passing
through R3 when each source is acting alone. Compare the values
obtained with the practical measurements.
2. Analyze the circuit using Kirchhoff’s rues. Find the values of the currents
passing through the three carbon resistors. Compare with the values
obtained from the experiment.
57
Experiment 8: Oscilloscope in AC Measurements
Introduction
In this experiment an AC sinusoidal signal will be investigated through Cathode
Ray Oscilloscope (CRO). The AC signal is the input to the CRO, and students will
do some changes manually to the input and they will notice these changes at the
CRO display. Then a diode and a resistor will be included in the circuit and the
output voltage across the resistor will be tested at the CRO to see the signal
produced from this circuit.
Apparatus:
 AC Power Supply, diode, CRO, 100 Ω, 1KΩ resistors
Theory
AC Voltage
Alternating current or voltage, periodically changes its magnitude and direction of
flow. It flows first in one direction and then in the opposite direction. The most
common AC voltage or current is a sinusoidal function of time. A sinusoidal
voltage may be described by an equation of the form:
V (t) = V0 sin(ωt)
(1)
where ω is the angular frequency, ω = 2πf, and f is the frequency of alternation
measured in Hz and equals to the reciprocal of the period T needed to complete
one cycle. Therefore, T = 1/f. The term V0, in Equation (1) above, represent the
maximum value of the voltage which means the voltage V at time t = π/(2 ω) or
the amplitude .The waveform that represent an AC signal described by Equation
(1) is shown below.
58
Figure 1. Sinusoidal wave with its peak-to-peak voltage Vp−p and its period T
AC voltage and currents cannot be measured with a scale voltmeter because the
pointer
cannot change direction quickly enough. They can be measured either with a
digital multimeter or an Oscilloscope.
Figure 2
59
The magnitude of an AC voltage may be characterized by the amplitude Vo, but a
more common practice is to use the “root-mean-square” voltage, which is
indicated by Vrms.
“Vrms : amount of AC power that produces the same heating effect as an
equivalent DC power”, The” root-mean-square” voltage depends on the type of
the AC signal. For
AC sinusoidal signal: Vrms 
Square signal:
Vrms 
Triangular signal: Vrms 
V p p
2 2
V p p
2
V p p
2 3
Diode in AC Circuit
From the previous experiment you have identified the diode, and you know that if
a diode is connected to a circuit with a DC power supply and your diode is in a
forward biasing then there will be a current passing through the circuit, but if the
diode is in a reversed biasing then there will be no current. In this part of the
experiment we will use the channels of the CRO to investigate the behavior of the
voltage across a resistor in an AC circuit that includes a diode as compared to the
voltage of the AC power supply in this circuit.
60
Figure 3
Connecting an AC Power supply to a circuit in which the diode is connected in
series to a resistor, and then you connect one of the channels of CRO across the
power supply input and the second channel across the resistor output of the circuit,
then you will see two signals on the CRO, first from power supply and second one
is from resistor. If you compare between the two signals you will see the difference
between both signals (input and output) because the diode is included in the circuit.
You can expect the shape of the output signal ”the output of the circuit” from the
concept of forward and reversed biasing, so either you will see only the positive
half of the cycle signal and negative half is cut, or you will see the negative half
only and the positive half is cut.
Procedure
Part One: AC Measurement
1. Connect the circuit of figure 2
2. Obtain various signals on the scope. Measure its amplitude, its peak to peak
voltage, frequency and period. Obtain other waveforms and determine their
characteristics.
3. Fix your signal generator at a frequency of 1KHz. Measure this frequency
using the oscilloscope.
4. Measure the amplitude of the signal with the help of the oscilloscope, then
with the digital multi-meter.
5. Repeat this for five different amplitudes and then plot a graph of the scope
measurement vs that of the multi-meter
Part Two: Diode in AC Circuit
1. Connect the circuit of figure 3.
2. Apply AC signal of small amplitude (3V) and 1KHz frequency
3. Draw the curve obtained to scale supplying proper units
Analysis of Results
1. What curve do you obtain from the plot of the scope measurement vs that of
the multi-meter? What is the significance of its slope?
61
Experiment 9: Charging and Discharging Of a Capacitor
The simplest form of a capacitor is two metal plates separated by an insulating
material. When connected to a power supply, positive charge will accumulate on
one of the capacitors plates and an equal negative charge will accumulate on the
other. This charge configuration results in the buildup of an electric field between
capacitors plates. This process is described as charging the capacitor. If the two
plates of a charged capacitor are connected together, the capacitor will discharge so
that each plate becomes neutral. Each capacitor is characterized by its plates
divided by the voltage difference across it,
C
Q
V
The unit of capacitor is the Farad (F).
Figure 8.8
Charging a capacitor:
When a capacitor is connected to an AC power supply, the charge in the simple RC
circuit shown in figure 6 builds up on the capacitor plates, during the positive half
period of the square wave, the charge builds up on the capacitor plates according
to the following formula,
Q(t )  C (1  e

t
RC
)
62
The voltage across the capacitor plates is defined by
Vc 
Q
C
Hence,
Vc   (1  e

t
RC
)
RC is usually called the time constant (τ) of the RC circuit, τ has the unit of time
(sec) and is a measure of how fast the voltage across the capacitors rises. When t =
τ,
Vc  0.63
Or the voltage across the capacitor rises to 0.63 of its maximum value. The current
passing through the circuit is given by
dQ   RC
I (t ) 
 e
dt
R
t
Therefore the voltage across the resistor is
VR  I (t ) R  e

t
RC
Discharging a capacitor:
During the negative half period of the square wave, the capacitor, in the RC circuit
of figure 6, discharge according to the following formula
Q(t )  Ce

t
RC
Hence, the voltage across the capacitor plates is given by
Vc  e

t
RC
63
RC is again called the time constant (τ) of the
circuit; it is a measure of how fast the voltage
across the capacitor plates decreases. When t=τ,
Vc  0.37
Or the voltage across the capacitor plates decays to
0.37 of its maximum value within a time τ.
The current passing through the circuit is
dQ
 
I (t ) 
  e RC
dt
R
t
Thus the voltage across the resistor is given by
Apparatus:
Figure 8.2
Resistor 5kΩ, Capacitor (0.01µF), Signal Generator and an Oscilloscope.
64
Procedure:
1. Connect the circuit of figure 8
2. Use a square wave from the signal generator to power your circuit
3. Display Vc on the oscilloscope screen. Measure τ for both charging and
discharging
4. Display VR on the oscilloscope screen. Measure τ for both charging and
discharging
Figure 8.3
Analysis of results:
1. Compare the values of τ obtained practically with the theoretical ly
predicted one.
2. Explain what happens when   0 and when    .
65
Experiment 10: Measurement of magnetic fields
Permanent magnet rod and a paper
Place a permanent magnet on a piece of paper. The magnetic field lines goes
through the points where the strength of the magnetic field has the same value. By
placing the magnetic field probe near the magnet, and adjusting the position, one
can find the position where the magnetic field is perpendicular to the probe, which
is the position where the maximum value is read in the display. Try to find
different places around the magnet with the same value for the magnetic field. A
curve drawn through these points is a magnetic field line.
Magnetic field from a coil:
Magnetic field vs. current in a coil
Direct Current through a coil results in a static magnetic field. The field strength
depends on whether the coil has an iron core, or just air, the number of windings
on the coil and the current.
The coil is connected to the power supply, and the Teslameter probe is placed at
the end of the coil, parallel to the direction of the windings (thus perpendicular to
the magnetic field).
Register the strength of the magnetic field at different currents. Make a table as the
follows:
I/A
B/T
66
Make a graph with the field strength on the y-axis, and the current on the x-axis.
Repeat with a coil with twice the number of windings. Try to hit the same values of
current as above. Make a table as follows:
I/A
B/T
Magnetic Fields of Coils
Introduction
The magnetic fields of various coils are plotted versus position as the Magnetic
Field Sensor is passed through the coils, guided by a track. It is particularly
interesting to compare the field from Helmholtz coils at the proper separation of
the coil radius to the field from coils separated at less than or more than the coil
radius. The magnetic field inside a solenoid can be examined in both the radial and
axial directions.
THEORY
R
Single Coil
x
For a coil of wire having radius R and N turns of wire,
the magnetic field along the perpendicular axis
through the center of the coil is given by
B
 o NIR 2

2 x R
2

3
2 2
(1)
67
Two Coils
Figure 2: Two Coils with Arbitrary Separation
R
R
x
R
R
B1 B2
x
R
x
d
For two coils, the total magnetic field is the sum of the magnetic fields from each
of the coils.
  
B  B1  B2 
o NIR 2
d

  x  R 2 

  2



2
3
2
 o NIR 2
xˆ 
d

  x  R 2 

  2



2
3
2
xˆ
(2)
For Helmholtz coils, the coil separation (d) equals the radius (R) of the coils. This
coil separation gives a uniform magnetic field between the coils. Plugging in
x = 0 gives the magnetic field at a point on the x-axis centered between the two
coils:
 8 NI
B o
xˆ
125R
(3)
68
Solenoid
For a solenoid with n turns per unit length, the magnetic field is
B  o nI .
(4)
The direction of the field is straight down the axis of the solenoid.
Single Coil Procedure
1.
Find the radius of the coil by measuring the diameter from the center of the
windings on one side across to the center of the windings on the other side.
2.
Set the Magnetic Field Sensor switch on Axial and x10 gain. With the DC
power supply off, set the Magnetic Field Sensor in the middle of the track
about 15 cm from the coil.
3.
Does the theoretical equation fit everywhere? If not, why not?
69
Helmholtz Coils Procedure
1.
Attach a second coil to the Helmholtz Base at a distance from the other coil
equal to the radius of the coil. Make sure the coils are parallel to each other.
See Figure 6.
2.
Connect the second coil in series with the first coil.
3.
Set the Magnetic Field Sensor switch on Axial and x10 gain. With the DC
power supply off, set the Magnetic Field Sensor in the middle of the track
about 5 cm from the first coil.
4.
Calculate the theoretical value for the magnetic field between the coils and
compare it to the measured value on the graph.
5.
Now change the separation between the coils to 1.5 times the radius of the
coils. Repeat steps 3 and 4.
6.
Now change the separation between the coils to half the radius of the coils.
Repeat steps 3 and 4.
70
Solenoid Procedure
1.
Connect the DC power supply in series with the digital ammeter and the
solenoid.
2.
Set the Magnetic Field Sensor switch on Axial and x10 gain. With the DC
power supply off, put the Magnetic Field Sensor inside the solenoid .
3.
Turn on the DC power supply and adjust it until the ammeter reads about
100 mA.
4.
Click on START and measure the magnetic field at various points all over
the inside of the solenoid, keeping the sensor probe parallel to the long axis
of the solenoid.
5.
Is the field inside the solenoid constant? What happens near the end of the
solenoid?
6.
Measure the length of the coil and using the given number of winds in the
coil, calculate the theoretical value of the magnetic field. Compare this
value to the value at the center at the coil.
7.
Set the Magnetic Field Sensor switch on Radial and x10 gain. With the DC
power switched off, put the Magnetic Field Sensor inside the solenoid.
8.
Turn on the DC power supply with the same current as before.
9.
Click on START and measure the magnetic field at various points all over
the inside of the solenoid, keeping the sensor probe parallel to the long axis
of the solenoid.
10.
Is the field inside the solenoid constant? What happens near the end of the
solenoid?
71
Lab Reports
Templates
72
Experiment #1
Measurements with Micrometer and Caliper
Group Number: ______________
Students Name:
ID
1
2
3
4
5
73
Objective:
__________________________________________________________
__________________________________________________________
Theory:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Data and Data Analysis: Measuring Metal blocks Dimensions
Data:
Trial
L( cm)
1
2
3
4
5
Average
W(cm)
T(cm)
The Mass of the Block: m= __________ , Δm=______________
74
Data Analysis:
1. finding the standard deviation of the mean for the length(L):
L(cm)
L-𝐿̅
(L-𝐿̅)2
(L-𝐿̅)2/N-1
Σ(L-𝐿̅)2/N-1=
σs(L)=
(Σ(L-𝐿̅)2/N-1) =
σm(L) =
2. finding the standard deviation of the mean for the width(W):
W(cm)
̅
W-𝑊
̅ )2
(W-𝑊
̅ )2/N-1
(W-𝑊
̅ )2/N-1=
Σ(W-𝑊
̅ )2/N-1) =
σs(W)= (Σ(W-𝑊
σm(W) =
75
3. finding the standard deviation of the mean for the Thickness:
T(cm)
T-𝑇̅
(T-𝑇̅)2
(T-𝑇̅)2/N-1
Σ(T-𝑇̅)2/N-1=
σs(T)=
(Σ(T-𝑇̅)2/N-1) =
σm(T) =
4. Calculate the volume :
̅ ∗ 𝑇̅ = _________________________
𝑉̅ = 𝐿̅ ∗ 𝑊
5. Calculate the uncertainty in the volume (Δ V)
Δ V=
So, the volume and its uncertainty:
»V±ΔV=(
)
6. Calculate the density (ρ) :
ρ= _______________
7. Calculate the uncertainty in the density (Δ ρ)
Δ ρ = _____________________________________________
So, the density and its uncertainty:
» ρ ± Δ ρ = _____________________
76
Questions and Conclusions:
Q1) what are the possible errors in this experiment?
__________________________________________________________
__________________________________________________________
_____________________________________
Q2) what is the best estimate of the real value of L, W and T?
__________________________________________________________
__________________________________________________________
________________________
Q3) what is the best estimate of the uncertainty in L, W and T?
__________________________________________________________
_____________________________________________
77
Experiment #2
Free Fall Motion
Group Number: ______________
Students Name:
ID
1
2
3
4
5
78
Objective:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Theory:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
___________________________________________
79
Data and Data Analysis:
1) Fill the table from the experiment and do the proper calculations:
Time t (ms)
Trial Y(cm)
t1
t2
2
2
t(sec) t (sec)
t3
g= 2y / t2
(m/s2)
1
2
3
4
5
6
7
g average=
2) Plot A graph y(m) vs. t2 (sec)2 find the slope
Slope=_____________________
3) From the slope calculate the experimental acceleration due to gravity
g exp= ____________________
4) Determine the percentage error
g( theory)−g(exp)
%error =|
| x 100% = ____________________
g (theory)
80
Conclusions:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
81
Experiment #3
Addition and Resolution of Vectors
Group Number: ______________
Students Name:
ID
1
2
3
4
82
Objective:
__________________________________________________________
__________________________________________________________
_____________________________________
Theory:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
83
Data
First Part: Resultant force of two forces
String 1
M1(kg)
Θ1(degree)
F1=M1g =
String 2
M2(kg)
Θ2(degree)
F2=M2g =
String 3
M3(kg)
Θ3(degree)
F3=M3g =
Second Part: Resultant force of two forces
String 1
M1(kg)
Θ1(degree)
F1=M1g =
String 2
M2(kg)
Θ2(degree)
F2=M2g =
String 3
M3(kg)
Θ3(degree)
F3=M3g =
Third Part: Resolving a vector
String 1
M1(kg)
F1=M1g =
Θ1(degree)
String 2
M2(kg)
F2=M2g =
Θ2(degree)
String 3
M3(kg)
Θ3(degree)
F3=M3g =
84
Data Analysis:
First Part: Resultant force of two forces
1-Determine the resultant force from Addition of components method.
Theoretical Part
String 1
F1x=F1cos(Ѳ1)
=
F1y=F1sin(Ѳ1)
=
F3=
F
2
3x
String2
F2x=F2cos(Ѳ2)
=
F2y=F2sin(Ѳ2)
=
 F 3 y  .............( N )
2
String3
F3x=F1x+F2x
=
F3y=F1y+F2y
=
 3  tan 1
F3 y
F3 x
 ...............
θ=
FR=
2-Determine the resultant force graphically (use graph papers)
F1=___________________________________________
F2=___________________________________________
FR=_______________________________________________
%𝑒𝑟𝑟𝑜𝑟 = |
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)−𝐹𝑅 (𝑔𝑟𝑎𝑝ℎ)
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)
| × 100% =_______________________
3-Record the value of the resultant force of the two forces on the force
table: Compare your results with your calculations.
Magnitude:
__________________________________________________
Direction:__________________________________________
%𝑒𝑟𝑟𝑜𝑟 = |
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)−𝐹𝑅 (𝑓𝑜𝑟𝑐𝑒_𝑡𝑎𝑏𝑙𝑒)
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)
| × 100% =____________________
85
Second Part: Resultant force of two forces
1-Determine the resultant force from Addition of components method.
Theoretical Part
String 1
F1x=F1cos(Ѳ1)
=
F1y=F1sin(Ѳ1)
=
F3=
F
2
3x
String2
F2x=F2cos(Ѳ2)
=
F2y=F2sin(Ѳ2)
=
 F 3 y  .............( N )
2
String3
F3x=F1x+F2x
=
F3y=F1y+F2y
=
 3  tan 1
F3 y
F3 x
 ...............
θ=
FR=
2-Determine the resultant force graphically (use graph papers)
F1=___________________________________________
F2=___________________________________________
FR=____________________________________________
%𝑒𝑟𝑟𝑜𝑟 = |
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)−𝐹𝑅 (𝑔𝑟𝑎𝑝ℎ)
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)
| × 100% =__________________
3-Record the value of the resultant force of the two forces on the force
table: Compare your results with your calculations.
Magnitude:
_________________________________________________
Direction:___________________________________________
%𝑒𝑟𝑟𝑜𝑟 = |
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)−𝐹𝑅 (𝑓𝑜𝑟𝑐𝑒_𝑡𝑎𝑏𝑙𝑒)
𝐹𝑅 (𝑡ℎ𝑒𝑜𝑟𝑦)
| × 100% =__________________
86
Third Part: Resolving a vector
1-Determine the magnitude of xy- components for the force
Theoretical Part
The force 𝐹⃗ :
𝐹⃗ =−𝐹⃗3 =
x-component
Fx=Fcosθ
=
y-component
Fy= Fsinθ
=
2-Determine the xy-components the force graphically by taking the
projections along the x and y-axes. (Use graph papers)
Fx=_________________
Fy=_________________
%𝑒𝑟𝑟𝑜𝑟 = |
%𝑒𝑟𝑟𝑜𝑟 = |
𝐹𝑥(𝑡ℎ𝑒𝑜𝑟𝑦)−𝐹𝑥(𝑔𝑟𝑎𝑝ℎ)
𝐹𝑥(𝑡ℎ𝑒𝑜𝑟𝑦)
𝐹𝑦(𝑡ℎ𝑒𝑜𝑟𝑦)−𝐹𝑦(𝑔𝑟𝑎𝑝ℎ)
𝐹𝑦(𝑡ℎ𝑒𝑜𝑟𝑦)
| × 100% =_____________________
| × 100% =______________________
Conclusions:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
87
Experiment #4
Measurements of g Using Simple Pendulum
Group Number: ______________
Students Name:
ID
1
2
3
4
88
Objective:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Theory:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
89
Data and Data Analysis :
The relationship between the length of the string and the period:
Tria
L(cm)
l
t1
Time t (s)
10 periods
t2
t3
t average
(sec)
One Period
Time
T(sec)
T2
(sec2)
1
2
3
4
5
Data Analysis:
1) Plot A graph T2 (sec2) vs. L (m) find the slope (use Grid Sheet)
Slope=_____________________
2) What is the unit of the slope?
_____________________________________
3) From the slope calculate the gravitational acceleration
g exp= ____________________
4) Determine the percentage error
%diff =
g( th)−g(exp)
g (th)
x 100% =________________________
90
Questions and Conclusions:
Q1) what are the possible errors in the experiment?
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Q2) what is the relation between the hanging mass and the period?
__________________________________________________________
________________________________________________________
Q3) if the angle θ not small, what will happen? What will be the Kind of
the motion?
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
91
Experiment #5
Hook’s Law
Group Number: ______________
Students Name:
ID
1
2
3
4
92
Objective:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Theory:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
_______________________________________________
93
Data:
Part1: the Investigation of Hookes Law
y0=_____________m
Trial m(Kg)
y(m)
Adding mass
L
Removing mass
R
y= (L+R)/2
Δy= y- y0
(m)
(m)
Part2: Studying of simple harmonic motion
Mass
(kg)
t1
Time t (s)
10 periods
t2
t3
t average
(sec)
One Period Time
T=t average/10
(sec)
T2
(sec2)
94
Data Analysis:
Part1: The Investigation of Hookes Law
1) Plot A graph Δy (m) vs. m (Kg) find the slope (use graph papers)
Slope=_____________________
2) From the slope calculate the experimental force constant (k)
k1 exp= ____________________
Part2: Studying of simple harmonic motion
1) Plot A graph T2 (sec2) vs. m(Kg) find the slope(use graph papers)
Slope=_____________________
2) From the slope calculate the experimental force constant k
k2 exp= ____________________
3) Compare between the two values of k that obtained from the two
methods?
%difference= |
2(k2 exp −k1 exp)
(k2 exp+k1 exp)
| ∗ 100% = _______________
4) Find the y-intercept of T2 vs. m graph.
____________________________________________________
5) Find the effective mass by connecting the best line to the y-axis in T2
vs. m graph. Where: 𝑚𝑒𝑓𝑓 =
𝑘×(𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡)
4𝜋2
___________________________________________________
95
Questions and Conclusions:
Q1) Can you explain: what is the meaning of the restoring force?
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__
Q2) Compare between the two values of k that you obtained by the two
methods, are they equal?
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__
96
Experiment #6
DC Circuits: Series and Parallel circuits
Group Number: ______________
Students Names: 12-
34-
Objective:
__________________________________________________________
__________________________________________________________
Theory:(Prove all the equations you used in this experiment )
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
_________________________________________________________
97
Part1: Series Connection:
Data and Data Analysis:
Part1: Series Connection
R1=_________±_________ R2=______________±_____________
is=__________±_________ Vs=______________±_____________
1) Rs (exp) =
2) ΔRs (exp) =
3) Rs (theory)=
4) Percentage difference =|
Rs(theor - Rs(Exp)
y)
Rs(theory)
| x 100%=
Part2: Parallel Connection:
Data and Data Analysis:
R1=________±__________ R2=_______________±____________
ip=_________±__________ Vp=_______________±____________
1) Rp (exp) =
2) ΔRp (exp) =
3) Rp (theory)=
4) Percentage difference = |
Rp(theor - Rp(Exp
y)
)
Rp(theory)
| x 100%=
98
Conclusions:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
99
Experiment #7
Kirchhoff’s Laws
Group Number: ______________
Students Names: 12-
34-
56-
Objective:
__________________________________________________________
__________________________________________________________
Theory:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
100
Part1:ε1 And ε2 connection
Data And Data Analysis:
R1=____________ R2=_______________
Δi = (
Δv = (
)
R3=______________
)
 Record the currents from the circuit and emf’s values:
 I1=(
)
Ɛ1=(
)

I2=(
)

I3=(
)
Ɛ2=(
)
 Theoretical part: Find all Currents using Kirchhoff’s Laws
101
 Compare the experimental value to the theoretical one :
𝑖 (𝑡ℎ)−𝑖1 (𝑒𝑥𝑝)
 %error for 𝑖1 =| 1
𝑖1 (𝑡ℎ)
| × 100%=__________________________
𝑖 (𝑡ℎ)−𝑖2 (𝑒𝑥𝑝)
 %error for 𝑖2 =| 2
𝑖2 (𝑡ℎ)
𝑖 (𝑡ℎ)−𝑖3 (𝑒𝑥𝑝)
 %error for 𝑖3 =| 3
𝑖3 (𝑡ℎ)
| × 100%=___________________________
| × 100%=___________________________
Part2:
A. ε1 connection
Record the current from the circuit:
» i31=(
)
B. ε2 connection
Record the current from the circuit:
» i32=(
)
Now: i3(exp)= i31 + i32 = _________________________________
𝑖 (𝑡ℎ)−𝑖3 (𝑒𝑥𝑝)
%error = | 3
𝑖3 (𝑡ℎ)
| × 100%=_______________
Conclusions
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
102
Experiment #8
Oscilloscope in Ac Measurements
Group Number: ______________
Students Names: 12-
34-
5-
Objective:
______________________________________________________
______________________________________________________
Data:
Part one: Ac Measurements:
a. Form the oscilloscope fills the table
Wave
Amplitude Vp-p
function
(V0(volt)) (Volt)
Sinusoidal
f(Hz)
ω(rad.Hz)
T(s)
Vrms (volt)
Triangular
Square
103
b. Oscilloscope vs. Multimeter measurements.
Record the voltage (Vp-p) from the oscilloscope and Vrms from the
multimeter on the below table for the same sinusoidal wave .
trial
Vp-p from the
oscilloscope
Vrms from the
multimeter
1
2
3
4
5
Data Analysis:
Part one: Ac Measurements
 Graphing (using graph papers)
Plot a graph of the peak –to - peak voltage vs. root mean square
voltage And Find the slope :( slope=
)
Q. What is the unit of the slope?
______________________________________________
 Compare between the theoretical value of the slope and the
experimental value
 %error=|
𝑉(𝑡ℎ)−𝑉(exp)
𝑉(𝑡ℎ)
| × 100%=___________________________
Part Two: Diode in Ac Circuit:
Draw the curve obtained from oscilloscope screen to scale supply proper
unit (using graph papers)
104
Conclusions
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
105
Experiment #9
Charge and Discharge a Capacitor
Group Number: ______________
Students Names: 12-
34-
Objective:
__________________________________________________________
__________________________________________________________
__________________________________________________________
Theory:
In charging capacitor prove that at t=τ ( Vc=0.632Vmax ) and (VR=0.37 Vmax)
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
106
Data And Data Analysis
 Record the magnitude of Resistor and capacitor:
R=_____________±____________, C=___________±___________
 Find τ (theory) : τ th = (
)
 Δ τ th=
Part one : charging capacitor Oscilloscope connected across capacitor
 Form the oscilloscope find τ(Exp) for charging process
Vc(volts)
τ (ms)
For C
%error= |
𝜏𝑒𝑥𝑝 −𝜏𝑡ℎ
𝜏𝑡ℎ
| × 100%=___________________________
Part two : Discharging capacitor: Oscilloscope connected across the
resistor.
 Form the oscilloscope find τ(Exp) for charging process
VR(volts)
τ (ms)
For R
%error= |
𝜏𝑒𝑥𝑝 −𝜏𝑡ℎ
𝜏𝑡ℎ
| × 100%=___________________________
107
Questions &Conclusions:
Q1: what are the charge and the voltage across the capacitor when
0 and when τ  ∞ in charging process?
τ
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Q2: what are the charge and the voltage across the capacitor when
0 and when τ  ∞ in Discharging process?
τ
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Q3: what is the possible source of errors in this experiment?
__________________________________________________________
__________________________________________________________
__________________________________________________________
108
Experiment #10
Magnatic Field Measurments
Group Number: ______________
Students Names: 12-
34-
56-
Objective :
__________________________________________________________
__________________________________________________
Data:
Part one: single coil:
I(A)
B(T)
Part Two: Double coils:
I(A)
B(T)
109
Data Analysis:
Part one: single coil
 Graphing (using Grid sheets)
Plot a graph of B vs. I And Find the slope :( slope=
)
 From the slope find μ0 = ___________________
Part two: double coils
 Graphing (using Grid sheets)
Plot a graph of B vs. I And Find the slope :( slope=
)
 From the slope find μ0 = ___________________
Questions &Conclusions:
Q1: theoretically, μ0 = 4π×10−7 V·s/(A·m) .find the percent error for the
two parts:
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
________________________________________________________
Q2. What is the unit of the slope obtained from the two graphs?
__________________________________________________________
______________
Q3. Give possible sources of errors for both parts.
__________________________________________________________
__________________________________________________________
_________________________________________________________
110