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ENGINEERING SCIENCE & PHYSICS DEPARTMENT
PHYSICS LAB MANUAL
C O L L E G E
O F
S T A T E N
I S L A N D
"There is no such thing as a failed experiment,
only experiments with unexpected outcomes."
--Richard Buckminster Fuller
PHY 206
SLS 261
DR. ANATOLY KUKLOV
ENGINEERING SCIENCE & PHYSICS DEPARTMENT
C I T Y U N I V E R S I T Y
O F
N E W
Y O R K
COLLEGE OF STATEN ISLAND
PHY 206/SLS 261
PHYSICS LAB MANUAL
ENGINEERING SCIENCE & PHYSICS DEPARTMENT
CITY UNIVERSITY OF NEW YORK
ENGINEERING SCIENCE & PHYSICS DEPARTMENT
PHYSICS LABORATORY
EXT 2978, 4N-214/4N-215
LABORATORY RULES
1.
No eating or drinking in the laboratory premises.
2.
The use of cell phones is not permitted.
3.
Computers are for experiment use only. No web surfing, reading e-mail, instant
messaging or computer games allowed.
4.
When finished using a computer log-off and put your keyboard and mouse away.
5.
Arrive on time otherwise equipment on your station will be removed.
6.
Bring a scientific calculator for each laboratory session.
7.
Have a hard copy of your laboratory report ready to submit before you enter the
laboratory.
8.
Some equipment will be required to be signed out and checked back in. The rest
of the equipment should be returned as directed by the technician. Remember,
you are responsible for the equipment you use during an experiment.
9.
After completing the experiment and, if needed, putting away equipment, check
that your station is clean and clutter free.
10.
Push in your chair.
11.
Before leaving the laboratory premises, make sure that you have all your
belongings with you. The lab is not responsible for any lost items.
Your cooperation in abiding by these rules will be highly appreciated.
Thank You.
The Physics Laboratory Staff
ENGINEERINGSCIENCE
SCIENCE&&PHYSICS
PHYSICSDEPARTMENT
DEPARTMENT
ENGINEERING
PHYSICSLABORATORY
LABORATORY
PHYSICS
EXT 2978, 4N-214/4N-215
EXT 2978, 4N-214/4N-215
10 ESSENTIALS of
writing laboratory reports ALL students must comply with
1. No report is accepted from a student who didn’t actually participate in the
experiment.
2. Despite that the actual lab is performed in a group, a report must be individually
written. Photocopies or plagiaristic reports will not be accepted and zero grade
will be issued to all parties.
3. The laboratory report should have a title page giving the name and number of
the experiment, the student's name, the class, and the date of the experiment.
The laboratory partner’s name must be included on the title page, and it should
be clearly indicated who the author and who the partner is.
4. Each section of the report, that is, “objective”, “theory background”, etc.,
should be clearly labeled. The data sheet collected by the author of the report
during the lab session with instructor’s signature must be included – no report
without such a data sheet indicating that the author has actually performed the
experiment is to be accepted.
5. Paper should be 8 ½” x 11”. Write on one side only using word-processing
software. Ruler and compass should be used for diagrams. Computer graphing
is also accepted.
6. Reports should be stapled together.
7. Be as neat as possible in order to facilitate reading your report.
8. Reports are due one week following the experiment. No reports will be
accepted after the "Due-date" without penalty as determined by the instructor.
9. No student can pass the course unless he or she has turned in a set of laboratory
reports required by the instructor.
10. The student is responsible for any further instruction given by the instructor.
PHY 206/SLS 261
CONTENTS:
1.
INTRODUCTION
1
2.
UNIFORM MOTION (1st Newton’s Law of Motion)
6
3.
FREE FALL
9
4.
FORCES AND ACCELERATION (2nd and 3rd Newton’s Laws)
12
5.
MECHANICAL WORK AND ENERGY
15
6.
HEAT AND INTERNAL ENERGY (CALORIMETRY)
18
7.
ELECTRIC FIELD AND ELECTRIC CURRENT (Ohm’s Law) 21
8.
WORK OF ELECTRIC FIELD (Joule experiment)
23
9.
REFLECTION OF LIGHT
26
10.
REFRACTION OF LIGHT
30
11.
SOUND WAVES
33
12.
ATOMIC SPECTRA
36
13.
RADIOACTIVE DECAY
39
DEMONSTRATIONS AND SUPPLEMENTARY EXPERIMENTS:
1.
CONSERVATION OF LINEAR MOMENTUM
41
2.
ELECTRIC FIELD AND ELECTRIC CHARGES
43
3.
ELECTROMAGNETIC FIELD
44
4.
COLOR AND WAVELENGTH OF LIGHT
46
5.
FORMATION OF AN OPTICAL IMAGE BY A CONVERGING
LENS
49
APPENDICES:
1.
MATHEMATICAL REVIEW
51
2.
ANALYSIS OF EXPERIMENTAL DATA
54
THE COLLEGE OF STATEN ISLAND
Department of Engineering Sciences & Physics
THE NATURE OF PHYSICAL PROCESSES
Introduction:
This one semester Physics course for future teachers has several objectives. One
of these is quite traditional -- giving an overview of the main physical facts and
explanations. This is the historical part of the course, and students may be tempted to rely
fully on just memorization of facts, laws and equations. However, the historical part is
not the main objective. A more important one is developing conceptual approach to
Nature as a whole in order to be able to recognize concepts which are relevant to a
particular problem. In other words, the most important goal is developing problem
solving skills.
Physics is not a collection of facts or equations. Physics looks for the most
general concepts which form the basis for the understanding of relationships between
various phenomena. In fact, only few general physical concepts can account for a huge
variety of natural phenomena. This is a direct manifestation of the unity of Nature.
Physics is the only intellectual achievement which, besides its cultural importance, is
endowed with an actual tremendous physical force. A few Physical concepts have
changed our world upon being implemented in industry and the military. No electronic
device could be manufactured without the understanding of the basics of electricity and
magnetism. No car engine would be possible without the knowledge of the laws of
mechanics. Even our social life is strongly affected by Physical concepts and their
implications. It is also good to realize that all medical imaging techniques such as X-rays,
ultrasound, MRI, isotopes etc. are all physics inventions. In addition, one of the most
critical cancer therapy – radiotherapy – is also the achievement of Physics.
At the same time, it is important to understand that not all Physical concepts are
equally general and are expected to be strictly valid. Some of them are only good
approximations to the actual behavior of physical objects. Such a difference will be
1
emphasized during the course. It is expected that most of the learning efforts are
devoted to the fundamental laws and their applications.
A central part of the conceptual approach is the understanding of the relationship
between Theory and Experiment, that is, observing what actually happens under given
circumstances and relating the results to theoretical predictions. A theory is considered
trustworthy if it allows us to explain coherently various phenomena and make predictions
with good accuracy. The following story sheds some light on this relationship:
Dogs do not like when anyone pulls their tail. Using this observation and the conceptual approach,
a theoretician John formulated a hypothesis saying: the number of times (B) a dog will
bite you
is proportional to the number of times (N) you pull the dog’s tail. Mathematically this
relationship is expressed as B/N=const. A brave experimentalist Steve was checking the
hypothesis by pulling his dog’s tail N times and counting the number B of bites he received. Steve
was going to see that the experimentally measured value of B/N was indeed nearly constant, as
predicted by John’s theory. However, no conclusion can be drawn because Steve was eventually
eaten by his dog. John is looking for another brave experimentalist who would continue
experimenting and either will confirm or disprove John’s theory.
This story is, of course, a joke. However, it contains the main elements of the relationship
between Theory and Experiment: Based on experimental observations, Theory formulates
a hypothesis which is supposed to be valid under certain circumstances. Experiment
realizes these circumstances in real life, and tries to compare the outcome with the
theoretical prediction. If the outcome is close to what is predicted, the theoretical
hypothesis can be considered reliable. Unfortunately, we are not always able to control
all the circumstances. Then more and more experiments are required in order to diminish
the role of these uncontrollable circumstances, and finally either to prove or to disprove
the hypothesis.
While working out a concept or preparing an experiment, it is crucial to simplify
the situation, so that unimportant factors are not interfering with the key factors. As you
will see, our laboratory experiments are composed in such a manner. However, not all the
unwanted factors can be completely eliminated. These introduce some uncertainty into
the measurements. Therefore, a very important issue which must be analyzed in each
experiment is the accuracy of the experimental result (see Appendix). So, estimating
errors of measured quantity is as important as finding the quantity itself.
2
II. Organizing the Laboratory Report
The laboratory report on the conducted experiment should contain the following:
1. Concise statement of objectives of a particular experiment;
2. A brief outline of the physical principles and assumptions used in the
experiment and a clear theoretical statement which is to be tested
experimentally;
Do not rewrite this manual ! Use your own words!
3. A short description of the experiment saying distinctly what (and how) is to be
measured and to be compared with the theoretical prediction;
4. Experimental data arranged in tabular form with labeled rows and columns, so
that the meaning of each number and its units in this table are evident;
5. Computations of the intermediate quantities and the final quantities which lead
to the main objective (the results of the computations are to be represented by
numbers and graphs);
6. Calculation of the percentage deviations (% difference) and errors (% error)
for each measured physical quantity;
7. Conclusion section which states clearly:
a) the value (-s) of the measured physical quantity (-ies);
b) the percent error (-s) and the percent difference (-s) for the
measured quantity (-ies) (see Appendix: Analysis of Data);
c) how reliably the experiment was done (refer to b));
d) what are the sources for the errors and the deviations which affect your
results;
e) based on b), your opinion on whether the experiment supports the
theory.
7. Answers to questions.
Technical Part
The laboratory report must be prepared personally by each student even though
the data was collected in group work. The report will not be accepted unless a student
3
personally took part in the measurements during the class work (or during a specially
arranged time). The report should be word processed, with the first page indicating the
name of the student and of the partners, the title of the experiment, the class, and the date
of the experiment. Graphs must be neatly drawn either by computer or by hand on graph
paper.
III. Report on Demonstrations
During each demonstration some topic will be discussed and corresponding
experiments or simulations will be performed. After each demonstration, a short report on
its essence must be written. This report should include a short description and physical
explanation of the experiments (or simulations) done during the demonstration. Finding
parallels and common ground with other experiments as well as facts and phenomena are
very welcomed. Extended conceptual discussion of the topic will be highly appreciated.
Problems related to the topic of the experiments will sometime be suggested for
solving. The solutions should be included in the report.
IV. Laboratory Rules and Regulations (not to be taken lightly)
The following rules and regulations are to be followed by anyone entering the
premises and using the laboratory facilities of the Nature of Physical Processes course
offered by the College of Staten Island.
1. Students are not to be in the laboratory unless the assigned instructor or
technician is present.
2. No drinking or eating are allowed in the laboratory at any time.
3. Order and decorum is to be maintained at all times.
4. The equipment should be left untouched until authorization for its use is given
by the instructor.
5. Tables and equipment are to be kept and left in a clean, neat and orderly
condition.
4
6. Each procedure which is to be performed on a computer or other equipment
must be considered first in order to avoid any harm to people, to the computers,
and to the software and other equipment.
The above rules and regulations are required primarily for the safety of all students, and
will also help the student to swiftly and correctly complete the laboratory assignments.
5
UNIFORM MOTION (NEWTON’S FIRST LAW)
One purpose of this experiment is to analyze the concept of Uniform Motion and to learn
how to perform measurements. A second purpose is to see the role of various factors in the
process of measurement which contribute to deviations (errors).
THEORETICAL BACKGROUND: An object executes Uniform Motion, that is, it moves
straight with constant velocity (or remains at rest), unless other bodies exert a finite resultant
force on the object. This statement is known as Ist Newton’s Law of motion. Thus, in order to
realize Uniform Motion it is crucial to eliminate (or strongly diminish) the main sources of such
forces. In this lab friction and gravity are of a particular concern.
Uniform motion can be described in terms of the distance S traveled during the time t by
a simple formula
S=Vt,
(1)
where V is the object velocity which stays constant during the motion.
Units used for S are meters (m) or centimeters (cm=1/100 m). Time is measured in seconds (s).
Correspondingly, the units for velocity are m/s or cm/s.
The formula (1) can be viewed graphically, if the horizontal axis of the graph is used for t and
the vertical axis for the distance S. Then, this graph is a straight line whose slope gives the
velocity V.
Graph: Distance vs time
The concept of Uniform Motion implies that:
For any measured pair of S and t the ratio S/t=V is independent of S and t, whenever
the object is not subjected to a net force.
6
Practically, all the objects around us experience external forces. This implies that in reality the
ratio S/t=V always deviates from a constant. In other words, the experimental data for S and t
represented on the graph by dots are in general not on the straight line.
If the external forces are reduced or compensated, an object’s motion becomes more and
more close to Uniform Motion. This is exactly what has to be tested by the experiment.
EXPERIMENT: Leveling the air track compensates for the gravity, and the air blowing through
the holes greatly reduces friction. Thus, after giving an initial push, the glider should move
nearly uniformly along the track in accordance with Newton’s First Law. In order to check this,
the glider coordinate as a function of time is to be measured by the synchronized timer with two
photogates set up along the track so that the distance between them is S. The glider will be
launched several times by the rubber band stretched to the same degree, so that the glider should
repeatedly perform the same motion. Once the glider passes through the two gates the time
reading on the timer screen gives the value of t.
PROCEDURE: Set the photogates the distance S apart (see the sketch). Make readings for
several distances S and the corresponding time intervals t. The velocity of the glider is to be
found in two ways:
a) by the formula (1); b) as the slope of the graph -- distance vs time.
ANALYSIS OF DATA: Arrange a table with the following columns: 1) distance traveled by the
glider; 2) time; 3) velocity; 4) deviations of the velocity from the mean velocity. Plot your data
for S and t on a graph distance vs. time. Make the linear fit of the data by drawing a straight line
through the data, and determine the velocity from the slope of this line. Also find the velocity as
an average of the entries in column 3). Calculate the mean deviation of the velocity (see the
Appendix: Analysis of Data).
DISCUSSION: Compare the results for V you obtained with what is expected for Uniform
Motion. Discuss reasons for the deviations of the velocity from the mean value.
QUESTIONS: 1) With what accuracy in percent does (or does not) your experiment
support the concept of the Uniform Motion?
2) The theory predicts that the velocity calculated by the slope and as the
average of column 3) should be the same. However, in practice these
two methods give slightly different results. Why is this so?
7
3) Will the coordinate vs time graph remain a straight line if the air track
becomes inclined? Explain.
4) What is the average velocity you measured if expressed in mm/s, m/s
and km/h?
5) For your one specific measurement of the velocity, make a
prediction what the glider displacement would be, if the
glider were moving with this velocity for 5 seconds.
8
FREE FALL
Uniformly accelerated motion will be analyzed and applied to an object falling freely due
to gravity.
THEORETICAL BACKGROUND: In the Uniform Motion experiment we learned that
if there is no external force, the distance covered by an object is proportional to the time
it moves. This statement is no longer true, if a constant force is applied to the object.
Earth’s gravity pulls all objects to the Earth’s surface. If gravity is the only force acting
on the object, the object’s motion is called Free Fall. In this case velocity varies in time.
Therefore, the distance S the object covers is not proportional to time t anymore. Instead,
the velocity change is proportional to time as
V-Vo = g t,
(1)
where Vo is the object’s initial velocity at the time moment t=0. In this relation, the
constant g shows how fast the velocity of the freely falling object changes in time. It is
called the acceleration of free fall. The value of g is expected to be independent of the
nature of the object. Close to Earth surface g = 9.80 m / s 2 =980 cm / s 2 . This value
will be considered as a known (or given ) value to be tested experimentally.
The Free Fall Motion is an example of Uniformly Accelerated Motion. The theoretical
graph V vs. t is a straight line.
V(cm/s)
Vo
t (s)
Graph: V vs. t
The distance covered by the object in the free fall is now related to g and t as well as to
Vo. Specifically,
1
S = Vo t + gt 2
2
.
(2)
While the graph of V vs. t is a straight line with the slope given by g, the graph of S vs. t
is not a straight line. The theoretical graph S vs. t is a parabola:
9
S
t
Graph: S vs t
Measurements are aimed at finding the value of g and comparing it with the given
value and also at how accurately the concept of Uniformly Accelerated Motion can be
applied to a freely falling object.
EXPERIMENT: The measurements will be done with the help of Behr Free-Fall
apparatus. The distance S(n) of the falling object traveled during n intervals of time is
tracked by the spark timer every ∆t =1/60 second on the waxed paper tape:
10
The displacement ∆S(n) during the nth time interval can be measured as the distance
between two successive spark traces:
∆S(n)=S(n+1) - S(n).
(3)
V(n)= ∆S(n)/∆t.
(4)
The velocity at the nth interval is
The acceleration can be obtained as a slope of the linear fit of the graph V vs. t=n∆t.
ANALYSIS OF DATA: Make a table containing four columns: 1) the time t=n∆t; 2) the
distance S(n) (assigning S=0 to the initial point n=0); 3) the velocity V(n) for each t.
Plot your experimental data for V on a V vs. t graph and make the linear fit of these data.
Find the acceleration g as the slope of the best linear fit line. Plot also a graph of your
experimental distances S vs. t.
DISCUSSION: Compare the value of g you found with the known one. Discuss the
reasons for the deviations of the V vs. t data from the corresponding points on the V vs. t
straight line graph. Evaluate the error of measuring g and compare it with the percent
difference. Discuss the shape of the S vs. t data graph and compare it with that obtained
in the experiment UNIFORM MOTION.
QUESTIONS: 1) What is the percent difference of the found g with the known?
2) What is the percent error of the measured g ?
3) How does the air resistance affect the measurement of g ?
4) If the same experiment were performed on the top of a mountain,
would you expect g to be larger or smaller than what you have obtained ?
5) Does this experiment prove that the value of g is independent of the
mass of the falling object?
6) Can a freely falling object be close to Earth and yet never hit the Earth?
11
FORCE AND ACCELERATION (NEWTON’S 2nd and 3rd LAWS)
The behavior of a physical object subjected to a constant external force is to be studied.
THEORETICAL BACKGROUND: The interaction between various objects is
responsible for a whole variety of phenomena in our Physical World. If no interaction
existed, our world would be a bunch of objects performing Uniform Motion in
accordance with Newton’s 1st law. We would not even exist because no forces would
bind the constituents of our organisms together. What a boring world!
So Force is one of the central physical concepts. It is not possible for us to trace out all
the possible means by which various forces act, and what are all the implications.
However, we will consider a specific situation which can be studied completely. This is
based on the observation that a force applied to a single object produces acceleration
of this object. Furthermore, a constant force produces a constant acceleration. In
accordance with Newton’s second law of dynamics, the acceleration is proportional to
the force. This law can be formulated as (Newton’s 2nd Law):
F=MA,
(1)
where A is the acceleration and F is the force. While F is essentially due to other objects,
the coefficient M is a property of the accelerating object. This property is called mass.
Applying this equation, we immediately deduce that due to gravity any object of the mass
m feels the force m ⋅ g close to Earth’s surface. This force will be applied to an object of
mass M in order to verify the equation (1) experimentally.
Unit of force is a Newton (N) and mass is measured in kilograms (kg). Accordingly,
1 N = 1 kg ⋅ m / s2 .
The 3rd Newton’s Law tells something very important about the action-reaction forces
between interacting objects: if the action force F is produced by some other object of
mass, say, m, then exactly the same reaction force in the opposite direction is produced
by the object M on the object m.
EXPERIMENT: A glider of mass M is placed on the air track with a string and hanging
weight of mass m attached. Gravity pulls the hanging mass m down with the force
W=mg. The action force on the glider is transferred from the hanging weight through the
string as the string tension F. The force F should produce the acceleration A in
accordance with the formula (1).
12
In order to measure A, two photogates are placed at a distance S apart so that the time t
the glider moves through this distance can be measured. If the glider starts moving from
rest with some constant acceleration A, this acceleration can be computed by the
following formula
A=2
S
t2
.
(2)
Finally, if indeed the formula (1) is valid, the mass M can be found as either the slope of
the linear fit of the graph F vs A or from the ratio
M=
F
A
(3)
In order to find F, we use 2nd and 3rd Newton’s Laws as: mg-F=mA and F=MA. From
these equations:
=
F m( g − A)
(4)
Summarizing, the purpose of these measurements is
to obtain data on F (from Eq.(4)) and A (from Eq.(2)), and check that these are
related to each other through the theoretical dependence given by equation (1). The
glider mass M is to be found and compared with the value obtained from the scale.
PROCEDURE: Set S at an appropriate distance (approximately 80-90 cm). For several
hanging weights mg measure time t . Do at least three measurements of t for each
weight. Compute the acceleration A in accordance with the formula (2). Do not forget to
13
measure the glider mass M on the scale in order to compare it with the value obtained
from equation (3).
ANALYSIS OF DATA: Arrange your data in columns: 1) hanging mass m ; 2) Force
F=m(g-A) ; 3) time t; 4) acceleration A (as given by equation (2)); 5) glider’s mass
M=F/A; 6) deviations of the mass M. Plot a graph of the tension force F vs. A, and find
the slope of the best linear fit of this graph. Compare the mass M obtained from the slope
with that obtained from the scale. Also, calculate the mass as the mean of the entries in
column 5). Find the mean deviation of M.
DISCUSSION: Give a conclusion as to whether this experiment supports Newton’s 2nd
and 3rd Laws of dynamics. Discuss the precision of your measurements for the glider
mass, and possible reasons for the errors.
QUESTIONS: 1) How close is the value of the glider mass obtained from the
slope to that obtained from the scale? What is the percent difference?
2) How could you check that the glider acceleration is constant during the
motion, and does not depend on the distance S ?
3) Use elementary algebra and express A in terms of m,M and g
from Newton’s equations mg-F=mA and F=MA.
4) For your one specific measurement of the acceleration, make a
prediction what the glider velocity and the displacement would be, if the
glider were moving with this acceleration for 4 seconds.
14
MECHANICAL WORK AND ENERGY
Conservation of mechanical energy will be studied as a fundamental law of Nature in a
simplest mechanical setup.
THEORETICAL BACKGROUND: The concept of Work and Energy is crucial for
understanding Nature. Its importance is not limited to the mechanics of a single object.
This concept can equally be applied to complex systems -- atoms, molecules, substances,
and even organisms. All acts of motion, transformation, and creation in our world are due
to Work and Energy. We will begin study of this concept for a simple mechanical system
-- the glider on the airtrack. In future experiments we will also analyze how the Work and
Energy concept can be applied to substances, electricity, and even our organisms.
A very special significance is endowed to the Law of Conservation of Energy which
states that total amount of Energy in the world is constant. That is, Energy can
neither be created or destroyed. It can be only transformed from one form to
another and redistributed between objects. The process of Energy transformation
and redistribution is called Work.
Here we will talk only about Mechanical Energy and Work. Any moving object possesses
Kinetic Energy KE =
MV 2
. The larger the object mass M or the velocity V, the greater
2
is KE. If two moving bodies are considered, then their total kinetic energy is
M 1V12 M 2V22
KE =
+
,
2
2
(1)
where the indices refer to the first and the second object, respectively. Forces produce
Work upon objects, and thereby change their KE.
Another form of mechanical energy is Potential Energy (PE). The PE depends on the
particular kind of force, acting on the object, as well as on the object’s position. It does
not depend on the velocity of the object. For example, in the gravitational field close to
Earth’s surface, an object of the mass M 2 has the PE
PE= M 2 gh,
(2)
where h is the height of the object’s position above some reference line. So the PE
changes with height. The unit of energy is the Joule (J) where 1 kg ⋅ m
2
s2
=1J.
If the mechanical system is isolated from the environment, the total Mechanical
Energy is
E=KE+PE=constant.
15
(3)
In other words, the total mechanical energy is conserved, in accordance with the general
statement given above.
Mechanical energy can be lost or acquired, if it is transformed to or from other
forms of energy. For example, friction decreases the KE and heats the environment. If,
however, no friction is present, the equation (3) says that any increase of the KE occurs at
the expense of the PE, or conversely any increase of the PE occurs at the expense of the
KE. Reformulated in terms of the KE change ∆KE and the change ∆PE, this statement
becomes
∆KE = - ∆PE
(4)
In this experiment, we will test the statement (4).
EXPERIMENT: The setup is similar to that we used in the earlier experiment we did on
“Newton’s 2nd Law”.
The first object with mass M1 is the glider on the level air track. The second object with
the mass M2 is the hanging weight. The total energy of this system -- the glider and the
hanging weight -- must remain the same during the joint motion of these two objects.
Thus, the graph ∆KE vs |∆PE| should be a straight line with slope one. Note that, while
the mass M2 changes its height by h, the mass M1 advances along the airtrack by the
same distance, that is, by h.
PROCEDURE: Make sure that the weight does not hit the floor before the glider passes
through the gate. Starting from rest (V=0), the velocity will increase uniformly due to the
object M 2 . For given M1 and M2, the velocity V will be measured by the timer in the
photogate in the regime gate. To calculate V use the formula:
V=d/t,
(5)
Where d=0.1 m which is the length of the flag on the glider and t is the time it takes for
the glider to cross through the photogate.
16
From equations (1), and (2), calculate the values of ∆KE and |∆PE|. Repeat this procedure
for several heights and masses M 1 and M 2 .
ANALYSIS OF DATA: Arrange the following data columns: 1) M 1 (kg) 2) M 2 (kg);
3) h (m); 4) time t (s); 5) V(m/s)=d/t (m/s); 6) ∆KE = ( M 1 + M 2 )V 2 / 2 (J); 7) |∆PE| =
M 2 g h (J). Plot the graph ∆KE vs |∆PE|. Find the slope of the best linear fit line for this
graph and compare it with the expected value 1. Compute the percent difference from the
expected value. Evaluate the percent error of the slope.
DISCUSSION: Conclude on whether your data supports the Law of Conservation of
Mechanical Energy and with what accuracy. Discuss possible physical reasons for the
deviations of the experimentally determined slope from the expected value.
QUESTIONS: 1) If the kinetic energy of an object has increased by 16 times, by how
many times has its speed increased?
2) If the slope of the graph ∆KE vs |∆PE| is greater than 1, does this mean
that the total energy has increased or decreased ?
3) Do frictional forces increase or decrease the total mechanical energy
of an object during its motion?
4) How would the slope (if compared to 1) of the best linear fit
graph change in the presence of friction?
5) Discuss possible deviations of the slope in cases when the air track was
not level.
17
HEAT AND INTERNAL ENERGY
The concept of Energy is applied to complex objects -- solids and liquids consisting of
huge amounts of atoms and molecules. Temperature is introduced as a parameter
characterizing the internal energy of substances. The Law of Conservation of Energy is to
be analyzed for the case of the internal energy of substances.
THEORETICAL BACKGROUND: In the work “Mechanical Work and Energy” the
energy of macroscopic objects -- glider connected to a weight -- has been discussed. The
Law of Conservation of Mechanical Energy was analyzed for a simple physical system. It
is not obvious that the same concept can be applied to atoms and molecules -- extremely
tiny objects. However, the notion of Energy turns out to be very general and applicable
for the whole Universe from galaxies to subatomic particles.
Temperature characterizes the kinetic energy of chaotic motion of atoms and
molecules. The hotter a piece of steel, the faster the motion of its atoms and electrons.
This means that the internal kinetic energy increases with the temperature. Nevertheless,
Temperature should not be identified with Heat. There is a simple relation between the
change of the internal energy Q of an object (or heat supplied to it) and the change of
temperature t of the object:
Q = Mc(t f − t i )
(1)
where the subscripts “i” and “f” refer to the initial and final states, respectively; M is the
object’s mass, and c denotes the Specific Heat for the material the object is made of. This
quantity indicates by how many Joules the heat energy of each kg of the body increases,
if the temperature is raised by one degree Celsius.
Temperature also controls the heat transfer from one object to another. Heat
normally flows from a hotter object to a colder one until temperatures of both objects
become equal.
If no mechanical work is being done on the system, the total internal energy
of this system will not change. This is a consequence of the Law of Energy
Conservation. In order to analyze the law, three objects are used -- calorimeter, water and
a piece of metal. If the metal is heated until some temperature t h and then immersed into
cold water, the metal will change will give up some heat
Qm = cm M m (t f − t h )
,
(2)
where the subscript “m” refers to the metal in equation (1) and t f is the final
temperature of the mixture -- metal, calorimeter and water in it. Accordingly, the water
and the calorimeter, which were initially at the temperature t i , will change their energy
so that their mutual change becomes
Qcw = cc M c (t f − t i ) + cw M w (t f − t i )
where the subscript “c” and “w” refer to the calorimeter and the water, respectively.
18
(3)
The total internal energy of the closed system which performs no mechanical
work does not change in time. In other words, the internal energy change must be
zero.
Expressing this as a formula, one has
Qm + Qcw = 0 .
(4)
From this equation it is possible to find the temperature t h the piece of metal had before
it was immersed into the water. If this temperature coincides with the actually known
temperature, then this will confirm the law of the energy conservation in this experiment.
The expression for t h which is obtained from equations (1)-(4) is
th = t f +
( M c cc + M w cw )(t f − t i )
M m cm
(5)
EXPERIMENT: A sample of metal is placed in boiling water so that its temperature t h
becomes 100 o C . This sample is then immersed into a calorimeter containing water at
room temperature, and the final temperature t f is measured. Finally, the initial
temperature of the sample t h can be calculated in accordance with the prediction (5)
made from the conservation law and then compared with the known value of 100oC .
PROCEDURE: Measure the quantities M m , M w , M c , and t i prior to immersing the
sample. Note that the mass of the inner cup plus the stirrer only should be measured for
Mc . Measure the final temperature t f after the sample was heated up in the boiling water
and then immersed into the calorimeter. Stir the water in the calorimeter a little, so that
the final temperature becomes the same for all three objects -- the sample, the water, and
the calorimeter. The three required values for the specific heats are
cc = 0.90
J
J
J
c
,
c
=
0
.
45
,
=
4
.
19
g ⋅ Co m
g ⋅ Co w
g ⋅ Co
19
ANALYSIS OF DATA: Arrange columns for t i , t h and deviations of t h . Repeat the
procedure several times, so that you will have several results for t h . Calculate the mean
of t h and the mean deviation of t h (see Appendix). Compare the mean of t h with the
known value 100 o C .
DISCUSSION: Discuss what you have tried to prove in this experiment. Make a
statement on the accuracy. Discuss possible physical reasons for the deviations.
QUESTIONS: 1) How close in percent is t h to the known value and what is the error of
th ?
2) If the calculated value of t h is always below 100 o C , what can you
say about the total internal energy of the system, -- was it lost or gained?
3) If the calculated value of t h is steadily well above 100 o C , what can
you say about the total energy of the system, -- was it lost or gained?
4) A 10 g bullet is stopped by 12 kg of water in a calorimeter. As a result
of this collision, the temperature of the water has increased by 0.05oC.
Assuming that all the kinetic energy of the bullet was transferred
to the water, calculate the velocity the bullet had before the impact.
20
ELECTRIC FIELD AND ELECTRIC CURRENT (OHM’S LAW)
Based on the demonstration “Electric Field and Electric Charges”, the concept of electric
current is introduced. A property of the conductor -- resistance and Ohm’s law are is
discussed. The difference between strict laws of Nature like the Law of Energy
Conservation and empirical rules is discussed.
THEORETICAL BACKGROUND: Materials containing electric charges which can
move freely are called conductors. Applying an electric field to such a material results in
the mechanical motion of the charges in one particular direction. This unidirectional
motion of electric charges is called electric current.
The applied electric field is characterized by the difference of the electric
potential V along the conductor. It is measured in Volts (V). The electric current I is
characterized by the amount of charge transferred through the conductor each second. It
is measured in Amperes (A).
For most conductors, the relation between V and I follows Ohm’s law
V=RI ,
(1)
where R is the resistance of the conductor, which does not depend on either V or I.
Resistance depends on various factors -- the material the wire is made of, the length of
the wire, its cross sectional area. Resistance also depends on the temperature of the wire.
The unit of R is the Ohm (Ω). If graphed on axes V vs I, the equation (1) yields a straight
line with the slope R.
However, not all conductors obey Ohm’s law. This means that, if plotted, the data
V vs I may be far from the straight line. Such conductors are called non-ohmic.
EXPERIMENT: To test the relation (1), a resistor which obeys Ohm’s law is used. It is to
be connected in series with the power supply and ammeter - the device measuring the
current I. The voltmeter is used to measure the voltage V across the resistor. Use this
chart for making the connections.
Subsequently, use the lamp as a non-ohmic resistor R, and repeat the measurements.
PROCEDURE: The current I should be measured for various V for the ohmic resistor and
the lamp. For each case, the graph V vs. I is to be plotted.
21
ANALYSIS OF DATA: For the ohmic resistor, arrange columns for V , I. Plot the data V
vs I and make the best linear fit. From the best linear fit line find the resistance R of the
ohmic resistor and compare it with the known value. For the lamp, set up three columns
V, I,R. Plot the graph V vs I for the lamp.
DISCUSSION: For the ohmic resistor, discuss how accurately your data supports (if any)
Ohm’s law. Discuss the percent error and percent difference. For the lamp, conclude
whether the direct proportionality indicated in equation (1) does or does not hold. Give
your opinion on the meaning of Ohm’s law within the context of general laws of Nature.
QUESTIONS: 1) For the ohmic resistor, how close in percent is R to the known value and
what is the error of R?
2) What is your physical explanation for the fact that the lamp data does
not follow Ohm’s law?
3) From your lamp data, find the lamp resistance at 15 V.
4) From your lamp data, find by how many times the resistance of the
lamp changes while the voltage changes from small to high values.
5) A 120Ω resistor is connected in series with another one, so that the
resulting resistance is 200Ω. What is the resistance of the second resistor?
22
WORK OF ELECTRIC FIELD
The work done by the electric field driving the electric current is analyzed. The Law of
Conservation of total Energy is tested for the case of electric energy conversion into
internal energy.
THEORETICAL BACKGROUND: In the two preceding experiments “Mechanical
Work and Energy” and “Heat and Internal Energy”, the concept of energy was discussed
and developed.
All physical objects and processes can be described in terms of the Energy content and
the Work done. The electric field possesses energy and can therefore perform work. If a
current I is driven by a voltage V across a resistor, some work is continuously being done.
The amount of this work during the time t is
W = IVt
(1)
This work heats the environment. The Law of Conservation of total Energy requires that
the heat Q acquired by the environment is exactly equal to the work W done by the
current
Q=W.
(2)
The relation (2) is of crucial importance. It says that the work done by the electric
field is not being lost. It is being converted into heat energy.
The purpose of the following analysis is to develop an experiment for testing the Law of
Energy Conservation represented by equation (2).
If the heat Q is delivered to a known amount M w of water contained in a
calorimeter of mass M c , the temperature increase of the water and the calorimeter can
be found from the formula
Qcw = cc M c (t f − t i ) + cw M w (t f − t i )
(see the experiment “Thermal Energy”; cw = 4.19 J
(3)
g ⋅ C o is the specific heat of water,
cc = 0.90 J g ⋅ C o is the specific heat of the calorimeter).
EXPERIMENT: A heat producing resistor is immersed into the water in the internal can
of the calorimeter. Voltage is applied to the resistor. The temperature of the water is
monitored by a thermometer. Time of current flow is also monitored.
23
PROCEDURE: 1) Weigh internal can of the calorimeter together with the stirrer. Fill the
can about 3/4 full of water, and weigh it again, and obtain the mass of the water.
2) Connect the voltmeter, ammeter, and the heating coil (resistor) according to the
diagram. Do not turn on the power supply until it has been checked by the
instructor.
3) Once the circuit has been checked, place the heating coil in the water and then
turn on the power supply by closing the switch. Record both the voltage V and the current
I. Open the switch once this is done. Stir the water gently and record the initial
temperature t i
4) Close the switch to reconnect the power supply, start the timer, and constantly
and gently stir the water. Observe the temperature rising. Take the readings of
temperature, current and voltage every 100 seconds for about 1000 s. Continue gently
stirring during the experiment. Avoid spilling the water from the internal can to the
external can!
ANALYSIS OF DATA: Organize your data as the following columns: 1) time; 2) voltage
; 3) current; 4) temperature; 5) work of electric field; 6) heat acquired by the calorimeter.
Do not forget to measure and record the other quantities entering equation (3), i.e. the
mass of the water and the mass of the calorimeter together with the stirrer.
Plot a graph of the experimentally determined values of the heat acquired by the
calorimeter vs the work of the electric field. If the experiment were absolutely precise,
the slope of the best linear fit line of this graph would be exactly 1. Find the actual slope
and compare it with 1.
DISCUSSION: Discuss what you have tried to prove in this experiment. Analyze
possible physical reasons for deviations of the measured slope from the predicted one.
24
QUESTIONS: 1) What is the percent difference of your measurement of the slope, and
what is the error of the experimental slope?
2) If the measured value of the slope of the Q vs W line is below the
theoretical prediction, is the work done by the electrical field greater than
or less than the heat transferred to the water in the calorimeter?
3) Given your data, calculate the time required to boil the water.
25
REFLECTION OF LIGHT
The law of light reflection is investigated and used for constructing images by mirrors.
THEORETICAL BACKGROUND:
Light is reflected by a mirror. The law of reflection states that
the angle of reflection (let us call it θr ) is equal to the angle of incidence θi, and the
three lines -- the incident ray, the reflected ray and the perpendicular (normal) to
the mirror -- are all in the same plane.
θr = θi
(1)
The image S’ of an object S giving off light is formed either by the intersection point of
the reflected rays (solid lines) or their geometrical continuations (dashed lines). In Fig.1
the point S’ is the image of the point S.
S
S’
θi
θr
Fig.1. Plane mirror (vertical solid line)
This image is the intersection point S’ of the rays sent by the object S. Here only two rays
are shown. It is important to note that any third ray sent by S will be reflected by the
mirror in such a way that its continuation will also pass through the image point S’. Such
an image is called a virtual image. We will use this method for constructing images in
more complex situations.
Concave mirrors can produce real images. An important characteristic of the
concave mirror is its focal point. This is the point F where the beams of light parallel to
the axis of the mirror converge, that is, they create an image of a very distant object (say,
a star). Convex mirrors produce virtual images. Accordingly, its focal point F lies behind
the mirror surface. The distance from the focal point to the mirror vertex is the focal
distance f. For the concave mirror it is positive and for the convex – negative. This
distance is given by half of the mirror curvature radius.
26
Fig.2.
In order to construct an image of any object placed in front of a mirror, two incident rays
can be used – one going parallel to the main axis (and then reflected through the focal
point) and the other going through the focal point (and then reflect parallel to the main
axis). This procedure is shown on Figs.3,4:
Fig.3: Real image S’ formation by the concave mirror
Fig.3: Virtual image S’ formed by the convex mirror: the incident ray 2 continued (dotdashed line) through the focal point F so that the reflected one goes parallel to the main
axis; the incident ray 1 is parallel to the main axis and the reflected one continued
backward through the focal point.
27
The distances do , di of the object S and the image S’ are related to the focal length f
through the equation
1 1 1
+ =
d 0 di f
(2)
The image can be larger or smaller than the object. The ratio M of the image height hi
to the object height ho is called magnification. The value hi is taken positive if the
image is upright and negative if inverted. Using simple geometry it is possible to show
that
M= −
di
do
(3)
Where di is positive for real images and negative for virtual.
EXPERIMENT: Using the ray box, the property of reflection of light will be studied for
the three types of mirrors – plane, concave and convex
.
Procedure I: Place the plane mirror on a sheet of paper and adjust the light source so as
to produce a single ray. Aim the ray toward the center of the mirror so that the ray is
reflected on itself. Trace the ray on the paper. This line which is perpendicular to the
plane mirror is the mirror axis. Now aim the ray at an angle to the mirror so that it strikes
the mirror at the point where the axis crosses the mirror. Draw both the incident and
reflected rays and measure the angle of incidence θi and the angle of reflection θr.
Repeat the procedure for two more angles of incidence. Compare the angles θi and θr
and conclude whether they obey the condition (1).
Procedure II: Locate the principle axis of the concave mirror as in the procedure I and
trace it. Adjust the ray box to produce a bundle of parallel rays. Aim the rays at the
mirror, parallel to its principle axis, so that the reflected rays cross each other at a point
on the principle axis. This point is the focal point. Trace these rays on the paper. Repeat
this procedure for the convex mirror. Take into account that in this case the reflected rays
must be continued backward until their intersection. Record the value of f.
Procedure III: Using the results of the procedure II, trace the rays through an object S
placed some distance do from the mirror (concave and, then, convex). Notice the
reflected beams and trace them on paper. Locate image S’. Record di. Repeat the
procedure for two more distances do. Use di and do in equation (2) and determine the
28
focal length f . Compare the focal length values obtained in the procedures II and III.
Record also the sizes of the object and image.
ANALYSIS OF DATA: For the procedure I arrange the data in 3 columns: 1. Angle of
incidence; 2. Angle of reflection; 3. Percent difference between the two.
For the procedure III put your data into 7 columns: 1. Object distance; 2. Image distance;
3. Focal length determined from the equation (2); 4. Deviations of the focal length from
the mean value; 5. The magnification determined by direct measurements of the object
and its image; 6. The magnification determined by the equation (3); 7. The difference
between the columns 5 and 6.
DISCUSSION: Discuss with what accuracy in % your data support the laws of geometric
optics. .
QUESTIONS: 1) How close to each other (in %) are the values of the angles
θi
and θr ?
2) How close (in %) is the mean focal length determined in the procedure
III to that found in the procedure II?
3) What is the error of the focal length found in the procedure III?
4) What is the mean percent difference between the magnification determined
directly and through equation (3)?
5) If di is twice of do, what are these in terms of the focal length f (hint: use the
equation (2))?
29
REFRACTION OF LIGHT
THEORETICAL BACKGROUND:
Refraction (bending) of light occurs at the boundary between two transparent media, e.g.,
air and glass. This is because the speed of light in the glass is smaller by a factor
n=n2/n1 , called the index of refraction, than the speed of light in the air. Snell’s Law
gives a relation between the angle of incidence θ1 , the angle of refraction θ2 and n
in two media:
n1 sin θ1 = n2 sin θ 2
(2)
Both angles are measured with respect to the normal (NN’ line in Fig.3):
Fig.3. Refraction through a prism
Refraction by convex or concave surfaces is used to create images of objects quite
similarly to mirrors. Even the equations relating the object and image distances and the
magnification are the same as for mirrors (see (2),(3) in the lab “Reflection of Light”).
Converging lens focuses a beam of light parallel to the main optical axis of the lens to the
focal point F behind the lens. A diverging lens, made of concave surfaces, diverges the
parallel beam of light so that their continuations converge at the focal point F in front of
the lens. The difference with mirror is that lens has two focal points – symmetrically at its
both sides:
30
Fig.4. Focal points of lenses.
EXPERIMENT: Using the ray box, the properties of refraction of light will be studied
for the three optical elements: the prism and converging and diverging lenses.
Procedure I: Draw a straight line (NN’) and place the trapezoidal prism (the frosted side
down) in such a way that the line is perpendicular to the base. Trace the outline of the
prism. Set up the light source to produce a single incident ray at some angle through the
point where the normal NN’ crosses the base so that the refracted beam goes through the
second base. Trace the incident and refracted beams. Measure the angles defined as in
Fig.1. Repeat the procedure for 3 more angles of incidence.
Procedure II: Using the ray box, trace a single ray on the paper. Place the converging
lens in the way of this ray so that the ray remains unrefracted. Trace the position of the
lens. These two traces on the paper give the position of the lens and its principle axis.
Now, aim a parallel bundle of rays at the lens so that the refracted rays intersect each
other at a point on the principle axis. Trace the rays. Their point of intersection is the
focal point. Find the second focal point by shining light from the opposite side. Repeat
this procedure with the diverging lens, keeping in mind that the refracted rays must be
continued backward in order to find the focal point F and its distance f from the center of
the lens.
ANALYSIS OF DATA: For the procedure I put your data into 4 columns: 1. Angle of
incidence; 2. Angle of refraction; 3. Index of refraction n2 determined from the equation
(1) (use n1 =1) ; 4. Deviations of n2 from the mean value.
For the procedure II, determine the focal lengths f and compare them with the known
values.
DISCUSSION: Discuss how good your data support the law of refraction.
QUESTIONS: 1) With what accuracy (percent error) have you found the trapezoidal
31
prism index of refraction?
2) What is the percent error of your measurements for n2?
3) Use the focal lengths obtained in the procedure II and determine the image
positions for an object ho = 2 cm tall placed do= 8 cm from a lens. What are the
sizes of the corresponding images (use equations (2),(3) from the lab “Relection
of Light”) ?
32
SOUND WAVES
Properties of sound as a wave phenomenon are studied. The speed of sound in air is
measured by employing the effect of Wave Interference.
THEORETICAL BACKGROUND: A sound wave is a process of propagation of
oscillations of the density of a substance. The wave transfers the energy of these
oscillations over long distances with the speed VS referred to as the speed of sound. The
speed of sound is a property of the particular substance. In the air at room temperature,
sound propagates with the speed
VS =330 m/s
.
(1)
Other characteristics of the wave are its frequency f and its wavelength λ. The frequency
is number of oscillations of the density per second. This unit is Hz=1/s. A duration of
one oscillation is called the period T. There is a simple relation T=1/f. The wavelength is
the distance a single oscillation propagates during one period T. Thus the relationship
between wavelength λ and period T is
λ= V T.
S
(2)
A property very specific for wave phenomena is wave interference. A simple way
to realize wave interference is to make two identical waves moving in opposite directions
meet each other. Then a standing wave is formed. It is characterized by specific points
called nodes where the two waves exactly cancel each other. The distance between two
adjacent nodes is λ/2. At the midpoint between two adjacent nodes, there is always an
antinode -- a place where the amplitudes of the two waves enhance each other (add
constructively). The distance between two adjacent antinodes is also λ/2.
Standing waves are always being formed in the strings or the pipes of musical
instruments. A typical resonant sound coming from the open end of the pipe is due to the
antinode formed exactly at the end. One can use this property to measure the speed of
sound. If the other end of the pipe is closed (as will be the case in this experiment), then
at this end a node is formed. Thus, if a strong (resonant) sound is heard from the pipe, it
must contain some whole number n of the halves of λ plus one quarter of λ. That is, its
length is
L=n λ/2+λ/4.
(3)
This situation is depicted on the sketch where the filled circles indicate the nodes and the
open ones – the antinodes. Here the pipe contains two antinodes (n=2) inside the pipe
and one antinode at the open end. It also has three nodes.
33
λ/4
λ/2
closed
end
λ/2
open
end
Fig. 1
Then, using equation (2) in (3), we find L=(n+1/2) VS /2f. For a given frequency f, the
length L for a particular n can be measured and the speed of sound can be found from the
expression
VS =2fL / (n+1/2).
(4)
EXPERIMENT: For the given frequency f of the tuning fork, the shortest length L of the
acrylic tube not filled with water (see the picture) should be found so that the first
resonant sound is clearly heard. This is an indication that the condition n=0 for equations
(3) and (4) is achieved. Increasing L will result in the occurrence of the second resonant
sound. Correspondingly, this implies n=1 in equations (3) and (4). Similarly, one can find
corresponding length L for higher n.
ANALYSIS OF DATA: Make a table with the following columns: 1) the number of the
antinodes n inside the pipe; 2) the length L of the corresponding empty part of the tube;
3) the speed of sound VS ; 4) deviations of VS. Find the speed of sound as the average of
the entries in column 3). Calculate the mean deviation as the average of the entries in
column 4), and find the percent error of your measurements of VS. Compare the speed you
obtained with the standard value of 330 m/s.
DISCUSSION: Discuss how close is the measured value of the speed of sound to the
standard value. Explicitly discuss whether the mean deviation of the measured speed is
larger or smaller than the difference between the measured VS and the standard value. If
these are close, what does this prove?
QUESTIONS: 1) Suggest another way to find VS which does not require measurements
34
of the wavelength.
2) During a thunderstorm the lightning is seen first and then the thunder is
heard. What can you say about the ratio of the speed of light to the speed
of sound?
3) If the time delay between the lightning and the thunder is 5 seconds,
how far from you is the place where the lightning and the thunder
occurred?
4) For the design of an organ pipe to play the musical note at 420 Hz, of
what length should the pipe be?
35
ATOMIC SPECTRA
The effect of light interference is used to measure the wavelengths of light emitted by
various atoms. This lab is based on the demonstration “Colors and Wavelength of
Light”.
THEORETICAL BACKGROUND: Every chemical element or compound, after being
excited by delivering energy to it, emits a unique pattern of wavelengths of light (or
colors). In fact, such a pattern makes it possible to recognize unambiguously the element
presence even in a very tiny amount. As discussed in the work “Color and Wavelength of
Light”, each color corresponds to a particular wavelength of the light wave. Therefore, to
analyze the pattern precisely, the wavelengths of light should be measured. This direct
measurement would be difficult to perform because of the very small values of the
wavelengths of visible light (400 - 700 nm). However, the wave properties -- interference
and diffraction -- turn out to be very helpful. The interference effect has already been
used to measure the wavelength of a sound wave (see the work “Sound Waves”). In fact,
standing waves can also be formed by light similarly to what happened with sound in the
resonating pipe. In this experiment, another device -- the diffraction grating -- will be
used to explore the interference of light. A key element of the grating are the very tiny
parallel lines set on the transparent slide. The distance d between these lines is a very
important parameter. Normally, each grating is labeled as to how many lines per mm it
has, from which the distance d can be determined.
Light striking the grating splits into many rays interfering with each other. As a
result, the interference antinodes (bright lines) are formed on the screen. The position of
the each antinode depends on the wavelength λ and the parameter d. This dependence is
given by the following equation
d sina=nλ
(1)
where a is the angle of diffraction (see the sketch below ) and n=0,1,2,3 stands for the
order of the antinode.
grating
screen
incident light
a
x
S
Fig.1. Diffraction and interference of light due to the diffraction grating
36
Measuring S -- the distance from the grating to the screen and x -- the distance from the
zero order antinode to the n-th order antinode, the sin(a) can be found
as
sin a =
x
x2 + S 2
.
(2)
The wavelength can then be determined as λ=d sin(a)/n from equations (1). A
substitution of the sin a from equation (2) gives
λ=
dx
n x2 + S 2
.
(3)
Such a method for measuring λ was used already to find the wavelength of light emitted
by a Laser in the work “Color and Wavelength of Light”.
EXPERIMENT: Each chemical element is excited by the High Voltage Electric
Discharge which is dangerous if handled improperly. Never touch the spectrum tube
or its holders ( SHOCK HAZARD !).
The human eye will play the role of the screen in this experiment. The grating
and a short ruler are to be placed on the meter stick. The ruler has a slit in its center. This
slit is to be aimed at the spectrum tube containing the glowing element.
Fig.2. Schematic of the meter stick optical bench with the spectrum tube .
First make sure you can find the spectrum of colors by looking through the
grating at the light emitted by the element. You will see the colored lines at the sides of
the light source. These lines form a group which repeats itself as one looks farther from
the center line. Each group corresponds to a different n in equations (1) and (3).
The first group of lines for which n=1 is to be analyzed. Record the distance x
along the short ruler for each distinct colored line, keeping the distance S along the meter
37
stick fixed. A normal sequence of the colors is violet- blue-green-yellow-red counting
from the slit. Note the names of the elements whose spectra you have analyzed. For each
element, identify its spectrum from the chart given in class and compare the
corresponding wavelengths with those of the lines you have measured.
ANALYSIS OF DATA: For each chemical element whose spectrum you have analyzed,
make a table with the following columns: 1) color of the line 2) position x (cm) of the line
3) the wavelength calculated from formula (3). For each color identify the corresponding
line from the chart and enter its wavelength in column 4). Do not forget to record the
distance S in cm.
For this work a grating having 600 lines per mm will be used. Find the
corresponding value of d which is to be used in the equation (3). Represent your answer
in nanometers (nm). Refer to the Appendix “Mathematical Review” in order to convert
correctly the value of d measured in mm into nanometers.
DISCUSSION: Discuss how precisely you have identified the wavelengths of the spectral
lines for each element you have studied.
38
RADIOACTIVE DECAY
Natural radioactivity is discussed and the process of radioactive decay is simulated. The
method of scientific simulation as a mean for analyzing a real situation is introduced.
THEORETICAL BACKGROUND: Each atom has a nucleus. Some of the nuclei are
unstable,-- these can break apart or transform into other nuclei. These processes are
called Radioactive Decay. As a result of the decay, various kinds of radiation are emitted.
The radioactive decay can be described by the following general model.
There is some probability p that an unstable atom decays during the time interval
T (specific for each element). For some elements this time can be very long (years and
even thousands of years), while for others it could be days or even seconds. If initially N
atoms existed, after the time interval T has elapsed the remaining number of atoms is (1-
p) N . After 2 intervals of T have elapsed, the remaining number is (1-p)(1-p) N.
After 3 intervals of T have elapsed, the remaining number is (1-p)(1-p)(1-p)
N, and
so on. Finally, we can arrive at the general expression for the number of atoms N(t)
remaining after n time intervals have passed:
N (t ) = (1 − p) n ⋅ N .
(1)
Radiation can be harmful if handled without caution. Therefore we will not use
any radioactive substance in the class. Instead, a simple simulation with dice will be
done.
A single die imitates a radioactive atom. Internal processes leading to the decay
are simulated by random throws of each die in the common large basin containing, e.g.
100 dice. We consider the “atom” having decayed, if the number 6 appears on the upper
side of the die. Then this “atom” should be removed from the basin. We can predict that
because of the random nature of the tossing, the probability for each die to display 6 is
p=1/6. Therefore, there is a high likelihood that 1/6 of all the atoms decays after a first
tossing, so that the remaining atoms are (1-p)=5/6 of 100. If the number of throws is
n=0,1,2,3.., we can make a table for the remaining number of atoms 100, 100× 5/6, (100
× 5/6)× 5/6, etc. (of course, the numbers are rounded off to make N(t) an integer):
n
N(t)
0
100
1
83
2
69
3
58
4
48
5
40
6
33
7
28
8
23
9
19
10
16
11
13
Algebraically, this dependence is given by the formula
5
N (t ) =   ⋅ 100
6
n
in accordance with the general expression (1).
39
(2)
EXPERIMENT - SIMULATION: Throw the 100 dice once and remove all the decayed
“atoms” (the “6” `s). Count the number N left and record it. Repeat the process for the
remaining atoms similarly until less than 10 atoms left. Repeat this procedure for 10-12
trials.
ANALYSIS OF DATA: Arrange the following columns: 1) throw number n; 2) N for the
first trial; 3) N for the second trial; 3) N for the third trial; 4) N for the fourth trial, etc.
Reserve a column for the average of N over all trials (not throws !). Finally, reserve a
column for the analytical dependence given by the formula (2). On a single graph plot a)
your data -- average of N vs. n and b) the dependence obtained from formula (2).
DISCUSSION: Discuss what you have tried to prove in this experiment. Make a
statement on the accuracy by estimating the deviations in your data as well as by
comparing the graph for the average of N vs n with the graph obtained from formula (2).
QUESTIONS: 1) Looking on your data, find the “time interval” (number of throws)
corresponding to the reduction by a factor of 1/2 of the number of
“atoms”.
2) Does this “interval” depend on the initial number of “atoms”? Give the
answer both by relying on your experimental data and by analysis of
formula (2).
40
CONSERVATION OF LINEAR MOMENTUM
Conservation of linear momentum will be studied as a fundamental law of Nature in a
simplest mechanical setup.
THEORETICAL BACKGROUND: Conservation of linear momentum is another
fundamental law of Nature – as important as the law of energy conservation. It says that,
if there is no net external force on a system of objects, the total momentum of such
system is conserved, that is, it does not change with time.
If there are two objects with masses M1 and M2 moving at the corresponding velocities
V1 and V2 the total momentum of the system P= M1 V1 + M2V2 is a constant,
provided all external forces are zero (or compensated). This law is a consequence of the
3rd Newton’s Law of motion.
In contrast with kinetic energy, momentum is a vector. This means that in the simplest
setup Fig.1 it can be positive or negative depending on the direction of the velocities.
Choosing positive direction to the right, velocity and momentum will be positive if an
object is moving to the right and -- negative otherwise.
EXPERIMENT: The setup consists of two gliders on leveled air track where two major
sources of the external net force – gravity and friction – have been greatly reduced. These
gliders will be allowed to collide with each other and the total momentum will be
measured before, Pb , and after the collision, Pa. According to the law: Pb =Pa, or
Pb =
Pa → M 1V1b + M 2V2b =
M 1V1a + M 2V2 a
(1)
Where the indexes “a,b” refer to the velocities after and before the collision, respectively.
This condition will be tested in the experiment by measuring momenta of each object
and, then, finding the totals.
Fig.1
41
PROCEDURE: For given M 1 and M 2 , the velocities before and after the collision will
be measured by the timers in the regime GATE, provided the collision occurs between
the photogates. The teacher will explain how to find the velocities before and after the
collision.
ANALYSIS OF DATA: Arrange the following data columns: 1) V1b ; 2) V2b ; 3) V1a;
4) V2a (kg); 5) Pb = M1 V1b + M2 V2b; 6) Pa = M1 V1a + M2 V2a. Record the velocities
according to their direction (positive if moving to the right and negative – if to the left).
Perform at least 10-15 measurements with different initial velocities, starting from the
case when one glider is stationary between the photogates. Use elastic and inelastic
collisions. Change gliders masses by adding weights to them. Make sure Pb varies within
sufficient margins so that the graph Pa versus Pb can be constructed.
Construct the graph Pa versus Pb. In the ideal situation this graph should be a straight line
going through the origin at the slope =1. Perform the linear fit and find the actual slope.
Compute the percent difference from the expected value of the slope.
DISCUSSION: Conclude on whether your data supports the Law of Conservation of
Linear Momentum.
QUESTIONS: 1) Can two objects moving at some finite velocities have the total
momentum zero? Explain and give an example.
2) Can two objects moving at some finite velocities have the total kinetic
energy being zero?
3) Two pieces of clay, with M1= 2kg and M2=3kg , move toward each
other at the speed of 12 m/s. Then, they collide and stick together. Find
their mutual velocity after the collision. What is its direction – continuing
in the direction of M1 or M2 before the collision?
42
ELECTRIC FIELD AND ELECTRIC CHARGES
Besides the contact interaction explored previously, we have heretofore
considered only gravity as a force which acts through empty space. We now consider
another very important force which acts through empty space – Electrostatic interactions.
Simple experiments on static electricity are to be performed. The following
instrumentation is used: ebony and glass rods, silk, fur, plastics (comb) etc. The effect of
the deflection of running water by an electrically charged rod (comb) is also to be
demonstrated. It is also to be observed that no static electricity effects can be produced
with massive objects made of metal.
The nature of electric charge and electric field is to be discussed, and the basics of
Coulomb’s law are to be outlined. Two effects should be clearly distinguished:
1) attraction or repulsion between objects carrying net charges;
2) the attraction of a substance carrying no net charge (running water, paper, hairs etc.)
by an electrically charged object due to the polarization effect of a neutral substance.
The structure of atoms is to be presented, and the electric charge of the electron is
to be introduced as the smallest unit of electric charge.
Different types of materials with respect to their ability to conduct electricity are
mentioned. Then, the concept of electric current is to be introduced as the background for
the experiments “ELECTRIC FIELD AND ELECTRIC CURRENT” and
“WORK OF ELECTRIC FIELD”.
43
ELECTROMAGNETIC FIELD
These demonstrations introduce students to electromagnetism as a basis for studying
wave properties of light and atomic spectra.
Magnetism is considered as a third example of an interaction through empty space in
addition to that of gravity and the electric field. Basic experiments with permanent
magnets are to be performed.
Special attention is to be paid to the fact that electric and magnetic effects are well
distinguished from each other. Crucial differences are to be emphasized:
1. The magnet does not affect paper, hairs and running water, while the rubbed comb
does;
2. Rubbing of the iron rod does not result in its activation for the attraction of polarizable
materials (paper, water, hairs); even without rubbing it attracts magnets.
It is to be stressed that despite the apparent differences, there is a common source
for electric and magnetic fields -- electric charges. It is to be pointed out that the electric
current -- a unidirectional motion of charges -- serves as a source for a magnetic field. A
demonstration of compass needle deflected by electric current is to be performed. The
reversal of the compass needle direction when the current is reversed should be pointed
out. Even stronger evidence for this effect will be demonstrated by a heavy iron rod being
pulled into the coil interior, when the current flows through the coil.
The concept of the electromagnetic field is to be introduced through the
demonstration of Faraday’s electromagnetic induction. It should be emphasized that if
the magnet and the coil were stationary, no induction current would have been created.
Only temporal changes of the relative position of the coil and the magnet result in
producing electric current.
Two coils are now to be placed in front of each other, so that turning on or
off the current in one of them induces a current in the other. Finally, the crucial link in the
Maxwell theory of the electromagnetic field is to be discussed -- the direct creation of a
magnetic field by a changing electric field. This effect, together with the Faraday’s
induction, leads to electromagnetic waves. Characteristics of electromagnetic waves - -
44
speed, frequency, period, wavelength, polarization -- are to be introduced and compared
with previously studied sound waves. It is pointed out on the electromagnetic wave
nature of light.
45
COLOR AND WAVELENGTH OF LIGHT
This demonstration serves as a basis for the lab “ATOMIC SPECTRA”
Light is a wave phenomena. In contrast to sound, it can propagate in empty space
-- vacuum. Light is a bundle of electromagnetic waves, each characterized by the same
speed C ≈ 30
. × 108 m / s in vacuum. Each wave is characterized by a wavelength λ and a
frequency f. The relation between all three parameters is exactly the same as for sound
λ=c/f.
Electromagnetic waves are perceived by our eyes, analogously to sound which we
can hear. However, the range of the wavelengths is very different. Visible light has
wavelengths in the range 450-650 nm which are too small to be seen directly – like waves
on the surface of water. In this range, light is percieved as various colors. White light is a
mixture of electromagnetic waves with wavelengths covering the entire visible range.
These colors are those which we see in a rainbow --- from short (violet) to long (red)
waves.
How can a single visible wave be produced? In analogy with sound, we might
look for some sort of “tuning fork” which is capable of giving off a single
electromagnetic wave. However, the frequency of such a wave is huge (about 1015 Hz);
thus no simple mechanical “tuning fork” is actually available. The LASER (Light
Amplification through Stimulated Emission of Radiation) is a device which allows us to
produce nearly perfect visible electromagnetic waves of a required wavelength. The
physics of this device is based on the quantum nature of light. This behavior is subtle,
but the outcome is pretty obvious -- the production of a visible wave characterized by a
single wavelength.
In this demonstration, three lasers, each producing different colors, are to be used.
A diffraction grating -- a device employing the effect of light interference -- will analyze
these colors so as to give the corresponding wavelengths. A key element of the grating
are very tiny parallel lines set on the transparent slide. The distance d between these lines
is a very important parameter. Normally, each grating is labeled as to how many lines per
mm it has, from which the distance d can be determined.
Light striking the grating splits into many rays which then interfere with each
other. As a result, the interference antinodes (bright lines) are formed on the screen. The
position of each antinode depends on the wavelength λ and the parameter d. This
dependence is given by the following equation
d sin(a)=nλ
where a is the angle of diffraction (see the sketch below ) and n=0,1,2,3 stands for the
order of the antinode.
46
(1)
grating
screen
incident light
a
x
S
Fig.1. Diffraction and interference of light on the diffraction grating
Measuring S -- the distance from the grating to the screen and x -- the distance from the
zero order antinode to the n-th order antinode, the sin(a) can be found
as
sin(a ) =
x
x2 + S 2
.
(2)
The wavelength can then be determined as λ=d sin(a)/n from equations (1). Substituting
the sin( a) from equation (2), we find
λ=
dx
n x2 + S 2
.
(3)
Using the equation (3), the wavelengths of all three lasers -- red, yellow and green,-- will
be found.
Here is the actual setup
47
48
FORMATION OF AN OPTICAL IMAGE BY A CONVERGING LENS
Image formation by a converging lens is to be studied.
THEORETICAL BACKGROUND: The effect of the refraction of light can be used to
create optical images. A converging lens focuses a beam of light which is parallel to the
main optical axis of the lens to one of the two lens focal points (see the work “Refraction
of Light”). The distance between each of the focal points and the lens is called focal
length f . The focal length is a specific property of the lens, which is determined by the
lens shape and the index of refraction of the lens material. If an object is placed on one
side of the lens (farther than the focal distance from the lens), a real and inverted image
of this object can be formed on the other side of the lens. To find the position of the
image, the ray tracing diagram can be used
ho
do
f
f
di
h
i
The image distance di obeys the lens formula
1 1 1
+ =
d 0 di f .
The image could be larger or smaller than the object. The ratio
(1)
M=
hi
ho
of the image
size to the object size is called the magnification. Referring to the above diagram, one
finds that the magnification can also be expressed as
M =−
di
d0
.
(2)
Where the sign convention is used: the image height hi is taken negative if the image is
upside down; di is positive for real image (and negative for virtual).
49
EXPERIMENT: A converging lens of known focal length together with an object and a
screen are placed on the optical bench. Measure the object size ho (it remains the same
throughout the experiment). The distance between the lens and the screen should be
adjusted until a sharp image of the object is formed on the screen. Then, the distances d0 ,
di
as well as the image size hi can be measured. From d0 and di, the focal length can
be found from equation (1), and then compared with the known value.
Dividing the image size hi by the object size ho , the magnification M can be
determined, and then compared with the prediction (2). Repeat this procedure for several
d0 .
ANALYSIS OF DATA: Arrange your data in six columns: 1) the object distance
the image distance di ; 3) the magnification
M=
hi
ho
d0 ; 2)
; 4) the magnification as predicted
by formula (2); 5) the sum 1/
do + 1/ di ;(it determines 1/f according to the
equation (1)); 6) deviations of the experimentally measured 1/f.
Plot the magnification |hi | / ho vs. the ratio di / dio . The slope of this graph
should be 1, as given by formula (2). Find the percent difference for this slope.
Determine the reciprocal of the focal length as the mean of the column #5.
DISCUSSION: discuss how well your data supports the lens equations (1) and (2).
Include in your statement the main values you have found. Evaluate the percent
differences for the slope and for the focal length. Estimate the error with which you have
found f , and compare the error with the percent difference.
QUESTIONS: 1) How close to each other is the value of the focal length you measured to
the known value?
2) If di is twice d0, what are these in terms of a)
f ; b) in cm for your particular lens?
3) If di is three times d0 , what is the magnification?
4) If the object is placed between the lens and its focal point, what kind of
image is the lens producing? Can it be captured on the screen?
50
Appendix: Mathematical Review (selected topics)
Scientific Notations
Instead of writing long numbers, it is convenient to use scientific notation. In this
notation
10 = 101 , 100 = 10 2 , 1000 = 10 3 , 1000000 = 10 6 , 1...( n zeros) = 10 n .
(1)
Very small numbers have negative powers:
01
. = 10 −1 , 0.01 = 10 −2 , 0.001 = 10 −3 , 0.000001 = 10 −6 , etc.
(2)
For example, we wish to round off the number 1574394908437269 so that only the first
three digits are displayed. We can do this by writing 1570000000000000 (the 13 digits
after the first three were replaced by zeros). This number is very inconvenient to handle.
Using scientific notation, 157...(13 zeros)=157 × 1..(13 zeros)= 157 × 1013 = 157
. × 1015 .
In the last part the identity 157 = 157
. × 10 2 and the rule 10 2 × 1013 = 1015 have been
used.
Now suppose that a very small number 0.000000023157 is to be rounded off so
that only the first three digits are to be displayed:
0.000000023157 ≈ 0.0000000232 = 2.32 × 10 −8 .
(3)
Note that the decimal point has been moved to the right by 8 places, and the power “-8”
of 10 was introduced in accordance with the rule (2).
The following describes how one multiplies and divides numbers in scientific
notation. The general rules are
10 a × 10b = 10 a +b , 10 a / 10b = 10 a −b
51
(4)
For example, let us find the ratio of the very large number 1574394908437269 to the very
small number 0.000000023157. Normally, the first three digits are sufficient. So these
numbers should be rounded off, as we did above, and represented in scientific notation:
1574394908437269
=
0.000000023157
=
157
1015
. × 1015 157
.
=
×
=
2.32 × 10 −8 2.32 10 −8
(5)
= 0.677 × 10
15− ( −8 )
= 0.677 × 10
23
Metric Units:
Length -- meter (m) is the basic unit.
1 centimeter (cm) = 1/100 m= 10 −2 m; 1 millimeter (mm) =1/1000 m= 10 −3 m
1 nanometer (nm) = 10 −9 m, 10 9 nm = 1m, 1km = 10 3 m .
Mass -- kilogram (kg) is the basic unit.
1gram (g) = 1/1000 kg = 10 −3 kg
Conversion of Units:
The following examples illustrate the procedure.
23 m = 23 × 1m=23 × 100 cm=2300 cm; 530
×10 −9 m = 530 × 10 −9 × 1m = 530 × 10 −9 × 10 9 nm = 530nm.
The following examples illustrate the conversion of complex units.To express a volume
of 25 m 3 in cubic centimeters, the relation between the meter and the centimeter is used:
25m3 = 25 × (1m) 3 = 25 × (100cm) 3 = 25 × (10 2 ) 3 cm3 = 25 × 10 6 cm 3
Here the general rule
(10 a ) b = 10 a×b
has been used.
As another example, let us express the density of oil 800 kg/ m 3 in grams per cubic
centimeter. We proceed
52
(6)
10 3 g
800kg / m = 800 × 6 3 = 800 × 10 3−6 g / cm3 =
10 cm
3
−3
= 800 × 10 g / cm = 0.8 × 10 3 × 10 −3 g / cm3 = 0.8 × 10 0 g / cm3 = 0.8 g / cm3
3
53
Appendix: Analysis of Data
Every measurement is subject to error. This results in deviations of the measured
quantity. For example, the length of the same pencil measured several times would come
out differently because of the manner in which the ruler was applied. Personal blunders
due to carelessness are also a source of error. Each particular instrument never gives the
result precisely. Many external factors which cannot be completely controlled change the
conditions of measurement and thereby affect the results. Thus, errors of measurement
and the associated deviations of the measured quantity are an inherent part of the
measurement process. Patience and experience are required in order to reduce the errors
and the deviations. What is especially important, is that the
deviations can be significantly diminished by the repetition of observations.
This means that the same quantity should be measured as many times as possible within a
reasonable duration of the experiment.
As an example, result of the measurement of a length of some object is given in
the table below
N
1
L (cm)
DL =|L-
L |(cm)
2
3
4
5
15.2
15.3
14.9
15.4
15.2
0.1
0.2
0.2
0.3
0.1
6
15.1
0.0
7
8
9
15.0
14.8
15.2
0.1
0.3
0.1
The upper row marked by N gives the number of the measurement. The second row
shows the object’s length obtained during each measurement (for example, the result of
the 4th measurement is 15.4 cm). The bottom row gives the deviations (or errors)
DL =| L − L |
of each measurement from the average value (mean value) of the length
L = avg ( L) =(15.2 + 15.3 + 14.9 +15.4 + 15.2 + 15.1 + 15.0 + 14.8 + 15.2 )/ 9 =
=15.1 cm
54
(1)
calculated from the 9 measurements. In calculating the average, the result must be
rounded off so that the number of significant digits is not more than that for each
measurement. The average deviation (mean deviation)
DL = avg ( DL)
(2)
indicates how the measured value varied due to all of the factors mentioned above. For
our example, DL = 0.2 cm. The final result for the object length is expressed as
L = L ± DL .
(3)
. ± 0.2) cm . This means that in the measurement of the length
For our example, L = (151
the result obtained was between 14.9 cm and 15.3 cm with high certainty.
Can we say that these measurements are reliable? In order to arrive at a
conclusion, we have to calculate the percent deviation (or the percent error). This is
DL
⋅100% .
L
(4)
The percent deviation indicates how small in percent the mean deviation DL is with
respect to the mean value L . For our example, the percent deviation is
0.2
⋅ 100% ≈ 1% . For our laboratory, measurements are considered accurate if the
151
.
percent deviation is less than 10%-15%. As long as 1% is less than 10%, we consider
the result of the measurement of the length L accurate. This sort of analysis should be
applied to measurements of other physical quantities.
Sometimes a purpose of the laboratory experiment is to measure some quantity
whose standard (given) value is well known. For example, we have measured the length
L of the object, and the given value of the length of this object is Lst . In this case it is
very important to compare these two quantities L and Lst in order to make a
conclusion on whether your experiment confirmed the value Lst and thereby supported a
55
theoretical concept underlying this value. An important quantity is the percent difference
between the measured (mean) value and the standard value
| Lst − L |
×100% .
Lst
(5)
In fact, if you are asked to compare the measured value with the standard (known) value,
you should calculate the percent difference (5) and make a statement on whether the
percent difference is less or bigger than 10%-15%. If it is less than 10%-15%, the
measured value is considered to be close to the standard one. For example, let us assume
that the standard value Lst for the length L is 15.0 cm. Then, the percent difference
between the measured value 15.1 cm and the standard value 15.0 cm is approximately
1%. Correspondingly, one can conclude that the measured value is close to the standard
value.
We can say that the experiment does confirm the concept within the
experimental percent deviation (or percent error), if the percent difference given by
equation (5) is not bigger than the percent deviation given by equation (4).
Returning to our example, we see that the percent deviation and the percent defference
are both about 1%. Accordingly, we conclude that the result of measurement of L has
confirmed the standard value within 1% of the percent error. Had the percent difference
been bigger than the percent deviation, we would have concluded that the standard value
has not been confirmed.
The deviations should be always estimated for the experimental data.
Furthermore, any experimental result for which no mean deviation is calculated is
considered as unreliable. Therefore, the report on the laboratory work must contain at
least two main results. The first one is the set of physical quantities determined
during this laboratory work. The second is the set of deviations for each quantity.
56