CHAPTER 1
Introduction
Major Points
1.
2.
3.
4.
The meaning of the terms model, principle, and theory.
The SI system of base nnits. The conversion of units.
The use of significant figures to indicate the precision of data.
The use of dimensional analysis to check equations and to obtain
relationships between physical quantities.
5. Reference frames. The Cartesian and polar coordinate systems.
1.1
WHAT IS PHYSICS?
Children have an insatiable curiosity about everything around them. Sights,
sounds, and smells are a constant source of wonder and amazement. They are
eager to learn about nature by looking at plants, birds, and insects, and by trying
all sorts of experiments with straws, bottles, pebbles, water, paint, balls, and of
course food and mud. They also love to take apart a watch or a mechanical toy to
see what is inside and how it works. A scientist is a person who retains some of
this childlike sense of curiosity and wonder about nature.
A scientist tries to make sense of how nature operates and to discover some
underlying order in the vast array of natural phenomena. This can be done at
various levels, each of which reveals a different layer of reality. Social science
deals with the behavior of groups, psychology with individuals, biology with the
structure and function of organisms, chemistry with the combinations of atoms.
Physics deals with the behavior and composition of matter and its interactions
at the most fundamental level. It is concerned with the nature of physical reality,
that is, only with things that can be measured by instruments. Its domain stretches
from inside the tiny nucleus of an atom to the vast expanses of the universe.
Geology, chemistry, engineering, and astronomy all require an understanding of
the principles of physics. Physics also finds many applications in biology, physiology, and medicine.
Between 1600 and 1900, three broad areas were developed in what is called
classical physics:
1. Classical Mechanics: The study of the motion of particles and fluids.
2. Thermodynamics: The study of temperature, heat transfer, and the properties
of aggregations of many particles.
3. Electromagnetism: Electricity, magnetism, electromagnetic waves, and optics.
The study of bodies such as the
Horsehead Nebula requires an
understanding of the principles of
physics.
Classical physics
2
CHAP. 1 INTRODUCTION
These three areas encompass virtually all the physical phenomena with which we
are familiar. However, by 1905 it became apparent that classical ideas failed to
explain several phenomena. Three important theories in modern physics are:
Modern physics
4. Special Relativity: A theory of the behavior of particles moving at high speeds.
It led to a radical revision of our ideas of space, time, and energy.
5. Quantum Mechanics: A theory of the submicroscopic world of the atom. It also
required a profound upheaval in our vision of how nature operates.
6. General Relativity: A theory that relates the force of gravity to the geometrical
properties of space.
The goal of physicists is to explain physical phenomena in the simplest and
most economical terms. For example, we want to discover the "ultimate" building blocks of matter. According to our present state of knowledge, ordinary matter is constructed from atoms, the atoms from nuclei and electrons, the nuclei
from neutrons and protons, the neutrons and protons from quarks. Indeed all
elementary particles (of which there are hundreds) can be constructed from just
two basic types of particle: quarks and leptons.
As another example of the drive for economy, consider the apparently wide
variety of forces we encounter in nature: forces exerted by ropes, springs, fluids,
electric charges, magnets, the earth and the sun, chemical forces, nuclear forces,
and so on. Despite this great variety, physicists can explain all physical phenomena in terms ofjust four basic interactions. Their ranges and relative strengths are
summarized in Table 1.1.
THE BASIC INTERACTIONS
Relative Strength
Interaction
Strong
1
10- 2
Electromagnetic
10- 6
Weak
10- 38
Gravitational
TABLE 1.1
The basic interactions
Range
10- 15 m
Infinite
10- 17 m
Infinite
The gravitational interaction produces an attractive force between all particles. It is responsible for our weight, causes apples to fall, and holds the planets in
their orbits around the sun. The electromagnetic interaction between electric
charges is manifested in chemical reactions, light, radio and TV signals, X rays,
friction, and all the other forces we experience every day. It also governs the
transmission of signals along nerve fibers. The strong interaction between quarks
and most other subnuclear particles holds particles within the nucleus. The weak
interaction between quarks and leptons is associated with radioactivity. In 1983 it
was confirmed that the electromagnetic and weak interactions are different manifestations of a more basic electroweak interaction. Progress has also been made in
attempts to combine the strong and electroweak interactions in a single grand
unified theory. Clearly, the dream of physicists is to discover a single fundamental
interaction from which all forces can be derived.
1.2
CONCEPTS, MODELS, AND THEORIES
In physics we deal with concepts, laws, principles, models, and theories. Let us
briefly consider the meaning of each of these terms.
Concepts
1.2 CONCEPTS, MODELS, AND THEORIES
3
A concept is an idea or a physical quantity that is used to analyze natural phenomena. For example, the abstract idea of space is a concept and so is the measurable
physical quantity, length. In physics we use concepts such as mass, length, time,
acceleration, force, energy, temperature, and electric charge. One can define a
physical quantity by the procedure used to measure it. For example, temperature
can be defined as the reading on a "standard" thermometer, or electric charge by
the force that electrified bodies exert on each other. Our intuitive understanding of
such operational definitions is often enhanced by their relation to human perception. For example, the concept of temperature is based on the sensations of hot
and cold, a force is a push or a pull, and so on. However, some concepts, such as
energy, are difficult to define precisely in words. A concept such as electric charge
is completely mysterious. One can measure charge and say what it does, but one
cannot say what it is.
Laws and Principles
A physicist tries to establish mathematical relationships, called laws, between
physical quantities through experimentation or theoretical analysis. Mathematics
is the natural language of physics because it allows us to state the relationships
concisely. Once a mathematical statement has been made, it can be manipulated
according to the rules of mathematics. If the initial equations in an analysis are
correct, then mathematical logic can lead to new insights and new laws.
Whereas a law may be restricted to a limited area of physics, a principle is a
very general statement about how nature operates. It spans the whole subject and
is part of its foundation. Consider a boat floating steadily down a river. The
principle of relativity states that the laws of physics deduced by people on the boat
must be the same as those discovered by people on land. It makes no reference to
specific laws, but does force us to ensure that the laws we formulate do not violate
this principle. As we will see, this seemingly innocuous statement has profound
consequences. Sometimes the terms law and principle are used interchangeably.
For example, we often refer to the law of conservation of energy, when really it
should be the principle of conservation of energy. Such subtle differences in
terminology are unimportant.
A law is a mathematical
relationship
A principle encompasses many
areas
Models
A model is a convenient analog or representation of a physical system. The
phenomena occurring in the system are analyzed as if the system were designed
according to the model. A model may merely replace the real thing to simplify the
analysis. For example, in some problems one might treat the earth and the moon
as ifthey were point particles. A model is often a mental picture of the structure or
workings of a system. Here are some examples: Light has been modeled as a
stream of discrete particles and also as a continuous wave; heat and electric
charge have been treated like fluids; matter was considered to be composed oftiny
indivisible atoms long before there was evidence for these entities; more recently
the atom itself was pictured as a tiny planetary system. Even great theoretical
physicists have used mechanical models as a way of relating abstract ideas to
something concrete and familiar. Once the theory is complete, the model may be
revealed to others, or quietly forgotten.
There are also purely mathematical models whose mathematical properties
reflect those of the real system. In some cases we begin to suspect that there is
This orrery is a mechanical model
of part of our solar system.
4
CHAP. 1 INTRODUCTION
A model can be useful even if it is
incomplete or incorrect
more to the model than pure mathematics. That is, perhaps the mathematical
entities represent actual physical quantities. Quarks, for example, were first proposed as part of a mathematical model of elementary particles. There is now so
much supportive evidence that we regard them as "real." But the reality of
quarks cannot be guaranteed since we cannot actually look inside a nucleus. It is a
quark model that successfully accounts for a range of phenomena.
A model can be useful as a stepping stone even if it is incomplete or later
shown to be incorrect. For example, in the model of the hydrogen atom proposed
in 1913 by Niels Bohr, an electron orbits a proton,just like a planet orbits the sun.
Although we now know that this is an unrealistic picture, Bohr used it to explain
features of the optical spectrum of hydrogen and other atoms. It was later refined
by the introduction of new concepts and used to explain the basis of the periodic
table. Its shortcomings gradually became apparent and it was superseded by
quantum mechanics around 1925. Although the models that considered heat and
electric charge to be fluids are incorrect, they nonetheless led scientists to establish important results. Unfortunately, concrete models are not always available.
The theory of quantum mechanics accounts for the strange behavior of atoms and
subatomic particles, but nothing in our everyday experience comes even close to
mimicking an atomic system.
Theories
Theories are always tentative
A theory uses a combination of principles, a model, and initial assumptions (called
postulates) to deduce specific consequences or laws. By organizing data from
different areas, or by tying together concepts mathematically, a theory reveals an
underlying unity in diverse phenomena. For example, Newton's theory of gravitation explained why an apple falls to the earth, the motion of the planets about the
sun, why the tides occur, and even the shape of the earth. His theory showed that
the same laws of physics apply to objects on earth as to the celestial bodies.
A physical theory must make precise numerical predictions, and its validity
rests ultimately on the experimental verification of these predictions. A theory is
considered to be plausible and accepted only if it has passed every experimental
test. Yet even if no disagreement has ever been found, one cannot be sure that a
theory has been proved "absolutely" correct. For over 200 years, classical mechanics was perfectly adequate for dealing with the motion of particles. Then, in
1905, the special theory of relativity showed that it is not correct when particles
move at very high speeds. Newton's law of gravitation explains the motion of the
planets perfectly well. However, the general theory of relativity is a more profound explanation of gravitation. Therefore, one should keep in mind that theories
are always tentative. Within their limits, classical mechanics and Newton's law of
gravitation are still extremely useful. In fact, they are precise enough for us to use
them to send a probe to another planet.
Contrary to common belief, theories do not follow inexorably from experimental observations. Consider the following observation: A rolling ball comes to a
stop. The Greek philosopher Aristotle (ca. 340 B.C.) noted that since the ball
eventually stops, it must need something to keep it going. The Italian physicist
Galileo (ca. A.D. 1600) was struck by the fact that the ball keeps going for so long.
He believed that if one could eliminate friction, it would go on forever. The same
"fact" is interpreted in ways that are diametrically opposed. Both views are
justifiable, but the second one marks the beginning of physics. Consider another
example: In the geocentric model of the universe, the sun, the stars, and the
planets revolve around a stationary earth. In the heliocentric model, the earth and
the other planets orbit the sun. The heliocentric model, which we now accept, was
1.3 UNITS
5
not inferred directly from the astronomical data, because the earth and the other
planets certainly do not appear to go around the sun.
Thus, although experiments serve to stimulate the creation of new theories
and also to test them, the "facts" alone do not lead to theory. The formulation ofa The formulation of a theory
theory requires a creative mind that can see beyond the facts to make intuitive requires both observation and
leaps and inspired guesses. Although science is a rational way of looking at the imagination
world, the creation of theories is not a rational procedure. This is the only way
that one can transcend the confines of existing knowledge. It may involve an
unexpected flash of insight whose origin even the scientist cannot explain. The
formulation of physical theories is often guided by such esthetic notions as
beauty, simplicity, and mathematical elegance. If two theories have the same
range and predictive power, the simpler, or more elegant one, is usually preferred.
Strictly speaking, a theory can only describe natural phenomena, not explain
them. But when a theory begins with a small number of assumptions and then
accounts for a wide range of phenomena, it is natural to say that it has explained
them. And indeed it has, but only in terms of the basic postulates and concepts.
Suppose one begins with Coulomb's law for the force between two charges and
derives some results that are confirmed experimentally. One could still ask why
charges attract or what charge is. One can explain only how the charges interact,
not why. A theory accounts for phenomena in terms of ultimately inexplicable
quantities such as mass and charge.
1.3
UNITS
The value of any physical quantity must be expressed in terms of some standard
or unit. For example, we might specify the distance between two posts in meters
or in feet. Such units are necessary for us to compare measurements and also to
distinguish between different physical quantities. All physical quantities can be
expressed in terms of three fundamental quantities: mass, length, and time. In the
Systeme International (Sf) the base units for mass, length, and time are the kilogram (kg), the meter (m), and the second (s). It is convenient to define additional
base units: the kelvin (K) for temperature, the ampere (A) for electric current, and
the candela (cd) for luminous intensity. A base unit must have a precise and
reproducible standard. For the moment we consider only mass, length, and time.
SI base units
Mass
The SI unit of mass, the kilogram (kg), was originally defined as the mass of one
liter of water at 4 0c. Practical difficulties in obtaining pure water and the fact that
this definition involved another quantity, namely temperature, led to its replacement. The SI unit of mass (1 kg) is now defined to be the mass of a platinumiridium cylinder kept in the International Bureau of Weights and Measures in
Sevres, France (see Fig. 1.1). With this standard one can measure mass to a
precision of 1 in 108 • At the atomic level it is convenient to have a secondary unit
of mass called the unified atomic mass unit (u). The mass of an atom of carbon- 12
is defined to be exactly 12 u. The relation between these units is 1 u = 1.66 X
10- 27 kg.
Time
The SI unit of time is the second (s). This was originally defined as 1184,600 of a
mean solar day. (The interval between the times at which the sun reaches the
highest point in the sky on successive days is called a solar day. Because of
FIGURE 1.1 The standard kilogram is
a platinum-iridium cylinder.
6
CHAP. 1 INTRODUCTION
seasonal variations and random fluctuations, the mean value over a year is taken.)
Because the rate of rotation of the earth has been gradually decreasing, the mean
solar day was chosen to be the value in 1900. This is hardly a reproducible
standard! In 1967, the second was redefined in terms of certain radiation emitted
by atoms of cesium-133. Specifically, in one second there are 9,162,631,770 vibrations in the radiation. The cesium atomic clock, shown in Fig. 1.2, is so stable that
it is accurate to within 1 s in 30,000 years. Secondary units of time include the
hour, the day, the year, and the century.
Length
FIGURE 1.2 A cesium atomic clock at
the National Bureau of Standards.
The value of the speed of light is
defined
The SI unit of length is the meter (m). The meter was originally defined (in the
eighteenth century) to be one ten-millionth (10- 7 ) of the distance from the equator
to the North Pole. In this century, but prior to 1960, it was defined as the distance
between two fine scratches on a platinum-iridium bar stored under controlled
conditions in Sevres, France. The use of the standard bar had two drawbacks.
First, although copies of the bar are available in major industrialized countries
(see Fig. 1.3), it is preferable to have a standard that can be produced in any wellequipped laboratory. Second, the width of the scratches became a limiting factor.
Thus, in 1960 the standard meter was measured as precisely as possible in terms of
the number of wavelengths of the orange light emitted by krypton-86. The meter
was then defined as 1,650,763.73 wavelengths of this light. As techniques improved (through the development of lasers), the precision with which the krypton
wavelength could be specified itself became a limitation. In 1983 the meter was
redefined as the distance traveled by light in a vacuum in 1/299,792,458 second.
This length standard, which depends on the definition of the second, effectively
defines the speed of light in vacuum to be exactly 299,792,458 m/s. The speed of
light has become a primary standard, and any improvement in measuring either
the meter or the second is automatically reflected in the other.
In the British system, still used in the United States, the base units are the
pound (lb) for force, the foot (ft) for length, and the second for time. Virtually all
scientific data are now expressed in SI units.
Derived Units
The units of physical quantities other than mass, length, and time are combinations of the base units and are called derived units. For example, the unit of speed
is mis, for acceleration it is m/s 2 , and for density (mass per unit volume) it is kg/m 3 •
Sometimes a derived unit is given a special name to honor someone. For example,
Newton's second law relates the acceleration a of a body of mass rn to the force F
acting on it: F = rna. The unit of force is kg·m/s 2• This combination is called the
newton (N).
Conversion of Units
It is often necessary to convert the unit of a physical quantity. Suppose we wish to
convert miles per hour (mi/h) to meters per second (m/s) given that 1 mi = 1.6 km.
The ratio (1.6 km)/(l mi), which has the value one, is called a conversion factor.
By proper use of such ratios, one can eliminate the unwanted unit and obtain a
new one. For example,
FIGURE 1.3 Prior to 1960 the meter
was defined as the distance between
two scratches on a platinum-iridium
bar.
5.0 mi
h
=
(5.0 mi) (1.6
1h
1 ml
3
(10 m)
1 km 3600 s
=
2.2 m
s
.........
When you substitute into any equation, do not mix SI units and British units.
1.5 ORDER OF MAGNITUDE
1.4
POWER OF TEN NOTATION AND SIGNIFICANT FIGURES
Suppose you were asked to compare the size of an atom specified as
0.000,000,000,2 m with that of a nucleus given as 0.000,000,000,000,005 m. In this
form, these numbers are difficult to handle. Very large or very small numbers
should be expressed in power of ten notation. The size of the atom is 2 x 10- 10 m
and that of the nucleus is 5 x 10- 15 m; thus the ratio of the sizes is
10
2 x 10- m
5 x 10 15 m
=
5 x
105
=
4
x
104
It is often convenient to denote the power of ten by a prefix to the unit. For
example kilo means thousand, so 2.36 kN = 2.36 x 103 N; milli means one
thousandth, so 6.4 ms = 6.4 x 10- 3 s. Other prefixes are listed inside the front
cover.
Numerical values obtained from measurement always have some uncertainty.
For example, the result of a measurement may be 15.6 m with an uncertainty of
2%. Since 2% of 15.6 is approximately 0.3, the result is 15.6 ± 0.3 m. The true
value is likely to lie between 15.3 m and 15.9 m. Instead of an explicit statement of
uncertainty, the precision of a result is often indicated by the number of digits
retained. We say that 15.6 m has three significant figures, with the understanding
that the last figure (6) may not be certain. The result 15.624 has five significant
figures, with the 4 uncertain. Zeros that serve only to indicate the power often are
not counted, but those at the end are. For example 0.002560 has four significant
figures. The number of significant figures in 12,000 is not clear, whereas 12,000.0
definitely has six significant figures. Power of ten notation is useful in such cases.
Thus, 1.2 x 104 has two significant figures, whereas 1.200 x 104 has four significant figures.
To ensure that the results of computations are not stated to unwarranted
precision, the following simple rule of thumb should be used: In products and
divisions, the number of significant figures in the final result should equal that of
the factor with the least number of significant figures. Thus, for example,
2.6
=
(6.387)
=
6.4
Although extra figures may be retained in the intermediate steps, we state the final
answer only to the two significant figures in 2.6. In additions and subtractions,
only the least number of decimal places should be retained. Thus, 17.524 + 2.4 3.56 = (16.364) = 16.4. Unless otherwise indicated in the text, you may assume
that all given values are precise enough for the final answer to be stated to three
significant figures. Thus 5 m may be taken to mean 5.00 m.
EXERCISE 1. Express the following in power of ten notation: (a) 1500 x 400; (b)
24,000/(0.006).
EXERCISE 2. Evaluate the following: (a) the volume V of a circular cylinder of
radius r = 1.26 em and height f = 7.3 em, where V = 7Tr 2f; (b) the sum 0.056 x 102
+ 11.8 X 10- 1 •
1.5
Power of ten notation
ORDER OF MAGNITUDE
We often hear people say there are' 'trillions of stars in the universe" or perhaps a
"mountain weighs a billion tons." The words "trillion" and "billion" are really
substitutes for "many, many, . . . ," without any intuitive feel for the reasonableness of the statements. Although such numbers are beyond our imaginations,
it is often possible to arrive at a rough estimate of the size of some quantity.
Significant figures indicate the
precision of data
7
8
CHAP. 1 INTRODUCTION
To do this, a scientist thinks in terms of orders of magnitude. This means that
he or she wants to "guesstimate" the size of something only to within a factor of
10. Obtaining an order of magnitude estimate for some complex phenomenon
often involves insight and experience concerning what is important and what is
irrelevant. It is ironic that in this "exact" science of physics, a physicist is often
respected most for the ability to give quick order of magnitude estimates, that is,
to be quite inexact. This ability allows him or her to cut through all the verbiage of
some presentation, and to decide with a "back of the envelope" calculation
whether a theory is reasonable.
To obtain an order of magnitude estimate, the input data need have just one
significant figure. For example,
An order of magnitude estimate
193.7 x 39.64
8.71
= (2
x 10 2)(4 x 10 1)
9
=
1
x
103
For certain purposes this would be close enough to the correct answer, which is
about 881.
A scientist or engineer may wish to make a measurement of some physical
quantity or to build an instrument. By making an order of magnitude calculation
based on the sensitivity of the instruments, the properties of the materials available, the size of the phenomenon itself, and so on, one can judge the feasibility of
the project. Here is an example.
EXAMPLE 1.1: An engineer is designing a pacemaker for cardiac patients. For a 20-year-old woman, how many times
should the device have to beat for her to have a normal life
expectancy?
(c) How many seconds in a year?
Solution: We require several estimates.
(Do the exact calculation and compare.) The total number of
beats is
(a) If she lives to 75 years, the device must last at least 60
years.
(b) How many times per second must the device beat? Our
normal pulse is about 76 beats per minute; so let's say I
beat per second.
(365 d/y)(24 h/d)(3600 s/h) = (400 d/y)(20 h/d)(4000 s/h)
= 3
x 107 sly
(I beat/s)(60 y)(3 x J07 sly) = 2 x J09 beats
It would be wise to include a safety factor of, say, 2. Therefore,
the cardiac pacemaker should operate for 4 x 109 beats before
breaking down.
You should cultivate the habit of knowing the order of magnitude value of
frequently encountered physical quantities, such as the size of an atom or a
nucleus, the mass and charge of an electron, the speed of light, the mass and
radius of the earth, the distance to the sun, and so on. This will help you develop
your insight, and will also prevent ludicrous answers. Quite often, after a calculation that involves a small mistake, a student will state that the deflection of an
electron in a TV tube is 10+ 12 m. A moment's reflection would reveal that this is
greater than the earth-sun distance!
1.6
DIMENSIONAL ANALYSIS
Each derived unit in mechanics can be reduced to factors of the base units mass,
length, and time. If one ignores the unit system, that is, whether it is SI or British,
then the factors are called dimensions. When referring to the dimensions of a
1.7 REFERENCE FRAMES AND COORDINATE SYSTEMS
quantity x, we place it in square brackets: [x]. For example, an area A is the
product of two lengths so its dimensions are [A] = U. The dimensions of speed
are [v] = LT-1, of force [F] = MLT-z, and so on.
An equation such as A = B + C has meaning only if the dimensions of all the
three quantities are identical. It makes no sense to add a distance to a speed. The
equation must be dimensionally consistent. Consider the equation s = 1atZ, in
which s is the distance moved in time t by a particle that starts from rest with
acceleration a. We have [s] = L, whereas [at Z] = (LT-Z)(TZ) = L. Both sides have
the dimension L, so the equation is dimensionally consistent.
When an algebraic expression has been derived, a check for dimensional
consistency should always be performed. This does not guarantee that the equation is correct, but at least it will eliminate any equation that is not dimensionally
consistent. Dimensional analysis can be used to obtain the functional form of
relations, as the following example illustrates.
9
An equation must be dimensionally
consIstent
EXERCISE 3. If P and Q have different dimensions, which of the following operations are possible: (a) P + Qi (b) PQ; (c) P - VQ; (d) I - PIQ?
EXAMPLE 1.2: The period P of a simple pendulum is the time
for one complete swing. How does P depend on the mass m of
the bob, the length e of the string, and the acceleration due to
gravity g?
Solution: We begin by expressing the period P in terms of the
other quantities as follows:
P = k m X ey gZ
where k is a constant and x, y, and z are to be determined. Next
we insert the dimensions of each quantity:
T = MXLYUT-2Z
= MXLy+zT-2z
and equate the powers of each dimension on either side of the
equation. Thus,
T:
M:
L:
1.7
1 = -2z
0 = x
These equations are easily solved and yield x = 0, z =
y = +1. Thus,
P
=k
-!, and
Ii
This analysis will not yield the value of k, but we have found,
perhaps surprisingly, that the period does not depend on the
mass. A derivation in terms of the forces acting on the bob
shows that k = 27T.
The results of such dimensional analysis depend on insight
into what the important parameters are. It would seem, at least
at first sight, that the angle through which the pendulum swings
should also be included. Since an angle is the ratio of two
lengths, it is a dimensionless quantity, so its effect would not
show up anyway. A careful derivation shows that the period
does depend to some extent on the angle of swing, but the
above expression is quite adequate for small angles.
O=y+z
REFERENCE FRAMES AND COORDINATE SYSTEMS
The position of a body has meaning only in relation to a frame of reference, which
is something physical, such as a tabletop, a room, a ship, or the earth itself. The
position is specified with respect to a coordinate system that consists of a set of
axes, each of which specifies a direction in space. In the Cartesian coordinate
system, the axes are labeled x, y, and z. They are mutually perpendicular and
intersect at the origin. In two dimensions, a point P may be located by its Cartesian coordinates (x, y), as shown in Fig. 1.4. A scale is marked on each axis.
Starting at 0, x and yare the number of units (+ or -) one must move in the
direction of each axis to reach P.
10
CHAP. 1 INTRODUCTION
p
In plane polar coordinates, the length of the line OP and the angle 8 with
respect to the reference direction (+ x axis) are given. These two types of coordinates are related as follows:
x = r cos 8
(l.l)
r sin 8
(1.2)
y
=
r
y
where
(1.3)
I Lt:..-..l...L-..L---'---------L--!.--'----'-_.T
o
------:x,....---
and
FIGURE 1.4 The Cartesian
coordinates of point P are (x, y). The
polar coordinates are (r, 0).
tan 8
=
x
(1.4)
Note that 8 is measured counterclockwise from the + x axis. We will not need to
use any other type of coordinate system in this book.
SUMMARY
The value of a physical quantity is expressed in terms of a unit of measurement.
The SI base unit of mass is the kilogram (kg), the unit of length is the meter (m),
and the unit of time is the second (s). Other physical quantities may be expressed
in terms of derived units, which are combinations of the base units. To convert
from one derived unit to another, for example, kmlh to mis, we use conversion
factors whose value is one, for example, (3600 sll h) or (I mi/l.609 km).
Numerical values obtained from experiment always have some uncertainty.
The uncertainty may be stated explicitly as a percentage or a range, as in 17 ±
2 m. Or, it may be implicit in the number of significant figures retained. In multiplication and division, the number of significant figures in the answer is that of the
factor with the least number of significant figures. In addition and subtraction, the
number of decimal places in the answer is that of the value with the least number
of decimal places.
In an order of magnitude calculation, one is concerned with estimating the
value of some quantity to within a factor of ten. The data that are used to obtain
the estimate need have only one significant figure.
A derived unit may be reduced to factors of the base units. The dimensions of
a physical quantity indicate the powers to which each base unit is raised. Any
equation must be dimensionally consistent.
ANSWERS TO IN·CHAPTER EXERCISES
1. (a) 6 x 105 ; (b) 4 x 106
2. (a) V = (3.142)(1.2W(7.3) = (36.4) = 36 cm 3 (two significant
figures); (b) 5.6 + 1.18 = (6.78) = 6.8 (one decimal place).
3. (a) No; (b) Yes; (c) Yes, but only if P and VQ have the same
dimensions; (d) No, PIQ must be dimensionless.
QUESTIONS
1. What features are desirable in the choice of a standard of
measurement?
2. What happens if someone steals the standard kilogram in
Sevres, France? Is this a safe standard? Can you suggest an
alternative mass standard?
3. The United States is among only about haifa dozen nations
that have not switched to the metric system. Are there any
advantages to the British system based on pounds, feet,
and seconds?
4. (a) What are the drawbacks in using a pendulum as a time
EXERCISES
standard? (b) List some natural phenomena that would be
better choices as time standards.
5. What are the problems in using a bar as a standard of
length?
6. Atomic clocks indicate that the length of a day fluctuates.
How do we know that it is not the rate of the clocks that is
fluctuating?
7. What is your height in meters?
11
8. It would be simpler to define the speed of light to be exactly
3 x 108 mls instead of 2.99792458 x 108 m/s. Why is this not
done?
9. Mass has been defined as the "quantity of matter" in a
body. Could you use this to set up a base unit? If so, how?
10. What is the difference between a reference frame and a
coordinate system? Give examples of each.
EXERCISES
1.3 Units
1. (I) Express the U.S. speed limit of 55 miles per hour in (a)
ft/s; (b) m/s.
2. (I) A furlong is 220 yards and a fortnight is 14 days. A
person walks at 5 mph. Express this in furlongs per fortnight.
3. (I) The density of water is about 1 g/cm 3 • What is this in SI
units?
4. (I) How many seconds are there in a year, which is 365.25
days?
5. (I) (a) The distance traveled by light in a year is called a
light-year. Given that the speed of light is 3.00 x 108 mis,
express the light-year in kilometers. (b) The average distance between the earth and the sun is called an astronomical unit (AU) and its value is about 1.5 x 1011 m. What is
the speed of light in AU/h?
6. (I) (a) Express the mass of a proton, 1.6726 x 10- 27 kg, in
terms of the unified mass unit (u). (b) The mass ofa neutron
is 1.00867 u. What is this in kilograms?
7. (I) A knot is a nautical unit of speed: 1 knot = 1.15 mph.
What is a knot in m/s?
8. (I) Given that 1 in. = 2.54 em exactly, express the speed of
light, 3.00 x 108 mis, in (a) ft/ns; (b) mils.
9. (I) A small car has a 2.2-L engine. Convert this to cubic
inches.
10. (II) The fuel consumption of cars is specified in Canada in
terms of liters per 100 km. Convert 30 miles per gallon to
this unit. Note that 1 gallon (U.S.) = 3.79 L.
1.4 Power of Ten Notation, Significant Figures
11. (I) Specify the number of significant figures in each of the
following values: (a) 23.001 s; (b) 0.500 x IQ2 m; (c)
0.002030 kg; (d) 2700 kg/s.
12. (I) Express the following values without prefixes to the
units: (a) 6.5 ns; (b) 12.8
(c) 20,000 MW; (d) 0.3 rnA; (e)
1.5 pA.
13. (I) Given that 7T = 3.14159, find: (a) the area of a circle of
radius 4.20 m; (b) the surface area of a sphere ofradius 0.46
m; (c) the volume of a sphere of radius 2.318 m.
14. (I) Express the following numbers in power often notation:
(a) 1.002/4.0; (b) (8.00 x 106 )-1/3; (c) 0.00076300.
15. (I) Evaluate [(3.00 x 10 12 )(1.20 x 10- 2°)/(4.00 x 10- 1)]-112.
16. (I) Evaluate (a) 1.075 x IQ2 - 6.37 x 10 + 4.18; (b) 402.1 +
1.073.
17. (I) Convert the following to scientific notation: (a) the distance to the sun, 149,500,000,000 m; (b) the wavelength of
yellow sodium light, 0.000,000,5893 m; (c) the radius of an
atom 0.000,000,000,2 m; (d) the radius of a nucleus
0.000,000,000,000,004 m.
18. (I) Evaluate (a) 15.827 - (2.30 x 10- 4)/(1.70 x 10- 3); (b)
88.894/11.0 + 2.222 x 8.00.
19. (I) One inch is defined to be 2.54 em exactly. Convert (a)
100.00 yd to meters; (b) one acre (4840 yd 2) to hectares
(W m).
20. (I) Express the precision of the following results by using just the appropriate number of significant figures:
(a) 6237 ± 42 m; (b) 27.34 ± 0.09 s; (c) 600 ± 0.003 kg.
21. (II) If the radius of a sphere is 10 ± 0.2 em, what is the
percentage uncertainty in (a) its radius; (b) its surface area;
and (c) its volume? (d) Do you perceive a pattern? If so,
what is it? (Hint: For (b) and (c) first find the minimum and
maximum possible values.)
22. (II) The dimensions of a board are measured to be 17.6 ±
0.2 em by 13.8 ± 0.1 em. What is its area?
1.5 Order of Magnitude
23. (I) (a) What is the surface area of the earth? (b) What is the
volume of the earth? (c) How many times larger is the
volume of the sun compared to that of the earth?
24. (I) How many hairs does a normal person have on the
head?
25. (I) A watch is advertised as being 99% accurate. Would you
buy it?
26. (I) How fast is a person at the equator moving relative to
the North Pole?
27. (I) How many frames are there in a 2-h feature film? (What
information do you need?)
28. (I) Use a meter stick to estimate the thickness of a sheet of
paper in this book.
29. (I) In an average lifetime: (a) How many kilometers does a
12
30.
31.
32.
33.
CHAP. 1 INTRODUCTION
person living in a city walk? (b) How many kilograms of
food are consumed by each person?
(I) How much more light does the 200-in.-diameter Mount
Palomar telescope collect compared to the pupil of your
eye?
(I) How many liters of water would be needed to raise the
level of Lake Superior by I cm?
(I) How many grains of uncooked rice are there in one cup?
(I) What is the volume of your body? How could you
roughly check your estimate?
1.6 Dimensional Analysis, 1.7 Reference Frames
34. (I) According to Newton's second law of motion, the force
F acting on a particle is related to its mass m and acceleration a according to F = ma. According to Newton's law of
gravitation there is an attractive force between particles
given by F = Gmlm2lr2, where r is the distance between
them. What are the dimensions of G?
35. (I) Check the following equations for dimensional consistency where v is speed (m/s), a is acceleration (m/s 2), and x
is position (m): (a) x = v 2/(2a); (b) x = tat; (c) t = (2xla) 1/2.
36. (I) The speed of a particle varies in time according to v =
At - Bt3 • What are the dimensions of A and B?
37. (I) Convert the following polar coordinates (r, 8) to Cartesian coordinates: (a) (3.50 m, 40°); (b) (1.80 m, 230°); (c)
(2.20 m, 145°); (d) (2.60 m, 320°).
38. (I) Convert the following Cartesian coordinates to plane
polar coordinates: (a) (3 m, 4 m); (b) (-2 m, 3 m); (c) (2.5 m,
-1.5 m); (d) (-2 m, -1 m).
39. (II) The argument of a trigonometric function must be a
dimensionless quantity. If the speed v of a particle of mass
m as a function of time t is given by v = wA sin[(klm) 1/2 t],
find the dimensions of wand k, given that A is a length.
ADDITIONAL EXERCISES
40. (I) A sphere has a volume of 3.2 L. What is its radius?
41. (I) An unopened cylindrical can has a radius of 3 cm and a
volume of 0.41 L. Find: (a) its height; (b) its surface area.
42. (I) A year is approximately 7r x 107 S. What is the percentage error in this value?
43. (I) A car can travel 11 km per liter of gasoline. Express this
in miles per (US) gallon.
44. (I) Suppose people were to stand along the equator with the
tips of their outstretched arms just touching. Roughly how
many people would be needed to encircle the earth (treated
as a smooth solid sphere)?
45. (I) Evaluate the following to the proper number of significant figures:
(al 3.88 x 10 1 + 4.57 x W; (b) 2.57r/(2.983 x 10- 4).
46. (I) A can of paint that covers 20 m2 costs $14.80. The walls
of a room 12 ft x 18 ft are 8 ft high. What is the cost of paint
for the walls?
47. (I) A 4-L can of paint covers 20 m2 of wall. What is the
thickness of the paint?
48. (I) According to Einstein's famous equation, E = mc 2 , the
product of a mass m and the square of the speed of light c is
equivalent to the energy E. (a) What is the unit of E? (b)
What is the energy equivalent to 1 g?
49. (II) Astronomers use a unit of distance called the parsec. It
is defined in terms of the astronomical unit (AU), which is
the mean radius of the earth's orbit around the sun: 1 AU =
1.49 X 1011 m. At a distance of one parsec, one AU subtends an angle of 1 second of arc (1" = 1°/3600); see Fig.
1.5. How many AUs are there in one parsec?
I'
1 AU
FIGURE 1.5 Exercise 49.
50. (I) Given that 1 inch = 2.54 cm exactly, what is the area (in
cm 2) of a rectangle with dimensions 4.1734 in. x 2.3846 in.?
51. (I) Convert the SI value of the acceleration due to gravity,
g = 9.80665 m/s 2, to the British unit ft/s 2.
52. (I) A carpet costs $17.60 per square yard. What is the cost
per square meter?
PROBLEMS
1. (I) What thickness of rubber is worn off a car tire in each
revolution? Given that the size of an atom is about 10- 10 m,
now many atoms does this correspond to?
2. (I) A particle moving in a circle of radius r at constant speed
v undergoes an acceleration a. Use dimensional analysis to
express the acceleration in terms of v and r.
3. (I) A block of mass m vibrates at the end of a spring. The
spring is characterized by a quantity k called the spring constant that is measured in N/m. Express the period P of the
vibration in terms of m and k. (1 N = 1 kg·m/s 2)
4. (I) Express the position x reached by a particle in terms of its
acceleration a and the elapsed time t in the form x = k amt".
Find m and n through dimensional analysis.
HISTORICAL NOTE
13
HISTORICAL NOTE: The Geocentric Theory Versus The Heliocentric Theory
The origins of physics as we know it today can be traced to the
confrontation between two views of the earth's place in the
universe. In the geocentric view, the earth is at the center of the
universe, with the sun, the planets, and the stars revolving
around it. In the heliocentric view, the earth and the planets
orbit the sun. The ingenious arguments produced by advocates on both sides of this great debate served to sharpen our
understanding of nature and its mechanisms
The Greek philosopher Plato (ca 400 B.C) advocated the
geocentric view. He believed that the celestial bodies (the
stars and planets) were "perfect." To him, this meant that their
natural motion had to be uniform (steady) motion in a circle.
However, planets sometimes appear to reverse their motion
temporarily, Clearly, a single uniform circular motion cannot
explain such retrograde motion Plato also believed that our
senses do not perceive the "real" world, and so truth should be
attained through reasoning alone. He posed the following
question: What combinations of uniform circular motions are
needed to reproduce the observed planetary paths? Or, as he
put it, "to save the appearances"
Aristotle, a student of Plato, suggested a system in which
each planet was assigned a number of spheres concentric
with the earth. The spheres (a total of 55) rotated about axes
oriented in various directions The combinations of different
axes and rates of rotation could produce quite complicated
motions, He still could not explain why the brightness of some
planets varies, but he dismissed this as a minor discrepancy.
(As we will see, one person's trivial detail can lead to glory for
another, not so cavalier.) Aristotle disagreed with Plato in a
fundamental way. He felt that it is through observation of nature, rather than pure reasoning, that one gains knowledge of
the world. Accordingly, he embarked on gathering an impressive collection of information in all fields Although his ideas
on motion and astronomy have proven to be false, many of his
contributions to science (especially biology), politics, ethics,
and law have stood the test of time.
To explain the variations in apparent brightness, speed,
and size of some planets, Hipparchus (ca, 150 B.C.) invented
a new system of uniform circular motions that were not all
concentric with the earth. The path of a planet was composed
of a deferent on which was superimposed an epicycle, as
shown in Fig 1.5a, Different rates of rotation could produce
various paths such as those in Fig. 1.5b. To improve agreement with observation, Ptolemy (ca A.D. 130) added other
refinements. For example, he shifted the center of the deferent
from the earth to another point called the eccentric. The Ptolemaic system was used by astronomers for many centuries
Earl ier, Aristarchus of Samos (ca. 310 B.C ) had proposed
a heliocentric theory that correctly placed the earth as the third
planet from the sun. The proposed orbital motion of the earth
around the sun should cause the apparent positions of the
stars to change during the course of the orbit But this socalled stellar parallax was not seen. (It requires careful measurements with a modern telescope.) Because of the apparent
absence of stellar parallax and any sensation of the earth's
motion, the idea lay dormant for 1800 years before it sparked
the imagination of Nicholas Copernicus,
THE COPERNICAN REVOLUTION
Copernicus left his native Poland in 1496 to bask in the light of
the Italian Renaissance. He studied law and astronomy in
Padua and Florence. He found the Ptolemaic system of deferents, epicycles, and so on, "not pleasing to the mind." He felt
that the hel iocentric system of Aristarchus was basically simpler in conception. It was clear to Copernicus that what appears to be the sun's circular motion around the earth could be
Planet
Earth
o
,
Eccentnc
I
.
f
\
I
Deferent
....
/
I
'-...,/
I
r
_........'"
".
/
(a)
\
I
I
I
I
I
(b)
FIGURE 1.5 (a) In the geocentric system each planet was assumed to move in a circular
epicycle superimposed on a deferent whose center was at the eccentric. (b) Various paths could
be produced by suitable choices for the rates of revolution.
14
CHAP. 1 INTRODUCTION
Cu)
(h)
FIGURE 1.6 (a) Nicholas Copernicus
(1473-1543). (b) In the heliocentric system
advocated by Copernicus, the planets orbit the
sun in circular paths. However, to improve
accuracy Copernicus had to employ the devices
of the epicycle and eccentric.
explained instead by a dai ly rotation of the earth. Still adhering
to Plato's rule of uniform circular motion, he developed a heliocentric theory that was published in 1543, just before he died
(see Fig. 1.6).
Copernicus easily explained retrograde motion and the
apparent variation in the brightness and sizes of the planets.
However, to improve agreement with observations, he was
forced to use epicycles and eccentrics. It was also realized
that if the Copernican model were correct, Venus should display phases: Its appearance should vary from a crescent to a
full circle, as we see in our moon. These phases, shown in Fig.
1.7, could not be observed without telescopes. In its final form
the Copernican system was neither simpler, nor more precise,
than the Ptolemaic system. The neat explanation of retrograde
motion was balanced by the apparent absence of stellar paral-
lax and the phases of Venus. Nonetheless, Copernicus had
demonstrated that the heliocentric system could also be used
to predict the positions of planets.
It is not surprising that people did not rush to embrace the
theory. The claim that the earth moves did violence to common
sense. People did not feel a strong wind as the earth rotates
under the atmosphere. Copernicus responded that the earth
drags along its atmosphere. He correctly explained the absence of stellar parallax by stating that the stars were simply
too far away for observers to detect it, but this was not a convincing argument at the time. Another important objection to
the idea that the earth moves was based on the fact that an
arrow shot vertically up falls back to the firing point. At the
time, people expected that it should be left behind. This objection Copernicus could not tackle. Different aspects of Coperni-
FIGURE 1.7 The phases of Venus. There was no explanation for this phenomenon in a geocentric
system.
HISTORICAL NOTE
FIGURE 1.8 Johannes Kepler (1571-1630).
cus' theory were taken up by the German astronomer Johannes
Kepler (Fig. 1.8) and the Italian physicist Galileo Galilei (Fig.
1.9).
The Danish astronomer Tycho Brahe spent twenty years
making very precise measurements (without a telescope) of
the positions of stars and the planets. (This was done partly to
enable him to cast more accurate horoscopes, which was part
of his duties.) After his death, his assistant Johannes Kepler
acquired the data. Kepler, who believed in the heliocentric
system, wanted to determine the exact orbit of Mars given data
taken from a moving earth, whose own orbit was unknown.
After six years of calculations he found a circle that looked
quite good. However, there were tiny discrepancies of 8 minutes of arc: Mars was either inside or outside the perfect circle
by this amount. A less dedicated person would have brushed
this aside as so much experimental error. But Kepler had faith
in Brahe's data, which were accurate to within 4 minutes of are,
which is close to the limit of resolution of the human eye. So
Kepler discarded those six years of work. (His writings do not
spare the reader any of the agony.) He came to suspect that the
idea of uniform circular motion was false. After another two
years of work, in 1609 he published two laws which stated: (i)
The planets move in elliptical, not circular, orbits; and (ii) their
speeds are not constant. (These laws are more fully discussed
later.) Kepler had replaced the 48 intertwined circles of Copernicus by just seven beautiful and unadorned ellipses (one
for each planet then known). As we will see, these laws (plus
one more) were extremely important for the development of
mechanics.
Galileo Galilei, a contemporary of Kepler, also advocated
15
FIGURE 1.9 Galileo Galilei (1564-1642).
the Copernican system Although he gained moral support
from Kepler, he was skeptical of the latter's interest in mysticism and numerology. Hence, he was also suspicious of Kepler's elliptical orbits. In 1609 word reached him that two lenses
could be combined to produce a magnified image. He soon
devised a telescope and immediately turned it skyward.
Among other things, he discovered moons orbiting Jupitersomething not envisaged either by Aristotle or by Copernicus.
When viewed through the telescope, the stars remained as tiny
spots of light. This observation lent support to Copernicus'
assertion that they are very far away. The most damaging evidence against geocentricity came when Galileo actually observed the phases of Venus. If Venus orbited the earth there
would be no reason for these phases to appear, or for the
apparent size of this planet to change.
Since the heliocentric theory conflicted with the Christian
belief that humans were at the center of the universe, Galileo's
brilliant defense of it made the Vatican uneasy. In 1618 it
passed an injunction that forbade Galileo to defend the Copernican system. In 1623, Barbarini, a friend of Galileo, became
Pope Urban VIII and agreed to let Galileo teach the new system-but only as a hypothesis. However, in 1632 Galileo published the Dialogues on the Two Chief World Systems, which
left little doubt as to his true opinions. The Vatican summoned
him before the Holy Inquisition and forced him to recant his
belief in the Copernican system as a true description of the
world. Because of his age (70 years) and his eminence, he was
sentenced to a relatively mi Id house arrest. It was then that
Galileo did his most important work: He overthrew Aristotle's
ideas on motion and thereby laid the foundation for mechanics.