Chapter 1
Units, Physical Quantities and
Vectors
1.1
Nature of physics
Mathematics. Math is the language we use to discuss science (physics,
chemistry, biology, geology, engineering, etc.) Not all of the mathematical
ideas were (so far ) applied to sciences, but it is quite remarkable to see how
very abstract mathematical concepts (one way or another) find their way into
science. It was even conjectured (relatively recently) that may be all of the
mathematics has a physical realization somewhere. There are even evidences
for that (i.e. multiverse theories), but it is perhaps too speculative to discuss
here.
Physics. Physics is built on top of mathematics and serves as a foundation for other sciences. Whether it bridges all of mathematics or only some
branches of mathematics we will never know. In fact one can never prove
anything in physics. This should not be surprising given that even pure
mathematics suffers from incompleteness (Godel’s theorems). The best that
we can do is to use the language of mathematics to construct physical theories to describe the world around us. The description might not be perfect,
but it is remarkable how well we can do.
So one can think of physics as a toolbox of ideas which can be used in
building scientific models in chemistry, biology, geology, astronomy, engineering, etc. Thus the more tools are in your toolbox the better scientist you will
be. In other words the more physics you know the better chemist, biologist,
geologist, engineer you will be. But if you really like is to design new “tools”
then you should think about becoming a physicist.
1
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
1.2
2
Solving physics problems
Concepts. In most disciplines the more material you can memorized the
better your final grade will be. This is not the case in physics. In fact it is
not the material that you have to learn, but the so-called physical concepts
and physical laws that you have to understand. (If you will decide to become
a physicists, then in few years you might learn that there is actually only
a single very deep concept, known as the least action principle, and almost
everything follows from it.) So the main issue in physics is to learn physics
concepts (e.g. Newton laws, conservation of energy, momentum, etc.). And
learning how to learn physics concepts will be your first and perhaps most
difficult task. I will try to help you with that, but it will be a lot more efficient
if you also help each other. Please discuss concepts, problems, solutions, etc.
with each other.
Problems. The only way to evaluate if you really understand concepts
is to solve problems. Understanding how someone else (e.g. classmates,
professors, etc.) solves problems is good, but definitely not enough. Yes, you
might have hard time solving problems at first, but it is absolutely essential
to learn how to solve problems on your own. The book provides useful
strategies and I encourage you to try to apply these strategies, but I am
afraid only practice can make you an expert. For your own record keep track
of how many problems you solved by yourself in each chapter (without anyone
helping or googling answers). I bet this number will be strongly correlated
with your grade on exams as well as your final grade.
1.3
Standards and Units
Units. Physics is an experimental science and experiments involves measurements of, for example, time, length, mass, etc. To express the results
of measurements we use units. Units of length, units of time units of mass.
There are some standard units just because historically we decided that these
units (e.g. seconds, meters, kilograms) are convenient in physical experiments. Not everyone (e.g. American and British vs. European) agrees what
should be the standard units and the history of such debates is quite interesting. One the other hand we should always be able to convert from one
system of units to another given the “dictionary”.
Time. What is time and arrow of time is a deep philosophical question.
We might discuss it a bit in the last day of classes in context of the second
law of thermodynamics. At this point all the we have to remember that time
is measured in units of seconds, milliseconds, microseconds, nanoseconds and
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
3
here is the dictionary:
1 nanosecond © 1 ns = 10≠9 s
1 microsecond © 1 µs = 10≠6 s
1 millisecond © 1 ms = 10≠3 s.
(1.1)
Length. Despite of the fact that length and time appear to us very
differently, there is a very deep connection (symmetry) between them. We
might discuss is briefly when we discuss gravitation, but you might not understand it completely until you take a course in special relativity (PHYS
2021). Meanwhile let just remember the conversion dictionary for units of
length
1 nanometer
1 micrometer
1 millimeter
1 centimeter
1 kilometer
©
©
©
©
©
1 nm = 10≠9 m
1 µm = 10≠6 m
1 mm = 10≠3 m
1 cm = 10≠2 m
1 km = 103 m.
(1.2)
Mass. Mass is also something very familiar to us in everyday life, but
also has very deep properties connecting it so length and time. We might
mention it briefly in connection to black-holes, but you would not appreciate
until you take a course in general relativity (PHYS 5551) or a math course
in differential geometry. As far as the conversion of units goes we have the
following dictionary
1 microgram © 1 µg = 10≠6 g = 10≠9 kg
1 milligram © 1 mg = 10≠3 g = 10≠6 kg
1 gram © 1 g = 10≠3 kg.
(1.3)
Other units can be formed from seconds, meters and kilograms, for example,
the unit of speed
units of volume
or units of force
m
s
m3
kg · m
© Newton.
s2
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
1.4
4
Converting Units
Dimension. Any physical quantity expressed in units of TIME, LENGTH or
MASS is said to have dimensions of time, length or mass respectively. More
generally one can have physical quantities which have mixed dimensions. For
example if d has units of LENGTH and t has units of TIME, then quantity
d
(1.4)
t
has units of LENGTH/TIME. Evidently Eq. (1.4) has quantities with the
same dimension (i.e. LENGTH/TIME) on both sides of the equation. So if
you are given that
v=
d = 30 m
t = 5s
(1.5)
then according to Eq. (1.4)
v=
d
30 m
m
=
=6 .
t
5s
s
(1.6)
This must be true for any equation that you write. Checking that the quantities on both sides of equation have the same dimension is a quick, but very
important test that you could do whenever you setup a new equation. If the
dimension is not the same than you are doing something wrong.
Conversion. Sometimes you will need to convert from one system of
units to another. This can always be done with the help of conversion dictionary. For the case of conversion from standard system of units to British
system of units the dictionary is:
1 in = 2.54 cm
1 pound = 4.448221615260 Newtons
(1.7)
Then, if we are given a quantity in units of speed, then we can convert it
from one system of units to another,
360
3
4 A
in
in
2.54 cm
1h
= 360
·
·
h
h
1 in
3600 s
B
= 0.254
cm
.
s
(1.8)
Note that although we have converted the units, the dimension of both sides
remains the same, i.e. LENGTH/TIME.
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
1.5
5
Uncertainty and Significant Figures
Experimental Measurements. Measurements are always uncertain, but
it was always hoped that by designing a better and better experiment we
can improve the uncertainty without limits. It turned out not to be the case.
There is a famous uncertainty principle of quantum mechanics, but you will
only learn it next year in (PHYS 2021) if you decide to take it. From our
point of view uncertainty is nothing but uncertainty in measurements. This
(as well as significant figures) will be discussed in your lab course.
1.6
Estimates and Order of Magnitude
Theoretical Estimates. Similarly to uncertainties in experimental measurements, theoretical predictions are never exact. We always make simplifying assumption and thus the best we can hope for is an estimate for
the physical quantities to be measured. A useful tool in such estimates
is known as order-of-magnitude estimate (also know as outcome of “back-ofthe-envelope calculations”). Such estimates are often done using the so-called
dimensional analysis - i.e. just use the known quantities to form a quantity
with the dimension of the quantity that you are looking for.
1.7
Vectors and Vector Addition
Vector. Some physical quantities are describe by a single real number. We
call these quantities - scalar quantities or scalars. Examples are temperature,
mass, density, etc.
T = 36.6 C
m = 78 kg
g
fl = 1.05 3 .
m
(1.9)
Other quantities also have a direction associated with them and thus are
describe by three real numbers.
˛ = (Ax , Ay , Az ).
A
(1.10)
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
6
We call these quantities - vector quantities or vectors. For example, position,
velocity, acceleration, etc.
˛r = (1 m, 2 m, 3 m)
m m m
˛v = (4 , 5 , 6 )
s
s
s
m m m
˛a = (7 2 , 8 2 , 9 2 ).
s
s
s
(1.11)
There are even more complicated physical quantities - called tensor quantities
or tensors - but we will not discuss them in this course. Apart from the
number of real numbers which describe these physical quantities, the scalars
and vectors (and also tensors) change very differently under the change of
coordinates.
Graphical representation. When dealing with vectors it is often useful
to draw a picture. Here is how it is done:
• Vectors are nothing but straight arrows drawn from one point to another. Zero vector is just a vector of zero length - a point.
• Length of vectors is the magnitude of vectors. The longer the arrow
the bigger the magnitude.
• It is assumed that vectors can be parallel transported around. If you
˛ to end of another vector B
˛ then the vector
attach beginning of vector A
˛
˛
˛
˛
A + B is a straight arrow from begging of vector A to end of vector B.
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
7
Coordinates. The space around us does not have axis and labels, but we
can imagine that these x, y and z axis or the coordinate system to be there.
This makes it possible to talk about position of, for example, point particles
using their coordinates - real numbers. Since one needs three real numbers
to specify position it is a vector. Similarly, velocity, acceleration and force
are all vectors.
Symmetries. You might complain that there is arbitrariness in how
one chooses coordinate system and you would be right. However, it turns
out that the physically observable quantities do not depend on the choice of
coordinate systems and thus one can choose it to be whatever is more convenient. Moreover, this symmetry is an extremely deep property which gives
rise to conservation laws that we will learn in this course. For example the
arbitrariness of choosing the x, y and z coordinates gives rise to momentum
conservation. And the arbitrariness in choosing time coordinate gives rise to
Energy conservation.
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
1.8
8
Components of Vectors
Magnitude. The length of vector or magnitude is a scalar quantity
or in components
˛ =A
˛ = |A|
A
(1.12)
Ò
(1.13)
(Ax , Ay , Az ) =
For example the length of vector
A2x + A2y + A2z .
˛ = (3, 4, 5)
A
is
˛ =
|A|
Ô
32 + 42 + 52 =
Ô
50 ¥ 7.07106781187.
(1.14)
(1.15)
Direction. One can also find direction of vector using trigonometric
identities. For example, in two dimensions
˛ = (Ax , Ay , 0)
A
(1.16)
the angle of the vector measure from x-axis is given by
tan ◊ =
Ay
Ay
or ◊ = arctan
.
Ax
Ax
(1.17)
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
9
Addition. Two vectors can be added together to get a new vector
˛ =A
˛ +B
˛
C
(1.18)
an in component form
(Cx , Cy , Cz ) = (Ax , Ay , Az ) + (Bx , By , Bz ) = (Ax + Bx , Ay + By , Az + Bz ).
(1.19)
For example, the sum of two vectors
is
˛ = (3, 4, 5)
A
˛ = (6, 7, 8)
B
(1.20)
(3, 4, 5) + (6, 7, 8) = (9, 11, 13).
(1.21)
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
1.9
10
Unit Vectors
Unit vectors. Is a vector that has magnitude one. Its is usually denoted
with a “hat”:
Ò
|û| = u2x + u2y + u2z = 1.
(1.22)
For example
or simply
Ò
Ô
Ô
|(1/ 2, 1/ 2, 0)| =
1/2 + 1/2 + 0 = 1
Ò
Ô
Ô
Ô
|(1/ 3, 1/ 3, 1/ 3)| =
1/3 + 1/3 + 1/3 = 1
Ô
|(1, 0, 0)| =
12 + 02 + 02 = 1
Ô
|(0, 1, 0)| =
02 + 12 + 02 = 1
Ô
|(0, 0, 1)| =
02 + 02 + 12 = 1.
(1.23)
(1.24)
In fact the last three vectors are so important that there are special letters
reserved to denote these vectors
î = (1, 0, 0)
ĵ = (0, 1, 0)
k̂ = (0, 0, 1).
(1.25)
Multiplication / division by scalar. Any vector can be multiplied by
a scalar to obtain another vector,
˛ = C B.
˛
A
(1.26)
(Ax , Ay , Az ) = C(Bx , By , Bz ) = (CBx , CBy , CBz ).
(1.27)
In components from
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
11
Thus one can make a unit vector out of any vector
Q
R
˛
A
(Ax , Ay , Az )
Ax
Ay
Az
b.
 =
=Ò
= aÒ
,Ò
,Ò
2
2
2
2
2
2
2
2
2
2
˛
|A|
Ax + Ay + Az
Ax + Ay + Az
Ax + Ay + Az
Ax + A2y + A2z
For example if
˛ = (2, 3, 4)
A
then
 =
A
(1.28)
B
2
2
2
Ô ,Ô ,Ô
.
29 29 29
(1.29)
Components. Note that any vector can be written in components in two
equivalent ways:
˛ = (Ax , Ay , Az ) = Ax î + Ay ĵ + Az k̂
A
(1.30)
Ax î + Ay ĵ + Az k̂ = Ax (1, 0, 0) + Ay (0, 1, 0) + Az (0, 0, 1).
= (Ax , 0, 0) + (0, Ay , 0) + (0, 0, Az )
= (Ax , Ay , Az )
(1.31)
just because
1.10
Product of vectors
Scalar (or dot) product. Dot product is a multiplication between two
vectors which produces a scalar:
˛ ·B
˛ =B
˛ ·A
˛ =C
A
(1.32)
In components
˛ ·B
˛ = (Ax , Ay , Az ) · (Bx , By , Bz ) = Ax Bx + Ay By + Az Bz = AB cos(„).
A
(1.33)
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
12
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
13
One can also derive multiplication table for unit vectors
î · î = (1, 0, 0) · (1, 0, 0) = 1 + 0 + 0 = 1
ĵ · ĵ = (0, 1, 0) · (0, 1, 0) = 0 + 1 + 0 = 1
k̂ · k̂ = (0, 0, 1) · (0, 0, 1) = 0 + 0 + 1 = 1
î · ĵ = (1, 0, 0) · (0, 1, 0) = 0 + 0 + 0 = 0
ĵ · k̂ = (0, 1, 0) · (0, 0, 1) = 0 + 0 + 0 = 0
k̂ · î = (0, 0, 1) · (0, 0, 1) = 0 + 0 + 0 = 0.
(1.34)
But then we can also apply the second component representation of vectors
(given by Eq. (1.30)) to check that Eq. (1.33) is indeed correct
˛ ·B
˛ = (Ax î + Ay ĵ + Az k̂) · (Bx î + By ĵ + Bz k̂) =
A
= Ax î · Bx î + Ax î · By ĵ + Ax î · Bz k̂ +
+Ay ĵ · Bx î + Ay ĵ · By ĵ + Ay ĵ · Bz k̂ +
+Az k̂ · Bx î + Az k̂ · By ĵ + Az k̂ · Bz k̂
= Ax Bx + Ay By + Az Bz .
(1.35)
Vector (or cross) product. Cross product is a multiplication between
two vectors which produces a vector:
˛ ◊B
˛ = ≠B
˛ ◊A
˛ =C
˛
A
(1.36)
In components
˛ B
˛ = (Ax , Ay , Az )◊(Bx , By , Bz ) © (Ay Bz ≠Az By , Az Bx ≠Ax Bz , Ax By ≠Ay Bx ) = C.
˛
A◊
(1.37)
One can also derive multiplication table for unit vectors
î ◊ î = (1, 0, 0) ◊ (1, 0, 0) = (0, 0, 0)
ĵ ◊ ĵ = (0, 1, 0) ◊ (0, 1, 0) = (0, 0, 0)
k̂ ◊ k̂ = (0, 0, 1) ◊ (0, 0, 1) = (0, 0, 0)
î ◊ ĵ = (1, 0, 0) ◊ (0, 1, 0) = (0, 0, 1) = k̂ = ≠ĵ ◊ î
ĵ ◊ k̂ = (0, 1, 0) ◊ (0, 0, 1) = (0, 0, 1) = î = ≠k̂ ◊ ĵ
k̂ ◊ î = (0, 0, 1) ◊ (0, 0, 1) = (0, 0, 1) = ĵ = ≠î ◊ k̂.
(1.38)
But then we can also apply the second component representation of vectors
CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS
14
(given by Eq. (1.30)) to check that Eq. (1.37) is indeed correct
˛ ◊B
˛ = (Ax î + Ay ĵ + Az k̂) ◊ (Bx î + By ĵ + Bz k̂) =
A
= Ax î ◊ Bx î + Ax î ◊ By ĵ + Ax î ◊ Bz k̂ +
+Ay ĵ ◊ Bx î + Ay ĵ ◊ By ĵ + Ay ĵ ◊ Bz k̂ +
+Az k̂ ◊ Bx î + Az k̂ ◊ By ĵ + Az k̂ ◊ Bz k̂
= (Ay Bz ≠ Az By ) î + (Az Bx ≠ Ax Bz ) ĵ + (Ax By ≠ Ay Bx )(1.39)
k̂.
Note that the magnitude of cross product is given by
˛ ◊ B|
˛ = AB sin „
|A
but the direction is determined by right-hand rule: