Chapter 1: Units, Physical
Quantities and Vectors
1
About Physics
2
What is Physics?

Phys’ics

Originally, natural sciences or natural philosophy

The science of dealing with properties, changes, interaction,
etc., of matter and energy
Physics is subdivided into mechanics, thermodynamics,
optics, acoustics, etc.


[Gr. Physika, physical or natural things]
From Webster's Unabridged Dictionary
3
Science

Science
[Latin scientia - knowledge]

Originally, state of fact of knowing; knowledge, often as
opposed to intuition, belief, etc.

Systematized knowledge derived from observation, study and
experimentation carried on in order to determine the nature or
principles of what is being studied.

A Science must have PREDICTIVE power
4
Physics: Like a Mystery Story

Nature presents the clues

Experiments

We devise the hypothesis

Theory

A hypothesis predicts other facts that can be checked
- is the theory right?

Right - keep checking

Wrong - develop a new theory

Physics is an experimental science
5
The Ancient Greeks
Aristotle (384-322 B.C.) is regarded as the
first person to attempt physics, and actually
gave physics its name.
On the nature of matter:
Matter was composed of:
Air
Earth
Water
Fire
Every compound was a mixture of these elements
Unfortunately there is no predictive power
6
On the Nature of Motion

Natural motion - like a falling body


Objects seek their natural place

Heavy objects fall fast

Light objects fall slow

Objects fall at a constant speed
Unnatural motion - like a cart being pushed

The moving body comes to a stand still when the
force pushing it along no longer acts

The natural state of a body is at rest
7
Aristotelian Physics

Aristotelian Physics was based on logic
o
It provided a framework for understanding nature
o
It was logically consistent
It was wrong !!!

Aristotelian physics relied on logic - not experiment
8
The Renaissance
Galileo Galilei (1564 -1642) was one of the
first to use the scientific method of observation
and experimentation. He laid the groundwork
for modern science.
9
Classical Mechanics
Mechanics: the study of motion
Galileo (1564 -1642) laid the
groundwork for Mechanics
Newton (1642-1727) completed
its development (~almost~)
Newton’s Laws work fine for

Large Objects - Ball’s, planes, planets, ...


Slow Objects - people, cars, planes, ...


Small objects (atoms)  Quantum Mechanics
Fast objects (near the speed of light)  Relativity
Classical Mechanics - essentially complete at the end
of the 19th Century
10
Why is Physics Important?
Newton’s Laws
and
Classical Physics





Planetary motion
Steam Engines
Radio
Cars
Television
Quantum
Mechanics




Microwaves
Transistors
Computers
Lasers
The Next
Great Theory

o
Teleportation
Faster than
light travel
(can’t exist
today)
"Heavier-than-air flying machines are impossible."
Lord Kelvin, president, Royal Society, 1895.
11
Mechanics

Physics is science of measurements

Mechanics deals with the motion of objects
o
What specifies the motion?
o
Where is it located?
o
When was it there?
o
How fast is it moving?
 Before we can answer these questions
 We must develop a common language
12
Units
13
Fundamental Units
Length [L]
Foot
Meter - Accepted Unit
Furlong
Time [T]
Second - Accepted Unit
Minute
Hour
Century
Mass [M]
Kilogram - Accepted Unit
Slug
14
Derived Units


Single Fundamental Unit

Area = Length  Length

Volume = Length  Length  Length
[L]2
[L]3
Combination of Units

Velocity = Length / Time
[L/T]

Acceleration = Length / (Time  Time) [L/T2]

Jerk = Length / (Time  Time  Time)
[L/T3]

Force = Mass  Length / (Time  Time) [M L/T2]
15
Units

SI (Système Internationale) Units:

mks: L = meters (m), M = kilograms (kg), T
= seconds (s)

cgs: L = centimeters (cm), M = grams (g), T
= seconds (s)

British Units:

Inches, feet, miles, pounds, slugs...

We will switch back and forth in stating problems.
16
Unit Conversion

Useful Conversion Factors:





1
1
1
1
inch = 2.54
m = 3.28
mile = 5280
mile = 1.61
cm
ft
ft
km
Example: convert miles per hour to meters per second:
mi
mi
ft
1 m
1 hr
m
1
 1  5280


 0.447
hr
hr
mi 3.28 ft 3600 s
s
17
Orders of Magnitude

Physical quantities span an immense range

Length
size of nucleus ~ 10-15 m
size of universe ~ 1030 m

Time
nuclear vibration ~ 10-20 s
age of universe ~ 1018 s

Mass
electron
universe
~ 10-30 kg
~ 1028 kg
18
Physical Scale

Orders of Magnitude Set the Scale




Atomic Physics ~ 10-10 m
Basketball ~ 10 m
Planetary Motion ~ 1010 m
Knowing the scale lets us guess the Result
Q: What is the speed of a 747?
Distance - New York to LA
4000 mi
Flying Time
6 hrs
= 660 mph
19
Dimensional Analysis


Fundamental Quantities

Length - [L]

Time
- [T]

Mass
- [M]
Derived Quantities

Velocity - [L]/[T]

Density
- [M]/[L]3

Energy
- [M][L]2/[T]2
20
Physical Quantities



Must always have dimensions
Can only compare quantities with the same
dimensions

v
=
v(0) +
a  t

[L]/[T] = [L]/[T] + [L]/[T]2 [T]
Comparing quantities with different
dimensions is nonsense

v
=
a  t2

[L]/[T] = [L]/[T]2 [T]2 = [L]
21
Provides Solution Sometimes

Period of a Pendulum
Period is a time [T] - t
Which of these could be
correct?
a ) t  lg 
2
b) t  mlg 
2
Can only depend on:
c) t 
l
g
Length [L] - l
d) t 
ml
g
e) t 
l
m
Mass [M] -
m
Gravity [L/T2] -
g
t  2
l
g
22
Solving Problems
23
Problem Solving Strategy

Each profession has its own specialized knowledge and patterns
of thought.

The knowledge and thought processes that you use in each of
the steps will depend on the discipline in which you operate.

Taking into account the specific nature of physics, we choose to
label and interpret the five steps of the general problem solving
strategy as follows:
24
Problem Solving Strategy



A. Everyday language:

1) Make a sketch.
2) What do you want to find out?
3) What are the physics ideas?
B. Physics description:
1) Make a physics diagram.
2) Define your variables.
3) Write down general equations.
D. Calculate solution:
1) Plug in numerical values.
E. Evaluate the answer:
1) Is it properly stated?
2) Is it reasonable?
3) Answered the question asked?
C. Combine equations:
1) Select an equation with the target variable.
2) Which of the variables are not known?
3) Substitute in a different equation.
4) Continue for all of the unknown variables .
5) Solve for the target variable.
6) Check units.
25
Problem Solving Strategy, Step
A
A. Everyday language description:
In this step you develop a qualitative description of the problem.

Visualize the events described in the problem by making a sketch. The sketch
should indicate the different objects involved and any changes in the situation
(e.g. changes in force applied, collisions, etc.) First, identify the different objects
that are relevant to finding your desired category. Next, identify whether there is
more than one stage (part) to the behavior of the object during the time from the
beginning to the end that is relevant for what you are trying to find out. Things
that would indicate more than one part would include key information about the
behavior of the object at a point between start and end of movement, collisions,
changes in the force applied or acceleration of an object.

Write down a simple statement of what you want to find out. This should be a
specific physical quantity that you could calculate to answer the original question.

Write down verbal descriptions of the physics ideas (the type of
problem). Identify the physics idea for each stage of each object. If the physics
idea is a vector quantity (motion, force, momentum, etc.) identify how many
dimensions are involved.
26
Problem Solving Strategy, Step
B
B. Physics description:

In this step you use your qualitative understanding of the problem to prepare
for the quantitative solution.

First, simplify the problem situation by describing it with a diagram in terms of
simple physical objects and essential physical quantities. Make a physics
diagram. You will need a diagram for each physics idea for each object, and
possibly one for each stage and for each dimension.

Define your variables (make a chart) of know quantities and unknown
quantities. Identify the variable you will solve for. Make sure variables are
defined for each object, stage, idea and dimension. Pay attention to units, to
make sure you have the right kind of units for each type of variable.

Using the physics ideas assembled in A-3 and the diagram you made in B-1,
write down general equations which specify how these physical quantities
are related according to the principles of physics or mathematics.
27
Problem Solving Strategy, Step
C
C. Combine equations:

In this step you translate the physics description into a set of equations
which represent the problem mathematically by using the equations
assembled in step 2.

Select an equation from the list in B3 that contains the variable you are
solving for (as specified in B2).

Identify which of the variables in the selected equation are not known.

For each of the unknown variables, select another equation from the list in B3
and solve it for the unknown variable. Then substitute the new equation in
for the unknown quantity in the original equation.

Continue steps 2 & 3 until all of the unknown variables (except the
variable you are solving for) have been replaced or eliminated.

Solve for the target variable.

Check your work by making sure the units work out.
28
Problem Solving Strategy, Steps D &
E
D. Calculate solution:

In this step you actually execute the solution you have planned.

Plug in numerical values (with units) into your solution from C-5.
E. Evaluate the answer:

Finally, check your work.

Is it properly stated? Is it reasonable?

Have you actually answered the question asked?
29
Problem Solving Strategy

Consider each step as a translation of the previous step into a slightly different
language.

You begin with the full complexity of real objects interacting in the real world and
through a series of steps arrive at a simple and precise mathematical
expression. The five-step strategy represents an effective way to organize your
thinking to produce a solution based on your best understanding of physics. The
quality of the solution depends on the knowledge that you use in obtaining the
solution.

Your use of the strategy also makes it easier to look back through your solution
to check for incorrect knowledge and assumptions. That makes it an important
tool for learning physics.

If you learn to use the strategy effectively, you will find it a valuable tool to
use for solving new and complex problems.
30
Vectors
31
Scalars & Vectors







A scalar is a physical
quantity that has only
magnitude (size) and
can be represented by a
number and a unit.

A vector is a physical
quantity that has both
magnitude (size) and
direction.

Examples of vectors?


Velocity
Force
Examples of scalars?
Time
Mass
Temperature
Density
Electric charge
32
Displacement Vector is a change in
position. It is calculated as the final
position minus the initial position.
Vectors are
•
•
represented pictorially by an arrow from one point to another.
represented symbolically by a letter with an arrow above it.
33
Some Vector Properties


Two vectors that have the
same direction are said to
be parallel.
Two vectors that have
opposite directions are said
to be anti-parallel.


Two vectors that have the
same length and the same
direction are said to be
equal no matter where they
are located.
The negative of a vector is a
vector with the same
magnitude (size) but
opposite direction
34
Magnitude of a Vector


( Magnitude of A)  A  A

The magnitude of a vector is a positive number (with
units!) that describes its size.

Example: magnitude of a displacement vector is its length.

The magnitude of a velocity vector is often called speed.

The magnitude of a vector is expressed using the same letter
as the vector but without the arrow on top of it.
35
Vector Addition

Vector C of a vector sum of vectors A and C.

Example: double displacement of particle.

Vector addition is commutative (the order of vector
addition doesn’t matter).
36
Vector Addition


CAUTION
Common error: to conclude that if C = A + B the
magnitude C should be equal the magnitude A plus
magnitude B. Wrong !
Example: C < A + B.
37
Vector Addition

Add more than two vectors:


     
R A B C  D C


 
   
R A B C  A E
38
Vector Subtraction

Subtract vectors:
  

A  B  A  ( B)
39
Vector Components

There are two methods of vector addition
Graphical  represent vectors as scaleddirected line segments; attach tail to head
 Analytical  resolve vectors into x and y
components; add components

40
Vector Components
 If R  A  B
 Then Rx  Ax  Bx and Ry  Ay  By
 Where Ax  A cos A and Ay  A cos  A
 Bx  B cos  B and By  B sin  B
41
Vector Components



R  Rx2  Ry2
  tan 1
Ry
R
Ry

Rx
Rx
If Rx< 0 and Ry > 0 or if Rx< 0
and Ry < 0 then    + 180o
42
Vector Components

CAUTION
The components Ax and Ay of a vector A are numbers; they
are not vectors !
43
Vector Components
  
R  A B
Rx  Ax  Bx
  

A  B  A  ( B)
Ry  Ay  By
44
Vector Components
45
VECTOR ADDITION
Problem Solving Strategy





IDENTIFY the relevant concepts and SET UP the problem:
Decide what your target variable is. It may be the magnitude of the
vector sum, the direction, or both.
Then draw the individual vectors being summed and the coordinate
axes being used. In your drawing, place the tail of the first vector at
the origin of coordinates; place the tail of the second vector at the
head of the first vector; and so on.
Draw the vector sum R from the tail of the first vector to the head of
the last vector.
By examining your drawing, make a rough estimate of the
magnitude and direction of R you’ll use these estimates later to
check your calculations.
46
Vector Components
There are two methods of vector addition

Graphical  represent vectors as scaled-directed
line segments; attach tail to head

Analytical  resolve vectors into x and y
components; add components

Components
 Component
 vectors
A  Ax  Ay

Ax  Ax

Ay  Ay
47
Vector Components

You can calculate components if its
magnitude and direction are known

Direction of a vector described by its
angle relative to reference direction

Reference direction  positive x-axis

Angle  the angle between vector A
and positive x-axis
y
90 < Θ < 180
cos (-) sin (+)
Θ = 90
Θ = 180
cos (+) sin (+)
x Θ = 0
270 < Θ < 360
180 < Θ < 270
cos (-) sin (-)
0 < Θ < 90
Θ = 270
cos (+) sin (-)
48
Vector Components
Ay
A
 sin 
Ax
 cos 
A
Ax  A  cos 
Ay  A  sin 
49
Vector Components
CAUTION

The components Ax and Ay of a vector A are numbers;
they are not vectors !

The components of vectors can be negative or positive
numbers.
90 < Θ < 180
cos (-) sin (+)
180 < Θ < 270
cos (-) sin (-)
50
Finding Vector Components
What are x and y components of vector
D? Magnitude of D=3.00m, angle is
=45.
IDENTIFY AND SET UP

Vector Components Trig Equations
EXECUTE

Angle here is measured toward
negative y-axis. But we need angle
measured from positive x-axis toward
positive y-axis. Thus, θ=-=-45.

Dx  D  cos   (3.00m)(cos( 45))  2.1m
Dy  D  sin   (3.00m)(sin( 45))  2.1m
51
Finding Vector Components

What are x and y components
of vector E? Magnitude of
D=4.50m, angle is =37.0.
IDENTIFY AND SET UP

Vector Components Trig
Equations
EXECUTE

Any orientation of axes is
permissible, but X- and Y-axes
must be perpendicular.

E is the hypotenuse of a right
triangle! Thus:
Ex  E  sin   (4.50m)(sin( 37.0))  2.71m
E y  E  cos   (4.50m)(cos( 37.0))  3.59m
52
Vector Components

Reverse the process: We know
the components. How to find
the vector magnitude and its
direction?

Magnitude: Pythagorean
2
2
theorem
x
y

Direction: angle between xaxis and vector
A A  A
tan  
Ay
Ax
  arctan
Ay
Ax
53
Vector Addition, Components
  
R  A B
Rx  Ax  Bx
Ry  Ay  By
Ax  A  cos  A

Ay  A  sin  A

Bx  B  cos  B

By  B  sin  B
54
Problem Solving Strategy
IDENTIFY AND SET UP

Target variable: vector
magnitude, its direction or
both

Draw individual vectors and
coordinate axes

Tail of 1st vector in origin, tail
of 2nd vector at the head of
1st vector, and so on…

Draw the vector sum from
the tail of 1st vector to the
head of the last vector.

Make a rough estimate of
magnitudes and direction.
EXECUTE

Find x- and y-components of each
individual vector

Bx  B  cos  B



By  B  sin  B
Check quadrant sign!
Add individual components
algebraically to find components of the
sum vector
Rx  Ax  Bx  Cx  ...
R y  Ay  B y  C y  ...


Magnitude
R  Rx  R y
Direction
  arctan
Ry
EVALUATE
 Check your results comparing them with the rough estimates!
Rx
55
Vector Components





θA=90.0-32.0=58.0
θB=180.0+36.0=216.0
θC=270.0
Ax=A cos θA
Ay=A sin θA
Distance
Angle
X-comp
Y-comp
A=72.4m
58.0
38.37m
61.40m
B=57.3m
216.0
-46.36m
-33.68m
C=17.8m
270.0
0.00m
-17.80m
-7.99m
9.92m
R  (7.99m) 2  (9.92m) 2  12.7m
  arctan 9.92m  7.99m  129
56
Unit Vectors
57
Unit Vectors

Unit vectors provide a
convenient means of notation
to allow one to express a
vector in terms of its
components.

Unit vectors always have a
magnitude of 1 (with no
units).

Unit vectors point along a
coordinate direction.

Unit vectors are written using
a caret (or "hat", ^ ) to
distinguish them from
ordinary vectors.

Ax  Axiˆ

Ay  Ay ˆj

A  Axiˆ  Ay ˆj
58
Unit Vectors


B  Bxiˆ  By ˆj
A  Axiˆ  Ay ˆj
  
R  A  B  ( Axiˆ  Ay ˆj )  ( Bxiˆ  By ˆj ) 
 ( Ax  Bx )iˆ  ( Ay  By ) ˆj  Rxiˆ  Ry ˆj


B  Bxiˆ  By ˆj  Bz kˆ
A  Axiˆ  Ay ˆj  Az kˆ

R  ( Ax  Bx )iˆ  ( Ay  By ) ˆj  ( Az  Bz )kˆ 
 Rxiˆ  Ry ˆj  Rz kˆ
59