Introduction &
Mathematical Concepts
Chapter 1
Outline
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Objectives
About Science
The Nature of Physics
Units
Vectors
Objectives
• Define physics and explain its role
and scope.
• Introduce the fundamental tools of
physics that will be the basis of all
further study: basic units, unit
conversions, scalars & vectors.
• Review basic trigonometric,
geometric and algebraic relations.
About Science
What is Science?
Science is a
(human) activity
that requires study
and method
Science is the body of
knowledge about nature
It is the observation,
identification, description,
experimental investigation,
and theoretical explanation
of natural phenomena
What is the
Scientific method?
It is an orderly method
for gaining, organizing
and applying new
knowledge.
Step 1: Recognize the problem
Step 2: Make a hypothesis
Step 3: Predict consequences
Step 4: Perform experiments
Step 5: Formulate a theory
The Nature of Physics
How is Physics related to Science?
Life Sciences
(living things)
Biology
Botany
Zoology
Science
Physical Sciences
(nonliving things)
Geology
Astronomy
Chemistry
Physics
Physics is the most basic science!?
It’s about the nature of
basic things (motion,
forces, energy, matter, heat,
sound, light,and the insides
of atoms
Physics
It’s about how matter
is put together
(molecules, matter)
Study of
matter that is
alive (cell)
Chemistry
Biology
Understanding science
begins with an
understanding of Physics
Describe all phenomena in the physical
world in terms of a few fundamental
relationships between measurable
properties of matter and energy
What is the goal of Physics?
Understand how things
happen in our natural
environment and why they
happen as they do.
Classical Physics
Modern Physics
Areas of
Physics:
Classical Mechanics: concerned with
the motion of objects moving at speeds
that are low compared to speed of light
Thermodynamics deals with heat, work,
temperature and the statistical behavior
of large number of particles
Electromagnetism: theory of electricity,
magnetism, and electromagnetic fields
Relativity: theory describing objects
moving at any speed, even those whose
speed approach the speed of light
Quantum mechanics: deals with the
behavior particles at the subatomic level
as well as the macroscopic world
Standards and Units
What are units for?
Physics laws are based on experiment
Measurements (of Physical quantities)
Comparison with some reference standard
Need units of measurements that
are invariable and can be
duplicated in various locations
(accessible)
The International System, or SI
SI Basic Units used in Mechanics
Length
meter (m)
Mass
kilogram (kg)
Time
Second (s)
Other Basic SI units
¾ Temperature Æ Kelvin
¾ Electric Current Æ Ampere
¾ Amount of matter Æ mole
¾ Luminous intensity ÆCandela
The Standard of Length: meter (m)
Current Standard (established in 1983):
The meter is the length of the path traveled by light in
vacuum during a time interval of 1/299 792 458 of a second.
Old Standards
-Yard (King of England A.D. 1120) = distance from the tip of his nose to
the end of his outstretched arm
- Foot = length of the royal foot of King Louis XIV (prevailed until 1799)
- Meter (as of 1799) = 1/107 the distance from the equator to the north
pole along a longitudinal line that passes through Paris
- Meter(as of 1960) is the distance between two lines on a specific
platinum-Iridium bar stored under controlled conditions
- Meter = 1 650 763.73 wavelengths of orange red light emitted from
krypton-86 lamp.
The Standard of Mass: kilogram (kg)
Current Standard (established in1887)
The kilogram is the unit of mass; it is equal to the mass of
the international prototype of the kilogram
The Standard of Time: second (s)
Current Standard (established in 1967):
The second is the duration of 9 192 631 770 periods of the
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the cesium 133 atom.
Old Standard before 1960:
The second was defined as 1/(60*60*24) the mean solar day
(The mean solar day is the average time between successive arrivals of
the sun to its highest point in the sky)
Other Systems of Units
The cgs system (used in Europe before the SI):
¾centimeter (cm),
¾gram (g),
¾ second
The British engineering system:
¾foot (ft), 1 ft = 0.3048 m
¾Slug, 1 slug = 14.59 kg
¾second
Derived Units
All other units may be expressed as a
combination of basic units.
Example:
Force Æ Newton N = kg*m/s2.
Energy Æ joule J = N . m = kg*m2/s2.
Pressure ÆPascal P = n/m2 = kg/(m*s2).
Unit Prefixes
Power
10-18
Prefix
atto(a)
Power
103
Prefix
kilo(k)
10-15
femto(f)
106
Mega(M)
10-12
pico(p)
109
Giga(G)
10-9
nano(n)
1012
Tera(T)
10-6
micro(µ)
1015
Peta(P)
10-3
milli(m)
1018
exa(E)
Dimensional Analysis
• Dimension denotes the physical nature
of a quantity
[L] Æ Length
[T] Æ Time
[M] Æ Mass
• Dimensional analysis is used to check
mathematical
relations
for
the
consistency of their dimensions.
Example 1:
Consider the equation v =zxt2. The
dimensions of the variables x, v,
and t are [L], [L]/[T], and [T],
respectively. What must be the
dimensions of the variable z, such
that both sides of the equation have
the same dimension?
Solution: dimension of z = [T]-3
The Nature of Physical Quantities:
Scalars and Vectors
Physical
Quantity
Scalar Quantity
Described by a single number (magnitude)
Example: mass, time, volume, density…
Vector Quantity
Described by both magnitude & direction
Example: Force, velocity, displacement …
The Components of a Vector
+y
A = Ax + Ay
A = (Ax2+Ay2)1/2
A
Ay
θΑ = tan-1(Ay/Ax)
Ay = A sin θA
θΑ
Ax
Ax = A cos θA
+x
Example 2:
Your friend has slipped and
fallen. To help her up, you
pull with a force F , as the
drawing
shows.
The
vertical component of the
force is 150 newtons, while
the horizontal component
is 150 newtons. Find (a)
the magnitude of F and (b)
the angle θ.
Solution: 2.0 102 N, 41 degrees
Vector Addition
+y
C =A+B
Cx = Ax + Bx
Cy = Ay + By
C
θC
A
Ax
Cx
B By C
y
Ay
Bx
C = (Cx2+Cy2)1/2
θC = tan-1(Cy/Cx)
+x
Vector Subtraction
+y
C
B
C =A+B
A
+x
+y
C
-B
A=C–B
A = C +(- B)
A
+x
Example 3:
The drawing shows a force vector that has a
magnitude of 475 newtons, find the (a) x, (b)
y, and (c) z components of the vector.
Solution to Example 3
F can be first resolved into two components; The z
component Fz and the projection onto the x-y plane, Fp.
Fp = F sin 54.0° = (475 N) sin 54.0°= 384 N.
z
y
F
54.0°
54.0°
F
F
p
F
y
z
33.0°
F
p
Fx
x
The projection onto the x-y plane, Fp, can then be resolved into
x and y components.
a. Fx = Fp cos 33.0° = (384 N) cos 33.0°= 322 N
b. Fy = Fp sin 33.0° = (384 N) sin 33.0°= 209 N
c. Fz = F cos 54.0° = (475 N) cos 54.0°= 279 N
Example 4:
A sailboat race course consists
of four legs, defined by the
displacement vectors A, B,
C, and D, as the drawing
indicates. The magnitudes
of the first three vectors are
A = 3.20 km, B = 5.10 km,
and C = 4.80 km. The finish
line of the course coincides
with the starting line. Using
the data in the drawing, find
the distance of the fourth leg
and the angle θ.
Solution: 6.88 km, 26.9 degrees