Introduction & Mathematical Concepts

Chapter 1

Outline

• 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 the body of knowledge about nature

It is the observation, identification, description, experimental investigation, and theoretical explanation

of natural phenomena

Science is a (human) activity

that requires study and method

What is the Scientific method?

It is an orderly method for gaining, organizing

and applying new knowledge.

Step 3: Predict consequences

Step 1: Recognize the problem

Step 2: Make a hypothesis

Step 4: Perform experiments

Step 5: Formulate a theory

The Nature of Physics

How is Physics related to Science?BiologyBotanyZoology

Life Sciences (living things)

ScienceGeologyAstronomyChemistryPhysics

Physical Sciences(nonliving things)

Physics is the most basic science!?

Biology

Study of matter that is alive (cell)

Chemistry

It’s about how matter is put together

(molecules, matter)Physics

It’s about the nature of basic things (motion,

forces, energy, matter, heat, sound, light,and the insides

of atoms

Understanding science begins with an

understanding of Physics

What is the goal of Physics?

Describe all phenomena in the physical world in terms of a few fundamental

relationships between measurable properties of matter and energy

Understand how things happen in our natural

environment and why they happen as they do.

Areas of Physics:

Relativity: theory describing objects moving at any speed, even those whose speed approach the 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

Quantum mechanics: deals with the behavior particles at the subatomic level as well as the macroscopic world

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Classical Mechanics: concerned with the motion of objects moving at speeds that are low compared to speed of light

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 Temperature Kelvin Electric Current Ampere Amount of matter moleLuminous intensity Candela

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 UnitsThe cgs system (used in Europe before the SI):

centimeter (cm), gram (g),second

The British engineering system:foot (ft), 1 ft = 0.3048 m Slug, 1 slug = 14.59 kg second

Derived Units

All other units may be expressed as a combination of basic units.

Example:Force Newton N = kg*m/s2.

Energy joule J = N . m = kg*m2/s2.

Pressure Pascal P = n/m2 = kg/(m*s2).

Unit Prefixes

Peta(P)1015micro(µ)10-6

exa(E)1018milli(m)10-3

Tera(T)1012nano(n)10-9

Giga(G)109pico(p)10-12

Mega(M)106femto(f)10-15

kilo(k)103atto(a)10-18PrefixPowerPrefixPower

Dimensional Analysis

• Dimension denotes the physical nature of a quantity

[L] Length[T] Time[M] 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 QuantityDescribed by a single number (magnitude)

Example: mass, time, volume, density…

Vector QuantityDescribed by both magnitude & directionExample: Force, velocity, displacement …

The Components of a Vector

+x

+y

AAy

Ax

θΑ

Ax = A cos θA

Ay = A sin θA

A = Ax + Ay

A = (Ax2+Ay

2)1/2

θΑ = tan-1(Ay/Ax)

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

C = A + BCx = Ax + Bx

Cy = Ay + By

C = (Cx2+Cy

2)1/2

θC = tan-1(Cy/Cx)

+x

+y

A

B

AxAy

Bx

ByCCy

Cx

θC

Vector Subtraction

A = C – BA = C +(- B)

C = A + B

+x

+y

A

B C

+x

+y

A

-B C

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 3F can be first resolved into two components; The zcomponent Fz and the projection onto the x-y plane, Fp.

Fp = F sin 54.0° = (475 N) sin 54.0°= 384 N.

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 Nb. Fy = Fp sin 33.0° = (384 N) sin 33.0°= 209 Nc. Fz = F cos 54.0° = (475 N) cos 54.0°= 279 N

54.0°

F

z

Fz

Fp

54.0° Fp

Fx

Fy

y

x

33.0°

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