1 Chapter 1: Units, Physical Quantities and Vectors

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Chapter 1: Units, Physical Chapter 1: Units, Physical Quantities and VectorsQuantities and Vectors

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About Physics

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What is Physics?

Phys’ics [Gr. Physika, physical or natural things]

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.

From Webster's Unabridged Dictionary

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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

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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

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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

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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

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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

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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.

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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, ... Small objects (atoms) Quantum Mechanics

Slow Objects - people, cars, planes, ... Fast objects (near the speed of light) Relativity

Classical Mechanics - essentially complete at the end of the 19th Century

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Why is Physics Important?

Newton’s Laws andClassical Physics

QuantumMechanics

The NextGreat Theory

Planetary motion Steam Engines Radio Cars Television

Microwaves Transistors Computers Lasers

Teleportationo Faster than

light travel(can’t exist today)

"Heavier-than-air flying machines are impossible." Lord Kelvin, president, Royal Society, 1895.

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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

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Units

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Fundamental Units

Length [L]FootMeter - Accepted UnitFurlong

Time [T]Second - Accepted UnitMinuteHourCentury

Mass [M] Kilogram - Accepted UnitSlug

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Derived Units

Single Fundamental Unit Area = Length Length [L]2

Volume = Length Length Length [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]

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Units

SISI (Système Internationale)(Système Internationale) Units:: mks: L = meters (m), M = kilograms (kg), T =

seconds (s) cgs: L = centimeters (cm), M = grams (g), T

= seconds (s)

British UnitsBritish Units:: Inches, feet, miles, pounds, slugs...

We will switch back and forth in stating problems.

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Unit Conversion

Useful Conversion Factors: 1 inch = 2.54 cm 1 m = 3.28 ft 1 mile = 5280 ft 1 mile = 1.61 km

Example: convert miles per hour to meters per second:

s

m447.0

s

hr

3600

1

ft

m

28.3

1

mi

ft5280

hr

mi1

hr

mi 1

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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 ~ 10-30 kg

universe ~ 1028 kg

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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

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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

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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]

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Provides Solution Sometimes

Period of a Pendulum

Period is a time [T] -

Can only depend on:

Length [L] - lMass [M] - mGravity [L/T2] - g

Which of these could be correct?

m

le

g

mld

g

lc

gmlb

gla

)

)

)

)

)2

2

g

l 2

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Solving Problems

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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:

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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.

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Problem Solving Strategy, Step AA. 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.

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Problem Solving Strategy, Step BB. 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. 

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Problem Solving Strategy, Step CC. 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.

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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?

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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.

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Vectors

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Scalars & Vectors

A scalar is a physical quantity that has only magnitude (size) and can be represented by a number and a unit.

Examples of scalars?

Time Mass Temperature Density Electric charge

A vector is a physical quantity that has both magnitude (size) and direction.

Examples of vectors?

Velocity Force

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Vectors are

• represented pictorially by an arrow from one point to another. • represented symbolically by a letter with an arrow above it.

Displacement Vector is a change in position. It is calculated as the final position minus the initial position.

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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

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Magnitude of a Vector

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.

AAAofMagnitude

)(

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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).

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Vector Addition C A U T I O N

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.

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Vector Addition

Add more than two vectors:

CDCBAR

EACBAR

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Vector Subtraction

Subtract vectors:

)( BABA

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Vector Components

There are two methods of vector additionGraphical represent vectors as scaled-

directed line segments; attach tail to headAnalytical resolve vectors into x and y

components; add components

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Vector Components

If R A B

Then and x x x y y yR A B R A B

Where cos and cosx A y AA A A A

cos and sinx B y BB B B B

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Vector Components

If Rx< 0 and Ry > 0 or if Rx< 0 and Ry < 0 then + 180o

2 2x yR R R

1tan y

x

R

R

R

xR

yR

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Vector Components C A U T I O N

The components Ax and Ay of a vector A are numbers; they

are not vectors !

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Vector Components

)( BABA

BAR

xxx BAR

yyy BAR

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Vector Components

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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.

VECTOR ADDITION

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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

yx AAA

xx AA

yy AA

Component vectors

Co

mp

on

ents

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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

Θ = 0

Θ = 90

Θ = 180

Θ = 270

x

y

90 < Θ < 180cos (-) sin (+)

180 < Θ < 270cos (-) sin (-)

0 < Θ < 90cos (+) sin (+)

270 < Θ < 360cos (+) sin (-)

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Vector Components

cosA

Axsin

A

Ay

cosAAxsinAAy

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Vector Components C A U T I O N

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 < 180cos (-) sin (+)

180 < Θ < 270cos (-) sin (-)

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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 EquationsEXECUTE Angle here is measured toward

negative y-axis. But we need angle measured from positive x-axis toward positive y-axis. Thus, θ=-=-45.

mmDDx 1.2))45)(cos(00.3(cos

mmDDy 1.2))45)(sin(00.3(sin

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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

EquationsEXECUTE Any orientation of axes is

permissible, but X- and Y-axes must be perpendicular.

E is the hypotenuse of a right triangle! Thus:

mmEEx 71.2))0.37)(sin(50.4(sin

mmEEy 59.3))0.37)(cos(50.4(cos

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Vector Components

Reverse the process: We know the components. How to find the vector magnitude and its direction?

Magnitude: Pythagorean theorem

Direction: angle between x-axis and vector

22yx AAA

x

y

A

Atan

x

y

A

Aarctan

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Vector Addition, Components

xxx BAR yyy BAR

Ax AA cos

Ay AA sin

Bx BB cos

By BB sin

BAR

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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

Check quadrant sign! Add individual components

algebraically to find components of the sum vector

Magnitude

Direction

Bx BB cos

By BB sin

yx RRR

... xxxx CBAR... yyyy CBAR

x

y

RR

arctanEVALUATE Check your results comparing them with the rough estimates!

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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

12999.792.9arctan

7.12)92.9()99.7( 22

mm

mmmR

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Unit Vectors

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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.

iAA xxˆ

jAA yyˆ

jAiAA yxˆˆ

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Unit Vectors

jBiBB yxˆˆ

jAiAA yxˆˆ

jRiRjBAiBA

jBiBjAiABAR

yxyyxx

yxyx

ˆˆˆ)(ˆ)(

)ˆˆ()ˆˆ(

kAjAiAA zyxˆˆˆ

kBjBiBB zyxˆˆˆ

kRjRiR

kBAjBAiBAR

zyx

zzyyxx

ˆˆˆ

ˆ)(ˆ)(ˆ)(