Chapter 1 & Chapter 3

Introductory Formalism and Vectors

• What is Physics?

• Representing and manipulating physical quantities

• Units. International System

• Dimensions and dimensional analysis

• Measurement and uncertainty. Significant figures

• Refresher of math formalism

• Vectors:

• Definition

• Components, unit vectors

• Vector addition: graphical and based on components

What is Physics? – Scientific Method. Branches of Classical Physics

SCIENCE is the activity for acquiring and organizing knowledge based on the scientific

method. Both in its physical and social forms, it employs systematically:

Observations: important first step toward scientific theory; require educated simplifications to

focus on what is important given the goals of the scientific study

Theories: formulated as hypotheses to explain observations and to conceptualize various

instances of nature. Must be: able to make predictions, falsifiable, and always perfectible

Experiments: Systematic tests of hypotheses, resulting into data which will tell if the

theoretical predications are valid within experimental limits

PHYSICS is the is the fundamental physical science:

Mechanics – the study of motion of physical bodies in its causal emergence. (PHYS

154)

Thermodynamics – the balance of heat, work and internal energy of an object

(PHYS 254)

Electricity and Magnetism – the study of the effects of the presence and motion of

electric charges (PHYS 155)

Optics – behavior and properties of light and its interaction with matter (PHYS 255)

Quantum and Relativistic Mechanics, and applications such as Nuclear,

Molecular, Solid State Physics, etc. (PHYS 255) Classical mechanics is just the

macroscopic limit of quantum mechanics and the small speed limit of relativistic

mechanics.

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What is Physics? – Structure of Matter

• Matter is made up of molecules

Molecules are made up of atoms

Atoms are made up of

1. Nucleons:

- protons, positively charged, “heavy”

- neutrons, no charge, about same mass as protons

- Nucleons are made up of quarks

Quarks may also have a structure

2. Electrons:

- negatively charges, “light”

- fundamental particle, no structure

• As the substance is probed deeper and

deeper, the matter obeys the laws of

quantum mechanics – a generalized

mechanics which at “macro” scales

becomes the Newtonian mechanics we’ll

be studying for most of this semester

Physical Quantities – Basic vs Derived

• Physics is an experimental science, that is, any of its statement must be verifiable

via an organized test upon nature.

• During an experiment one measures physical quantities

Ex: mass, length, time, temperature, current, etc.

• The physical quantities describe an objective reality

• Some quantities are considered as basic physical quantities: for instance, in

mechanics

are considered basic since the other physical quantities are derived from them

Ex: velocity, acceleration, energy, momentum, etc.

• Consequently, the units for the derivable quantities can be expressed in units of

length, mass and time

But what are “units”?

Quantity Notation

Length 𝐿, 𝑙

Mass 𝑀,𝑚

Time 𝑇, 𝑡

Quiz 1: Basic quantities: Why

do you think these particular

quantities are fundamental in the

material universe?

Units – Standards

• Any measurement makes necessary a standardized system of units.

Ex: kilograms, slugs, meters, inches, seconds, hours etc.

• Defining units allows a consistent way of providing numerical values for physical

quantities measured in an experiment

• The unit standardization is just a convention agreed upon by some authority.

• Examples of unit standards:

Système International (SI) (International System)

Gaussian System (cgs)

British System

In our course we shall be working

exclusively in the SI (MKS) system,

where the basic units are tabulated

as following:

Quantity Unit Notation

Length meter 𝐿 𝑆𝐼 → m

Mass kilogram 𝑀 𝑆𝐼 → kg

Time second 𝑇 𝑆𝐼 → s

Quantity Unit Standard

Length [L] Meter, m Length of the path traveled by light in 1/299,792,458 second.

Time [T] Second, s Time required for 9,192,631,770 periods of radiation emitted by

cesium atoms

Mass [M] Kilogram,

kg

Formerly: platinum cylinder (International Prototype IPK) kept

in the International Bureau of Weights and Measures, Paris

Currently (2019): The kilogram will be related to a fixed value

for Planck's constant h, a fundamental quantity of quantum

physics. It will be measured using an electromagnetic Kibble

balance

Units – Definitions of basic units

Système International - SI

Old SI New SI Comments:

• When using the inexorably changing

International Prototype, the integrity

of the SI system was affected

• The system of unit dependencies was

reformatted in the new system such

that the kg-definition depends on the

definition of length and time units

Dimensions and Dimensional Analysis

• The dimension of a quantity is given by the basic quantities that make it up; they

are generally written using square brackets

Ex: Speed = distance / time

Dimensions of speed: [L]/[T]

• Quantities that are being added or subtracted must have the same dimensions

• Any physical equation must always be dimensionally consistent (i.e. all terms must

have the same dimension)

• A quantity calculated as the solution to a problem should have the correct

dimensions. This can be used to verify the necessary (but not sufficient) validity of

a certain result

Quiz 2: Dimension of derived quantity: The mass density of an object is defined as the

mass of the object (quantity of substance) per the stretch of space occupied by the substance.

For instance, volume density ρ rho) of an object is given by 𝜌 = 𝑚𝑎𝑠𝑠 𝑣𝑜𝑙𝑢𝑚𝑒

Which of the following represents the dimension of this density in terms of basic quantities?

a) kilograms/meters

b) mass/meters cubed

c) mass/length cubed

d) weight/length cubed

Problem:

1. Dimensional analysis: A student solves a physics problem trying to find the speed of an

object. She winds up with the speed, v, of the object given by equation

where t is time. Her teacher tells her that the equation is correct dimensionally.

a) What are the dimensions of the quantities A and B? What are their respective units? Can

you dare to suggest what could be the physical nature of these quantities?

b) Has the student solved the problem correctly?

2Av Bt

t

Measurement and Uncertainty – Significant Figures

• No measurement is exact; there is always some uncertainty due

to limited instrument accuracy and difficulty reading results

• Every measuring tool is associated with an uncertainty which

can be used to specify the instrument’s accuracy

Ex: The width of a plank cannot be measured to better than a 1 mm both

due to the roughness of the edge and the accuracy of the instrument used

• The uncertainty can be indicated using the number of significant figures: the

reliably known digits in a number directly or indirectly measured

• Then, one knows the uncertainty in the physical quantities given numerically

Ex: 23.21 cm = 2.321×10-1 m has 4 significant figures

0.062 cm = 6.2×10-4 m has 2 significant figures (the initial zeroes don’t count)

Ex: Mass = 148 kg (3 s.f.) Uncertainty ≈ ± 1 kg

(mass between 147 and 149 kg)

Speed = 2.2 m/s (2 s.f.) Uncertainty ≈ ± 0.1 m/s

(speed between 2.1 and 2.3 m/s)

• Writing out the numbers in scientific

notation helps delineate the correct

number of significant figures:

Measurement and Uncertainty – Derived values

• N.B calculators will not give you the right number of significant figures; they

usually give too many but sometimes give too few (especially if there are trailing

zeroes after a decimal point)

• Results of products or divisions retain the uncertainty of the least certain term

• Results of summations or subtractions retain the least number of decimal figures

• Numeric integer or fractional coefficients in equations have no uncertainty.

Ex: 255 × 2.5 = 640 Uncertainty ≈ ± 10

7.68 + 5.2 = 12.9 Uncertainty ≈ ± 0.01

Ex: A calculator will provide wrong significant

figures for the result of operations such as

2.0 / 3.0 or 2.5 × 3.2

Quiz 3: What should be the answers with the correct

number of significant figures in the two cases ?

Mathematical refresher – Mind your language…

• The idiom of this physics course will be a mixture of natural language and algebraic

formalism requiring a certain attention. So, treat your algebra with the same respect that you

offer to your everyday parlance. Here is an indispensable albeit incomplete list of

requirements:

• Try to use symbols consistently throughout your solution, and avoid using the same symbol

for different quantities in the same argument

• Adapt the generic equations to the language of the problem and always show symbolic

expressions before feeding in the numbers

Ex: 𝐹 = 𝑚𝑎 is a generic formula for force. If in a problem two masses 𝑚1,2 are acted by

forces 𝐹1,2, write distinct expressions: 𝐹1 = 𝑚1𝑎1 and 𝐹2 = 𝑚2𝑎2

• Avoid using numbers in algebraic manipulations. Carry out your argument using symbols and

feed the numbers only in the final expression

• Build your arguments in clear, complete, and meaningful sentences

• Make sure that the terms on both sides of the “=“ sign are indeed equal, including all terms in

a chain of equalities. For instance, make sure that simplifying terms on two sides of one

equality in a chain doesn’t falsify another equality in the chain

Ex: This succession of equalities may be true:

… but it becomes false if you simplify carelessly:

Ex: 𝐹 𝑚 = 𝑎 is a formally correct statement meaning that the ratio between 𝐹 and 𝑚 is

equal to 𝑎. However, a stray 𝐹 𝑚 followed by no operator doesn’t state anything!

F ma mv t

F m a m v t

Vectors – Definition and representation

Scalars are physical quantities completely described only by a number.

Ex: time, mass, temperature, etc.

Vectors describe physical quantities having both magnitude and direction.

Ex: position, displacement, velocity, acceleration, force, etc.

magnitude

θ

θ or direction

direction

• The direction of a vector depends on the arbitrary system of coordinates

• However, the magnitude does not depend on how you choose to span the space

N

𝑉 or V

y

x

y

x

Vectors – Properties

• Vectors can be added or subtracted in any order, but, if the vectors represent

physical quantities, they must have the same nature

• Multiplying a vector by a positive number multiplies its magnitude by that

number (if the number is negative the vector flips in the opposite direction):

• Therefore, any vector can be written as a number (its magnitude) times a unit

vector with the direction of the vector:

V

V

2V

2V

2V

2V

V

V ˆV Vv v

ˆ 1v

ˆVv

unit vector

• The simplest physical situations that we are going to encounter will involve vectors

along the same straight line, such that they can have only two directions which can be

arbitrarily considered as “negative” and “positive”

• Moreover, for simplicity, the arrows on top of the symbols can be dropped:

• In these cases, vectors can be added graphically and as numbers:

Vectors – One Dimensional

Ex: Say that we have 3 arrows (vectors) along the same line (1D) with magnitudes provided

in arbitrary units on the diagram, and we want to add them:

• Graphically, chain the vectors tail to tip: the resultant connects the tail of the first on to the

tip of the last on in the chain

• Numerically: add together the vectors represented by their respective magnitude and the sign

1v + 2v 3v+ = R

1 2 3 1 2 3 6 3 5 4R v v v v v v

1v v 2v v v v

6 3 5

this sign means “equivalent to” not “equal”: never use an equal sign

between a vector and a number

A vector of magnitude 4 units

pointing to the right

• In general, even if the vectors are not along the same axis, they can be added

graphically by using the same tail-to-tip method:

Vectors – 2D Vector Graphical Addition

• The method offers a qualitative idea about the resultant: in order to obtain the

resultant numerically (magnitude and direction), one has to use scaled grid paper

which is a rather cumbersome technique

The vector sum can be obtained graphically by chaining the vectors each with the

tail to the tip of the previous: then the vector resultant connects the tail of the first

vector to the tip of the last one. The operation can be done in any order.

Ex: Say that we have 3 arrows (vectors) in a plane (2D) and we want to add them up:

1v + 2v 3v+ = R

1v2v

3v

1 2 3R v v v

Notice that in 2D, the arrows above the vector symbols cannot be skipped since a vector can

have an infinity of directions not only two as in the 1D case: the operation between the arrows

cannot be reduced to an immediate algebraic addition or subtraction

Ex: Physical example: Successive 2D displacements can still be added to obtain the

total displacement

• An application of vector

summation in mechanics is

calculating the net

displacement of an object

traveling from an initial

position to a final one via

several successive partial

displacements

• If we denote d1, d2 and d3

three successive displacements

the net displacement is

• It is given by the vector sum

(or resultant) of the partial

displacements

• Notice that adding the partial

displacement follows the logic

of tail-to-tip method

1 2 3netd d d d

initial final

netd

1d

2d

3d

Vectors – Graphical Subtraction

• In order to subtract vectors, we can still use the addition procedure by adding the

negative of the arrow being subtracted

• We define the negative of a vector to be a vector with the

same magnitude but pointing in the opposite direction. v v

1v

_

2v

= R

1v

2v

1 2 1 2R v v v v

Ex: Say that we have 2 arrows (vectors) in a plane and we want to subtract them:

=

1v

+

2v

Ex: Physical example: linear displacement is defined as the final position minus the initial

position

2 1r r r

2 1r r r

• If we denote r1 and r2 two

positions successively

occupied by a moving

objects, the displacement is

reference

initial

1r

final

2r

r

Vectors – Components

• Note that, in order to obtain magnitudes and directions, the graphical methods

should be used on grid paper.

• A more computational way to get magnitudes and directions is by using vector

components in arbitrary systems of coordinates:

y

x

V

xV

yV

2 2

1

cos Components from

sin direction and magnitude

Direction and magnitude

from componentstan

x y

x

y

x y

y

x

V V V

V V

V V

V V V

V

V

θ

Notation: ,x yV V V

Caution: The components are not are not vectors or vector magnitudes. They can be

negative if the corresponding vector components point in the negative direction of the

respective axis.

y

x

ˆy yV V y

vector component

component

ˆx xV V x

Vectors – Axial unit vectors

• For any system of coordinates (1-D, 2-D or 3-D), one can use unit vectors to define

positive “directions” pointing along the axes.

• Popular notations: 𝑖 , 𝑗 , 𝑘 or, more intuitively, 𝑥 , 𝑦 , 𝑧

• 2-D case:

ˆ ˆ ,x y x y x yV V V V x V y V V

V

magnitude 2 2

x yV V V direction 1tan y xV V

y

x

Quiz 4: Characterizing a position: Recall that the position of an object is a vector

connecting a reference (such as the origin of a coordinate system) to the location of the object.

Which of the following does not represent the position of the point P on the figure?

P

r = 11 m

θ = 49o

a) (11 m, 49o)

b) (7 m, 8 m)

c) 7𝑥 + 8𝑦

d) about eight meters above the turtle

e) all of the above

Vectors – Addition and subtraction using vector components

• The addition and subtraction of vectors can be reduced to addition and subtractions

of components

• Recall that the components depend on the system of coordinates, so the operation

first demands picking a SC. However, the resultant will be the same in any SC.

• Given n coplanar vectors, the addition can be solved in 2D as following:

1 2 1 1 2 2ˆ ˆ ˆ ˆ ˆ ˆ... ...n x y x y nx nyV V V V x V y V x V y V x V yR

1 2 1 2 .. ˆ.. ˆ ˆ ˆx x n y nyx x yyx yR V V V V VR Vx y

2 2

x yR R R

1tan y xR R

magnitude:

angle with respect to positive x:

Ex: The procedure can be visualized graphically: the

components (Rx, Ry) of the resultant R are aligned with

the components of the vectors involved so they can be

added as numbers

R A B

Problem

2. Operating with vectors: Given the two vectors in the figure, find the following vector

resultants

where 𝐴 and 𝐵 are vectors with magnitudes 4 and 5 units respectively, by using

a) Graphical method

b) Vector components

1 2R A B

A

B

30

4

5

2 2R A B