What is Physics?

• Physical quantities, units and

unit standards

• Dimensions and dimensional

analysis

• Measurement and uncertainty;

significant figures

• Coordinate systems

• Solving physics problems: A

basic algebra refresher

Chapter 1

Introduction

A primer of scientific method

Physics represents reality using

physical quantities

Physical quantities reproduce the

coherence of reality

To confirm its hypotheses, Physics

must measure quantities

To express itself, Physics uses math

as a language in which physical

quantities are words and its grammar

reproduce the clockwork of reality

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

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

What is Physics? – The Conditions of Science

Observations: important first step towards formulating a scientific theory. Scientific

observations require educated simplifications of the observed reality meant to focus

on what is important given the goals of the scientific endeavor

Theories: formulated as hypotheses to explain observations and to conceptualize

various instances of nature. A scientific theory must be:

• capable to make predictions

• falsifiable

• always perfectible

Experiments: Systematic and intelligent tests resulting into data which will tell if the

theoretical predication is valid within the limits set by the experimental setup. Then

the cycle goes on.

What is Physics? – Branches of Classical Physics

PHYSICS is the is the fundamental physical science:

Mechanics – the study of motion of physical bodies in its causal emergence. This is

our focus in PHYS 107

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

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

electric charges

Optics – behavior and properties of light and its interaction with matter

Quantum and Relativistic Mechanics, and applications such as Nuclear,

Molecular, Solid State Physics, etc. 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

• Other unit systems can have significant, albeit local, importance:

Ex: cgs units are

𝐿 : centimeters (cm)

𝑀 : grams (g)

[T]: seconds (s)

British system has force (weight) instead of mass as one of its basic quantities:

[L]: feet (ft)

force: pounds (lb)

[T]: seconds (s)

• Since units are conventional, they can be easily converted using conversion factors

Units – Comments

Ex: 6.00 meters ~ 6.00 meters foot

3.28 meter

19.7 feet

Ex: micro → 10-6, mili → 10-3,

kilo → 103, mega → 106

• However, conversion should be performed consistently, such that all quantities

involved in an operation are specified by numbers with the same units

• When a certain quantity is very large or very small in direct SI units, it is

customary to use powers and ten and corresponding prefixes

Dimensions – Dimensional Analysis

• The dimension of a quantity is a characteristic of its physical nature, and is given by

the basic quantities that make it up; can be written using square brackets

Ex: Speed = distance / time

Dimensions of speed: [v] = [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 added 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 ?

• The motion of an object can be seen as the change of the object’s location in space

• In order to quantify the “location” we need a reference frame: a coordinate system

• A coordinate system (SC) structures out the space by introducing:

1. a reference point (origin) considered fixed,

2. a set of mutually perpendicular directions (axes) labeled and scaled

• Then the location of a point in space is given by a number of coordinates equal to

the number of directions, or the dimension of the space of interest:

• We employ the SC based on the geometry of the problem we have to analyze

• For instance, for a one-dimensional space (a line) one axis is sufficient:

Coordinate Systems – One-Dimensional

-10 -5 5 10

P (x = 9) Q (x = -7)

Negative x Positive x

7 units in negative

x-direction

9 units in positive

x-direction

• For a two-dimensional space (a plane) we need two parameters to describe a

location. Two popular types of coordinates are:

Coordinate System – Two-Dimensional

3 units

5 units

P (11,49o)

y

x

r

θ x

y

2 2r x y 1tany

x

b) polar coordinates (r,θ): useful in

rotational motions.

a) rectangular coordinates (x,y) (or

Cartesian): useful in linear motions

Quiz 3: What are the Cartesian

coordinates of point N?

N

Problems:

2. Rectangular Coordinates: Two points in a rectangular coordinate system have the

coordinates 𝑥1, 𝑦1 and 𝑥2, 𝑦2 .

a) Determine the distance between these points in terms of the coordinates

b) Use as an example points P and Q on the left frame of the previous slide.

3. Polar Coordinates: A toy car moves around a loop, as

in the figure.

a) What are the polar coordinates (𝑟, 𝜃) of the car in

points A and B?

b) What are the Cartesian coordinates of the car in an

arbitrary point (𝑟, 𝜃) in terms of the respective polar

coordinates?

𝑟

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

Mathematical refresher – Some algebraic rules

Ex: Try to isolate the term that contains your unknown on one side of the “=“ sign. If

the unknown is in two terms, factor it out. For instance, if x is the unknown and a, b, c

are some constants

isolate terms in factor out solve for x x x bax b cx ax cx b x a c b x

a c

• Recall how to solve simple systems of two simultaneous equations with two

unknowns:

Ex: Most of the time you can use the method of substitution. For instance, if x,y are

unknowns and a, b, c, d are some constants:

substitute factor out solve for and y x x y

dx

ax y b cabx cax d x b ca d

bx cy d ady

b ca

• Whatever you do to one side of the equation, you must do to the other

• If there are two terms that are added on one side of the equation, then subtract from

both sides of the equation to eliminate that term from that side of the equation.

• If two variable are multiplied, then divide (every term in the equation on both

sides) to eliminate that variable from that term

• Train your skills at rearranging terms in equalities to solve simple linear equations:

changed sign

divide by coefficient on both sides

divide by coefficient on

both sides

Mathematical refresher – Some standard symbols

• In our class we are going to encounter a few recurrent standard symbols:

𝑡 is an arbitrary time

𝑡0,1,2… are sequential times with respect to the same initial moment

Δ𝑡 = 𝑡1 − 𝑡0 would be a time interval

𝑥, 𝑦 are coordinates along x- and y-axes

Δ𝑥 and Δ𝑦 would be changes of coordinates

𝑣 is a speed or velocity

𝑣0,1,2… are sequential velocities at times 𝑡0,1,2…

Δ𝑣 would be a change of speed or velocity

𝑎 is an acceleration

𝐹 is a force

𝐹0,1,2… are several forces

𝑚 is a mass

The Delta symbol 𝚫 in front of a

quantity means a change of that quantity

between an initial and final value: Δ𝑐 = 𝑐𝑓𝑖𝑛𝑎𝑙 − 𝑐𝑖𝑛𝑖𝑡𝑖𝑎𝑙

Solving physics problems

... means really understanding a technique and the theoretical overlay and being able

to handle the necessary instrumental mathematics to reach a meaningful result

• Tentative steps:

• Identify the relevant concepts and the target variable

• Set up the problem by making simplifying assumptions, drawing diagrams and

graphical representations, and choosing the relevant equations

• Execute by solving equations. There may be new target variables (unknowns

which are not specifically required in the statement of the problem) to be

identified in order to identify the final one

• Evaluate by solving numerically, performing a dimensional analysis and trying

to see if the result makes sense physically – for instance by considering

particular cases or comparing numerical results to the common sense knowledge