Preamble: 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:

• Use symbols consistent 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

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

m is equal to a. A stray F/m followed by no operator doesn’t state anything!

•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:

F ma mv t= =

F m= a m= v t

Chapter 1 & Chapter 3

Introductory Formalism and Vectors

• What is Physics?

• Physical quantities

• Units. International System

• Dimensions and dimensional analysis

• Measurement and uncertainty. Significant figures

• Vectors:

• Components, unit vectors

• Vector addition: graphical and based on components

• Vector product: scalar and vectorial

•Modeling Motion

• Solving problems

•Motion diagrams

• Kinematic vectors

What is Physics? – The �ature of Science

Observations: important first step toward scientific theory; requires imagination to

tell what is important.

Theories: created to explain observations and to conceptualize various instances of

the Nature; will make predictions; must be falsifiable and always perfectible.

Experiments: Systematic and intelligent observations organized into data which

will tell if the prediction is accurate within some limits. Then the cycle goes on.

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

method developed primarily during the last few centuries. It employs systematically:

Physics (from the Greek, φύσις (phúsis), "nature") is the fundamental physical

science concerned with the structured understanding of the underlying principles of

the natural world. Physics deals with the elementary constituents of the Universe, that

is, all classes of matter and energy, and their interactions, as well as the analysis of

systems which are best understood in terms of these fundamental principles.

Branches of Classical Physics:

• Mechanics – the study of motion of physical bodies in its causal emergence:

classical (or Newtonian) mechanics is just the macroscopic limit of quantum

mechanics, and the small speed limit of relativistic mechanics

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

• Electricity and Magnetism – the study of electric and magnetic fields

Physical Quantities – Basic quantities

• 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

Length [L]

Time [T]

Mass [M]

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

• What are “units”?

We shall be working in the SI system, where the basic units are kilograms [kg],

meters [m], and seconds [s].

Physical Quantities – Standards of units

• 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)

Gaussian System (cgs)

British System

Quiz 1: Why does the SI provide units only for mass, length and time?

Physical Quantities – the International System

Quantity Unit Standard

Length [L] Meter, mLength of the path traveled by light in

1/299,792,458 second.

Time [T] Second, sTime required for 9,192,631,770 periods of

radiation emitted by cesium atoms

Mass [M] Kilogram, kg

Platinum cylinder in International

Bureau of Weights and Measures,

Paris

Système International - SI

• 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

force: pounds

[T]: seconds

• Since units are conventional, they can be easily converted

Physical Quantities – Writing and converting units

Ex: 6.00 meters ~ 6.00 meters( ) feet3.28

meters× 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

Physical Quantities – 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.

Problem:

1. Dimensional analysis: The speed, v, of an object is given by the equation

where t is time. What are the dimensions of the quantities A and B?

2Av Bt

t= +

• Uncertainties are inherent to any measurement

• Every measuring tool is associated with an uncertainty which can be used to specify

the instrument’s accuracy

Ex: Time = 1.67 ± 0.05 s Length = 1.2 ± 10% m

• The uncertainty can be indicated using significant figures

Ex: Mass = 148 kg⇒ Uncertainty ≈ ± 1 kg

Speed = 2.21 m/s (3 s.f.) ⇒ Uncertainty ≈ ± 0.01 m/s

Speed = 0.21 m/s (2 s.f.) ⇒ Uncertainty ≈ ± 0.1 m/s

• Writing out the numbers in scientific notation clearly delineates the correct number

of significant figures:

Physical Quantities – Measurement and Uncertainty. Significant Figures

Physical Quantities – Derived significant figures

• Caution: 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) the calculator shows the result of 2.0 / 3.0

b) the calculator shows the result of 2.50 × 3.2

Quiz 2: What should be the answers with the correct

number of significant figures in the two cases ?

Vectors – Definition and representation

Scalars are physical quantities completely described only by their magnitude.

Ex: time, mass, temperature, etc.

Vectors describe physical quantities having both magnitude and direction.

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

magnitude

θ

θordirection

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

V

Vr

Vor

y

x

y

x

Vector – 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:

Vr

V

2Vr

2V

2V−r

2V

Vr

V ˆV Vv≡r

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

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

1vr

+ 2vr

3vr

+ = Rr

1 2 3 1 2 3 6 3 5 4R v v v v v v= + + ≡ + + − = ++ − =r r r r

1v v≡ +r

2v v≡ −rv 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:

1vr

+ 2vr

3vr

+ =Rr

1vr

2vr

3vr

1 2 3R v v v= + +r r r r

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

initial

final

netdr

1dr

2dr

3dr

• 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= + +r r r r

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.vr

v−r

1vr

_

2vr

= Rr

1vr

2v−r

( )1 2 1 2R v v v v= − = + −r r r r r

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

=

1vr

+

2v−r

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

position

reference

2 1r r r∆ = −r r r

initial

1rr

final

2rr

r∆r

( )2 1r r r∆ = + −r r r

1r−r

r∆r

• If we denote r1 and r2 two

positions successively

occupied by a moving

objects, the displacement is2rr

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

Vr

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

θ

θ −

= +

= ⇒=

= +⇒

=

r r r

θ

�otation: ( ),x yV V V≡r

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 j=r

vector component

component

ˆx xV V i=r

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: ,

• 2-D case:

i

j

( )ˆˆ ˆ, ,i j k ( )ˆ ˆ ˆ, ,x y z

( )ˆ ˆ ,x y x y x yV V V V i V j V V= + = + ≡r r r

Vr

magnitude2 2

x yV V V= + direction 1tan y xV Vθ −=

Vectors – Components and unit vectors in 3D

y

Vr

ˆx xV V i=r

ˆy yV V j=r

i j

( )2 2 2

ˆˆ ˆ , ,x y z x y z x y z

x y z

V V V V V i V j V k V V V

V V V V

= + + = + + ≡

= + +

r r r r

z

x

k

ˆz zV V k=r

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 i V j V i V j V i V jR = + + + = + + + + + +

r r rr

( ) ( )1 2 1 2ˆ ˆ ˆ ˆ .. .. x x n y nyx x yyi jR V V V i V VR V j+ = + + + + + + +

2 2

x yR R R= +

( )1tan y xR Rθ −=

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= +rr r

Problem

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

resultants

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

a) Graphical method

b) Vector components

1R A B= +rr r

Ar

Br

30θ = °

4

5

Ar

Br

2 2R A B= −rr r

Vectors – Types of products

• There are two types of product of vectors, each with a specific applicability:

1. Scalar product → results in a number

2. Vector product → results in a vector

• Unlike in the case of vector addition, vectors with different physical nature

can be multiplied via either of the two products. The result will be a different

physical quantity.

Ex: a) The energy exchanged by the action of a force is called work and is given

by a scalar product

b) The rotational effect of applying a force is determined by torque – a vector

product:

W F r= ⋅r r

r Fτ = ×rr r

• Results in a scalar

• $otation:

“dot product”

• Definitions:

• Interpretation: the scalar product between two vectors is the product between any

of the vectors and the projection (component) of the other vector along the first one:

or

Vectors – Dot product

( ), ,x y zA A A A=r

( ), ,x y zB B B B=r

θ

A B⋅r r

cosA B AB θ⋅ =r r

x x y y z zA B A B A B A B⋅ = + +r r

Ar

Br

θ

cosA θ

Given vectors:

Ar

Br

θ

cosB θcosA B BA θ⋅ =

r rcosA B AB θ⋅ =

r r

Problem:

3. Application of dot product: Consider the following pair of vectors:

a) Sketch the vectors in the system of coordinates corresponding to the given components

b) Calculate the angle between the two vectors

( )2.00,6.00A = −r

( )2.00, 3.00B = −r

Application: the scalar product can be used to find the angle between two vectors

with given components:

3D:

2D:

2 2 2 2 2 2

2 2 2 2

arccos

cos

arccos

x x y y z z

x y z x y z

x x y y

x y x y

A B A B A B

A A A B B BA B

ABA B A B

A A B B

θ

θ

θ

+ + = + + + +⋅

= ⇒ + = + +

r r

• Results in a vector

• $otation:

“cross product”

• Definitions:

or, in determinant form,

A B×r r

sinA B AB θ× =r r

( ) ˆy z z yA B A B iA B× = − +

r r

( ) ˆz x x z jA B A B− +

( ) ˆx y y xA B A B k−

ˆˆ ˆ

x y z

x y z

A A A

A B B B B

i j k

× =r r (No problem if don’t know

how to handle this; use the

previous form!)

( ), ,x y zA A A A=r

( ), ,x y zB B B B=r

θ

Given vectors:

Vectors – Cross product

Vectors – More about Vector Product…

• Interpretation for the magnitude: the

magnitude of the cross product between

two vectors is the product between one of

the vectors and the component of the other

vector perpendicular on the first one:

Ar

Br

θ

sinA θ

Ar

Br

A B×r r

Ar

Br

B A×rr

• Comment: Notice that, unlike the dot product, the vector product is not commutative

A B B A× = − ×r rr r

sinA B BA θ× =r r

• Direction: given by a right-hand rule: Align your fingers along the first vector such

that you can curl them toward the second vector. The vector product is perpendicular

onto the plane of the vectors in the direction indicated by the thumb

outin

Problems:

4. Cross product of unit vectors: For a concrete feel of what unit vectors and the cross

product are, cross-multiply the unit vectors (i, j, k) between them.

5. Product vector operations: Consider again the vectors

a) Calculate the magnitude of the vector using

- first the scalar product

- and then the component definition of the vector product .

b) Calculate

( )2.00,6.00A = −r

( )2.00, 3.00B = −r

A B×r r

A B⋅r r

( )A A B⋅ ×r r r

Modeling Motion – 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

•Mechanics is the science of motion: this

semester we’ll focus on two types of

motion: translational and rotational

•Most of the time, the motion of a certain

object is better analyzed by splitting a

more complex motion into simple ones,

and by making simplifying assumptions

Modeling Motion – Simplifying assumptions

Ex: Projectile motion is a

2D translational motion

Ex: A spinning top

performs rotational motion

• Two simplifying assumptions that we’ll consistently make are:

1. When analyzing translational motion, moving objects will be assumed point-like:

objects with mass concentrated in their centers of mass, that is, they are particles

2. When analyzing rotational motion, moving objects will be assumed as being made

of rigidly bounded particles, that is, they are rigid bodies

• Other simplifying assumptions will be more particular: gravity in proximity of the

Earth’s surface will be considered constant, sometimes surfaces will be assumed very

smooth in order to neglect friction, in some introductory systems pulleys will be

considered as very light, springs will be also light and perfectly elastic, etc.

• These assumptions can be successfully removed when building more complex models

Modeling Motion – Motion diagrams

•Motion can be made easier to visualize and solved by representing it schematically

• The particle model in combination with the vector formalism are the backbone of

motion diagrams:

1. The motion is represented by dots at the particle location at equal time intervals

2. Then the motion is characterized using motion vectors in every location: position,

displacement, velocity, acceleration

Ex: Say that a car moving in a

straight line is filmed using a

camera taking snapshots at equal

time intervals

• Each frame is going to show the

car at a different location

•When combined, the frames

merge into a motion diagram

• In case we want to use the

diagram to describe the translation

of the car, it can be simplified by

replacing the car with a particle

and the ground with an axisx

Modeling Motion – Position and Displacement

How do we characterize the motion using kinematic vectors?

• Kinematic vectors:

1. Position, r: a vector connecting a point of reference with the location of the particle

reference

point3rr

2rr

0rr

position vectors shifted

vertically for visibility1rr

2. Displacement, ∆r: vector difference between two positions

• Displacement can be represented as a vector connecting an initial to a final location

• The dots on a motion diagram can be connected by successive vector displacements,

with the net displacement being the vector sum of partial displacements

1 1 0r r r∆ = −r r r

2 2 1r r r∆ = −r r r

3 3 2r r r∆ = −r r r

Ex: In the case of

the car moving in

a straight line:

Ex: In the case of the car moving in

a straight line, the successive

displacements are equal

1 2 3r r r r∆ = ∆ + ∆ + ∆r r r r

Ex: Projectile motion

final initialr r r∆ = −r r r

Modeling Motion – Velocity

• The average velocity between two locations on the trajectory has the same direction

as the associated displacement, so the successive velocities can be represented using a

similar succession of vectors

• If the displacement changes from segment to segment, the average velocity will

change accordingly

_1 _ 2 _ 3 avg avg avgv v vr r r

Ex: In the case of the car moving in a straight line, the

average velocities between any two successive

locations is the same, equal to the average velocity in

the whole interval

3. Average Velocity, vavg: a vector given by the displacement

divided by the elapsed time

_1 _ 2 _ 3avg avg avg avg

rv v v v

t

∆= = = =∆

rr r r r

Ex: Say that an object slides down an incline. Since it

speeds up, it covers larger and larger displacements in

equal intervals of time

_1 _ 2 _ 3 _ 4avg avg avg avgv v v v< < <r r r r

_1avgvr

_ 2avgvr

_ 3avgvr

_ 4avgvr

avg

rv

t

∆=∆

rr

Modeling Motion – Acceleration

• So, the acceleration is a measure of the change in velocity

• Since velocity is a vector, it can change in magnitude, in direction, or in both of them

• The direction of the acceleration is given by the direction of the change in velocity:

1. If the particle slows down in a line, the acceleration opposes the velocity

2. If the object speeds up in a line, the acceleration has the same direction as velocity

3. If an object accelerates in plane, the acceleration is inside the curvature of the

trajectory

• On the motion diagrams, the direction of the acceleration can be estimated by

building graphically the vector difference between two successive velocities

3. Average Acceleration, aavg: a vector given by the change in

velocity divided by the elapsed time

Ex: Object sliding down an incline. The

acceleration is a vector down the incline

1vr

2vr

3vr

2v−r

3 2v v v∆ = −r r r

acceleration has the

direction of ∆v

Ex: Projectile motion. The acceleration is a

vector vertically downward

1vr

2vr

2vr

1v−r

2 1v v v∆ = −r r r

avg

va

t

∆=∆

rr

Problem:

6. Motion diagram: A skier slides down the constant slope of a hill, then moves on a flat

surface, then up the constant slope of another hill. Sketch qualitatively the motion diagram

of the skier’s translational motion using average velocity and acceleration vectors.