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
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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.
CL
AS
SIC
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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