Chapter 1: Measurements in Physics – Models, Units, and Analysis

What is Physics?

Introduction to Physics

Physics is a natural science that studies the universe, focusing on matter, energy, space, and time. It seeks to understand the fundamental laws and patterns that govern natural phenomena.

• Patterns in nature are described by models and theories.

• Well-established theories become physical laws or principles.

• Major branches of physics include Mechanics, Electromagnetism, Quantum Physics, Thermodynamics, Optics, and Relativity.

Physics ≡ Nature: Physics describes phenomena, relationships, and equations that explain the workings of the natural world.

Models, Theories, and Laws

Scientific Reasoning in Physics

• Models are simplified representations that help in understanding phenomena. They create mental pictures but have limitations.

• Theories are broader and more detailed than models, providing testable predictions and explanations.

• Physical Laws are concise statements describing how nature behaves under certain conditions.

• Principles are similar to laws but apply to a wider range of phenomena.

Example: The atomic model helps visualize atoms, while Newton's laws provide mathematical descriptions of motion.

Measurement

Quantifying Physical Phenomena

To describe natural phenomena, physicists make measurements of various physical quantities such as mass, length, time, and temperature.

• Each measurement consists of a number (magnitude) and a unit.

• Example: L = 2.3 m (quantity: length, number: 2.3, unit: meter).

• Physics is an experimental science that relies on accurate measurements.

Units

The SI System and Base Quantities

To ensure consistency, scientists use the SI (Système International d'Unités) system, adopted globally in 1960. The SI system defines seven base quantities and their corresponding base units:

Base Quantity	Base Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	Ampere	A
Temperature	Kelvin	K
Amount of substance	mole	mol
Intensity of light	candela	cd

Other quantities (e.g., velocity, area) are derived quantities formed by combining base units (e.g., m/s, m2).

SI Units in Mechanics

Quantity	SI Unit	Other Units	Definition of SI Unit
Length	meter (m)	centimeter (cm), foot (ft)	The meter is defined as the distance traveled by light in vacuum during a time interval of 1/299,792,458 of a second.
Mass	kilogram (kg)	gram (g), slug	The kilogram is defined as the mass of a specific platinum–iridium alloy cylinder.
Time	second (s)	minute (min), hour (h)	The second is the time taken by 9,192,631,700 oscillations of the radiation emitted by the cesium atom.

Scientific Notation and Prefixes

Expressing Large and Small Numbers

• Scientific notation is used to write very large or very small numbers in the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

• Example: $0.0015$ cm = $1.5 \times 10^{-3}$ cm; $20,000$ m = $2.0 \times 10^4$ m.

• Scientific notation helps maintain the correct number of significant figures.

SI Prefixes

Prefixes are used to represent powers of ten for units:

Prefix	Symbol	Multiplying Factor
Giga-	G	$10^9$
Mega-	M	$10^6$
Kilo-	k	$10^3$
Centi-	c	$10^{-2}$
Milli-	m	$10^{-3}$
Micro-	μ	$10^{-6}$
Nano-	n	$10^{-9}$
Pico-	p	$10^{-12}$
Femto-	f	$10^{-15}$

Significant Figures

Precision in Measurement

• Significant figures are the digits in a measurement that are known with certainty plus one estimated digit.

• Rules for significant figures:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros after a decimal point are significant.

  • Trailing zeros without a decimal point are not significant.

• Example: 0.008 (1 significant figure), 3.0027 (5 significant figures), 5700 (2 significant figures if no decimal point).

Calculations with Significant Figures

• Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

• Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: $4.75 - 0.4168 = 4.33$ (rounded to 2 decimal places).

Converting Units

Changing Between Units

• Unit conversion involves multiplying by conversion factors to change from one unit to another.

• Example: $25$ mm = $0.025$ m; $0.02$ kg = $20$ g.

• For time: $2$ days = $172,800$ s (since $2 \times 24 \times 60 \times 60 = 172,800$).

• For area: $880$ ft2 = $81.75$ m2 (using $1$ ft = $0.3048$ m).

• For speed: $55$ mi/h = $24.6$ m/s (using $1$ mi = $1609$ m).

Dimensional Analysis

Checking Equations and Units

• Dimensional analysis uses the dimensions of physical quantities to check the correctness of equations and to derive units.

• Common dimensions: Length [L], Mass [M], Time [T].

• Example: Speed has dimensions $[L][T]^{-1}$; Area has dimensions $[L]^2$.

• Equations must be dimensionally consistent: both sides must have the same dimensions.

Examples of Dimensional Analysis

• Kinetic energy: $KE = \frac{1}{2}mv^2$ has units $kg \cdot m^2/s^2$.

• Newton's second law: $F = ma$ has units $kg \cdot m/s^2$.

• Gravitational constant $G$ in $F = G\frac{Mm}{r^2}$ has units $m^3 \cdot kg^{-1} \cdot s^{-2}$.

Additional info: Dimensional analysis is also used to derive relationships between physical quantities and to convert between units in complex equations.