Introduction to Physics: Fundamental Concepts and Tools (Chapter 1 Study Notes)

Introduction to Physics

What is Physics?

Physics is the study of the fundamental laws of nature. These laws can be expressed in the form of mathematical equations, allowing us to describe, predict, and understand the behavior of the physical universe.

• Mechanics: The study of motion and its causes.

• Kinematics: The study of the laws that describe motion.

• Dynamics: The study of the laws of causes of motion.

Fundamental Physical Quantities of Mechanics

Physics relies on a set of fundamental quantities, each with a defined dimension, SI unit, and abbreviation:

Quantity	Dimension	SI Unit	Abbreviation
Length	[L]	Meter	m
Mass	[M]	Kilogram	kg
Time	[T]	Second	s

Mass vs. Weight

• Mass: The amount of matter comprising an object (measured in kilograms).

• Weight: The force of gravity acting on an object (measured in newtons).

• Note: Mass and weight are not the same; mass is an intrinsic property, while weight depends on gravitational acceleration.

SI Prefixes

Standard prefixes are used to designate common multiples in powers of 10:

Power	Prefix	Abbreviation
1012	tera	T
109	giga	G
106	mega	M
103	kilo	k
10-2	centi	c
10-3	milli	m
10-6	micro	μ
10-9	nano	n
10-15	femto	f

Dimensional Analysis

Dimensions and Physical Quantities

The dimension of a physical quantity refers to its type, denoted by square brackets (e.g., [L] for length).

Quantity	Dimension
Distance (and Position)	[L]
Area	[L]2
Volume	[L]3
Velocity	[L]/[T]
Acceleration	[L]/[T]2
Energy	[M][L]2/[T]2

Dimensional Consistency

Physics equations must be dimensionally consistent: each term in an equation must have the same dimensions.

Example 1: Dimensional Consistency

Show that the equation $x - x_0 = v_0 t + \frac{1}{2} a t^2$ is dimensionally consistent.

• $x, x_0$ have dimension [L]

• $v_0 t$ has dimension $[L][T^{-1}][T] = [L]$

• $a t^2$ has dimension $[L][T^{-2}][T^2] = [L]$

All terms have dimension [L]; the equation is dimensionally consistent.

Example 2: Dimensional Consistency

Show that $v^2 = v_0^2 + 2a(x - x_0)$ is dimensionally consistent.

• $v^2, v_0^2$ have dimension $([L][T^{-1}])^2 = [L]^2[T]^{-2}$

• $a(x - x_0)$ has dimension $[L][T^{-2}][L] = [L]^2[T]^{-2}$

All terms have dimension $[L]^2[T]^{-2}$; the equation is dimensionally consistent.

Significant Figures and Scientific Notation

Uncertainty in Measurement

• Every measurement has an associated uncertainty, indicating its accuracy.

• Absolute Uncertainty: The margin of error in a measurement (e.g., $21.5 \pm 0.1$ cm).

• Percentage Uncertainty: $\frac{\text{Absolute Uncertainty}}{\text{Value of Quantity}} \times 100\%$

• Example: $\frac{0.1\ \text{cm}}{21.5\ \text{cm}} \times 100\% = 0.5\%$

Significant Figures

• The number of digits reliably known in a physical quantity.

• Examples:

  • 2.56 s → 3 significant figures

  • 0.000256 s → 3 significant figures

  • 21.5 m → 3 significant figures

  • 1.3 kg → 2 significant figures

  • 1.30 kg → 3 significant figures

Rules for Significant Figures

• Multiplication/Division: The result should have as many significant figures as the least accurately known quantity.

• Addition/Subtraction: The result should have as many decimal places as the term with the fewest decimal places.

Examples

• Multiplication/Division: $\frac{107.2\ \text{m}}{16.3\ \text{s}} = 6.58\ \text{m/s}$ (3 significant figures)

• Addition: $6.15 + 4.59 + 2.5 = 13.2$ (1 decimal place)

Scientific Notation

• Used to avoid ambiguity in the number of significant figures, especially with zeros.

• Format: Number between 1 and 10 × Power of 10 (e.g., $3.412 \times 10^3$ m)

Examples

• 3412 m → $3.412 \times 10^3$ m (4 significant figures)

• 16,006 m/s → $1.6006 \times 10^4$ m/s (5 significant figures)

• 0.000389 s → $3.89 \times 10^{-4}$ s (3 significant figures)

• 1200 kg → $1.2 \times 10^3$ kg (2 significant figures if zeros are not significant)

Converting Units

Unit conversion is often necessary in physics. Use unit conversion factors to change from one unit to another.

• Example: Convert 15 inches to centimeters.

  • 1 inch = 2.54 cm

  • $15\ \text{in} \times \frac{2.54\ \text{cm}}{1\ \text{in}} = 38.1\ \text{cm}$

• Example: The Eiffel Tower is 301 m tall. Convert to feet, kilometers, and nanometers:

  • Feet: $301\ \text{m} \times \frac{3.281\ \text{ft}}{1\ \text{m}} = 987\ \text{ft}$

  • Kilometers: $301\ \text{m} \times \frac{1\ \text{km}}{1000\ \text{m}} = 0.301\ \text{km}$

  • Nanometers: $301\ \text{m} = 3.01 \times 10^{11}\ \text{nm}$

Density Conversion Example

• Density of silver: $1.05 \times 10^4$ kg/m3

• Convert to g/cm3:

  • 1 kg = 1000 g, 1 m3 = 106 cm3

  • $1.05 \times 10^4\ \frac{\text{kg}}{\text{m}^3} \times \frac{1000\ \text{g}}{1\ \text{kg}} \times \frac{1\ \text{m}^3}{10^6\ \text{cm}^3} = 10.5\ \frac{\text{g}}{\text{cm}^3}$

Order of Magnitude Calculations

An order of magnitude calculation is a rough estimate of a quantity, accurate to within a factor of about 10. These are useful for checking the reasonableness of answers.

• If your answer differs greatly from your order of magnitude estimate, re-examine your calculations.

Scalars and Vectors

• Scalar: A physical quantity represented by a number with appropriate units (e.g., distance, speed, temperature).

• Vector: A physical quantity with both a numerical value and a direction (e.g., displacement, velocity, acceleration).

• Vectors in One Dimension: Only two possible directions (e.g., left/right), typically indicated by positive and negative signs.

Example: Velocity in One Dimension

• Two cars move in opposite directions at 25 m/s.

• Assign right as positive, left as negative.

• Car #1: $+25$ m/s (right), Car #2: $-25$ m/s (left)

• The magnitude of velocity (speed) is always positive.

Physics Problem Solving

Section 1-8 of the textbook provides strategies for solving physics problems, including careful analysis, use of units, and estimation techniques.