Chapter 1: Units, Physical Quantities, and Vectors

Learning Outcomes

This chapter introduces foundational concepts in physics, focusing on the nature of physical quantities, units, vectors, and problem-solving strategies. Students will learn:

• Four-step problem-solving strategy for physics problems

• Three fundamental quantities and their units

• How to work with units and significant figures

• Methods for adding and subtracting vectors graphically and using components

• Two ways to multiply vectors: scalar (dot) product and vector (cross) product

The Nature of Physics

Physical Theories and Laws

Physics is an experimental science that seeks to discover patterns in nature, which are formulated as physical theories. When a theory is well-established and widely used, it is called a physical law or principle.

• Physical theory: A model or explanation that describes observed phenomena.

• Physical law/principle: A theory that has been extensively validated and is broadly accepted.

• Example: Galileo's investigation of falling objects and pendulum motion led to foundational physical laws.

Solving Problems in Physics

Four-Step Problem-Solving Strategy

Effective problem-solving in physics involves a systematic approach:

1. Identify: Determine relevant concepts, target variables, and known quantities.

2. Set Up: Select appropriate equations and create a sketch of the situation.

3. Execute: Perform mathematical calculations to solve the problem.

4. Evaluate: Compare the answer with estimates and check for consistency.

Idealized Models

Simplifying Physical Systems

Physicists often use idealized models to simplify complex systems for analysis. For example, a real baseball in flight is affected by air resistance and wind, but an idealized model may treat it as a point object with constant gravitational force and no air resistance.

• Real system: Includes all complexities and forces.

• Idealized model: Ignores secondary effects to focus on primary forces.

• Application: Modeling a baseball as a point mass to analyze its trajectory.

Standards and Units

Fundamental Quantities and SI Units

Physics relies on three fundamental quantities, each with standard units in the International System (SI):

• Length: measured in meters (m)

• Time: measured in seconds (s)

• Mass: measured in kilograms (kg)

Unit Prefixes

Prefixes are used to express multiples or fractions of units:

• Micro- (μ):

• Kilo- (k):

• Milli- (m):

• Nano- (n):

Unit Consistency and Conversions

Equations must be dimensionally consistent; only quantities with the same units can be added or equated. Always carry units through calculations and convert to standard units as needed.

• Example: To convert 3 minutes to seconds:

Uncertainty and Significant Figures

Measurement Precision

The uncertainty of a measurement is indicated by its number of significant figures.

• Multiplication/Division: The result has as many significant figures as the factor with the fewest significant figures.

• Addition/Subtraction: The result is limited by the term with the fewest digits to the right of the decimal point.

• Application: Small errors in measurement can lead to significant consequences in physical systems.

Vectors and Scalars

Definitions

• Scalar quantity: Described by a single number (magnitude only), e.g., temperature, mass.

• Vector quantity: Has both magnitude and direction, e.g., displacement, velocity.

• Notation: Vectors are represented in boldface italic with an arrow, e.g., .

Drawing Vectors

• Drawn as lines with arrowheads; length indicates magnitude, direction indicates orientation.

• Vectors with equal magnitude and direction are equal; opposite direction indicates a negative vector.

Vector Addition and Subtraction

Graphical Methods

• Head-to-tail method: Place the tail of one vector at the head of another; the resultant vector is drawn from the tail of the first to the head of the last.

• Parallelogram method: Place vectors tail-to-tail and complete the parallelogram; the diagonal represents the sum.

• Order independence: Vectors can be added in any order.

Special Cases

• If vectors are parallel:

• If vectors are antiparallel:

Subtracting Vectors

• Subtracting from is equivalent to adding to :

Multiplying a Vector by a Scalar

• Multiplying a vector by a scalar gives a vector with magnitude and direction depending on the sign of .

Addition of Vectors at Right Angles

Trigonometric Methods

• For vectors at right angles, use the Pythagorean theorem to find the magnitude:

• Use trigonometry to find direction:

• Example: A skier moves 2.00 km north and 1.00 km east; resultant displacement is 2.24 km at 63.4° east of north.

Components of a Vector

Resolving Vectors

• Any vector can be expressed in terms of its components along the x and y axes:

• Components may be positive or negative depending on direction.

Finding Magnitude and Direction from Components

• Magnitude:

• Direction:

Unit Vectors

Definition and Use

• Unit vector: A vector with magnitude 1 and no units, used to specify direction.

• Common unit vectors: (x-direction), (y-direction), (z-direction)

• Any vector can be written as:

Multiplying Vectors

Scalar (Dot) Product

• The scalar product of vectors and is:

• Result is a scalar quantity.

• In terms of components:

• Can be positive, negative, or zero depending on the angle between vectors.

Vector (Cross) Product

• The vector product of and is:

• Magnitude:

• Direction: Determined by the right-hand rule.

• Anticommutative property:

Right-Hand Rule

• Point fingers of right hand along , curl toward ; thumb points in direction of .

Summary Table: SI Base Quantities and Units

Summary Table: Vector Operations

Additional info: Some explanations and examples have been expanded for clarity and completeness.