Units, Physical Quantities, and Vectors

Introduction to Physics

Physics is the study of the fundamental laws of nature, seeking to understand how and why things work the way they do. It spans the smallest scales of particle physics to the largest scales of the cosmos, and its principles are foundational in engineering, architecture, medicine, and technology.

• Physical theories are patterns that relate natural phenomena.

• A physical law or principle is a well-established theory.

• Physics emphasizes problem-solving and conceptual understanding over rote memorization of formulas.

Problem-Solving in Physics

Effective problem-solving in physics involves a systematic approach:

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

2. Set Up the problem by choosing appropriate equations and drawing diagrams.

3. Execute the solution by performing calculations.

4. Evaluate the answer by checking for consistency and reasonableness.

Standards and Units

SI Units and Prefixes

Physics relies on standardized units for measurement. The International System of Units (SI) is the most widely used system:

• Length: meter (m)

• Time: second (s)

• Mass: kilogram (kg)

Prefixes are used to denote multiples or fractions of units, such as kilo- (103), milli- (10-3), and micro- (10-6).

Unit Conversions and Dimensional Consistency

Equations in physics must be dimensionally consistent; only quantities with the same units can be added or equated. Unit conversions are performed using conversion factors to ensure consistency.

• Example: To convert 8.144 miles/hour to meters/second, use the appropriate conversion factors for miles to meters and hours to seconds.

Uncertainty and Significant Figures

Accuracy, Precision, and Error

Measurements in physics are subject to uncertainty. Understanding the difference between accuracy and precision is essential:

• Accuracy: How close a measurement is to the true value.

• Precision: How close repeated measurements are to each other.

Significant Figures

Significant figures reflect the precision of a measurement. The rules for determining significant digits are:

• All nonzero digits are significant.

• Zeros between significant digits are significant.

• Trailing zeros to the right of the decimal are significant.

• Leading zeros are not significant; they only indicate the position of the decimal point.

Examples:

• 200 (1 significant figure)

• 200.2 (4 significant figures)

• 0.00203 (3 significant figures)

Significant Figures in Calculations

• For multiplication and division, the result has as many significant figures as the factor with the fewest significant figures.

• For addition and subtraction, the result is limited by the term with the fewest digits to the right of the decimal point.

Uncertainty Types

• Absolute uncertainty: Expressed in the same units as the measurement.

• Relative uncertainty: Expressed as a percentage.

• Random errors: Can be reduced by repeated measurements (decrease as $1/\sqrt{N}$).

• Systematic errors: Consistent biases that do not average out.

Vectors and Scalars

Definitions and Notation

Physical quantities are classified as either scalars or vectors:

• Scalar: Described by a single number and unit (e.g., mass, temperature).

• Vector: Has both magnitude and direction (e.g., displacement, velocity, force).

Vectors are typically denoted in boldface with an arrow, such as $\vec{A}$, and their magnitude as $A$ or $|\vec{A}|$.

Vector Representation and Operations

• Displacement vector: Represents a change in position.

• Vectors can be parallel or antiparallel depending on their direction.

Vector Addition and Subtraction

Vectors are added graphically by the head-to-tail method or using the parallelogram method. Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

Vector subtraction is performed by adding the negative of a vector: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

Multiplying a Vector by a Scalar

Multiplying a vector by a positive scalar changes its magnitude but not its direction; multiplying by a negative scalar reverses its direction.

Components of a Vector

Any vector in a plane can be resolved into perpendicular components, typically along the x- and y-axes:

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

Components can be positive or negative depending on the vector's direction.

Vector Addition Using Components

The sum of vectors can be found by adding their respective components:

• $R_x = A_x + B_x$

• $R_y = A_y + B_y$

Products of Vectors

Scalar (Dot) Product

The scalar product (dot product) of two vectors $\vec{A}$ and $\vec{B}$ is defined as:

• $\vec{A} \cdot \vec{B} = AB \cos \phi$

• It yields a scalar quantity.

• If vectors are perpendicular, the dot product is zero; if parallel, it is maximized.

In component form:

• $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

Vector (Cross) Product

The vector product (cross product) of two vectors $\vec{A}$ and $\vec{B}$ is a vector perpendicular to both, with magnitude:

• $|\vec{A} \times \vec{B}| = AB \sin \phi$

• Direction is given by the right-hand rule.

Component form for $\vec{C} = \vec{A} \times \vec{B}$:

• $C_x = A_y B_z - A_z B_y$

• $C_y = A_z B_x - A_x B_z$

• $C_z = A_x B_y - A_y B_x$