Units, Physical Quantities, and Vectors

The Nature of Physics

Physics is a foundational natural science that seeks to understand the patterns and laws governing the phenomena of the universe. It relies on observation, experimentation, and mathematical modeling to describe and predict physical behavior.

• Experimental Science: Physics is based on experiments and observations.

• Physical Theories: Patterns discovered are formulated as physical theories.

• Physical Laws and Principles: Well-established theories become physical laws or principles, such as Newton's Laws of Motion.

• Example: Galileo's experiments with falling objects led to the understanding of uniform acceleration due to gravity.

Problem-Solving in Physics

Effective problem-solving in physics follows a systematic approach. This ensures clarity and accuracy in finding solutions to physical problems.

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

• Step 2: Set Up – Choose appropriate equations and draw a sketch of the situation.

• Step 3: Execute – Perform mathematical calculations.

• Step 4: Evaluate – Compare your answer with estimates and check for consistency.

• Example: Solving for the time it takes an object to fall from a certain height using kinematic equations.

Idealized Models

Physicists often use idealized models to simplify complex real-world systems, making analysis more manageable.

• Idealization: Ignoring factors such as air resistance or treating objects as point masses.

• Example: Modeling a baseball in flight by neglecting air resistance and assuming constant gravitational force.

• Application: Idealized models are used in introductory physics to focus on fundamental principles before adding complexity.

Physical Quantities and SI Units

Physics uses standard units to measure fundamental quantities, ensuring consistency and clarity in communication and calculation.

• Fundamental Quantities: Length, time, and mass.

• SI Units:

  • Length: meter (m)

  • Time: second (s)

  • Mass: kilogram (kg)

• Unit Prefixes: Used to express very large or small quantities (e.g., milli-, micro-, kilo-, giga-).

• Example: 1 gigasecond (Gs) = s

Unit Consistency and Conversions

Equations in physics must be dimensionally consistent, meaning all terms must have the same units.

• Dimensional Consistency: Only quantities with the same units can be added or equated.

• Unit Conversion: Use conversion factors to change units (e.g., minutes to seconds).

• Example: To convert 3 minutes to seconds:

Uncertainty and Significant Figures

Measurements in physics are subject to uncertainty, which is reflected in the number of significant figures reported.

• Significant Figures: Indicate the precision of a measurement.

• Rules:

  • Multiplication/Division: Result has as many significant figures as the least precise measurement.

  • Addition/Subtraction: Result is limited by the least number of decimal places.

• Example: Approximating as (3.14286) or (3.14159); the latter is more precise.

Density Example

Density is a physical property defined as mass per unit volume.

• Formula:

• Example: A rock with mass 1.80 kg and volume m3 has density kg/m3.

Vectors and Scalars

Physical quantities are classified as either scalars or vectors.

• Scalar: Described by magnitude only (e.g., temperature, mass).

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

• Notation: Vectors are denoted by boldface letters with arrows (e.g., ).

• Magnitude: Written as or .

Drawing and Adding Vectors

Vectors are represented graphically as arrows. Their addition follows specific rules.

• Graphical Addition: Place vectors head-to-tail; the resultant is drawn from the tail of the first to the head of the last.

• Parallelogram Method: For two vectors, construct a parallelogram; the diagonal is the sum.

• Properties:

  • Commutative:

  • Resultant magnitude depends on the angle between vectors.

• Example: Adding displacement vectors to find total displacement.

Subtracting Vectors

Subtracting a vector is equivalent to adding its negative.

• Rule:

• Graphical Representation: Reverse the direction of the vector being subtracted.

Multiplying a Vector by a Scalar

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative).

• Positive Scalar: Lengthens or shortens the vector in the same direction.

• Negative Scalar: Reverses the direction of the vector.

• Example: is twice as long as ; is three times as long but in the opposite direction.

Addition of Vectors at Right Angles

When vectors are perpendicular, their sum can be found using the Pythagorean theorem and trigonometry.

• Magnitude:

• Direction:

• Example: A cross-country skier travels 2 km north and 1 km east; resultant displacement is km at east of north.

Components of a Vector

Vectors can be broken down into components along the x and y axes, facilitating calculation and analysis.

• Formulas:

• Positive/Negative Components: Components may be positive or negative depending on the vector's direction.

• Example: A vector at clockwise from the y-axis with m; and can be found using trigonometric relationships.

Unit Vectors

Unit vectors have a magnitude of 1 and indicate direction along coordinate axes.

• Notation: (x-direction), (y-direction), (z-direction)

• Expressing Vectors:

• Example:

Products of Vectors

Scalar (Dot) Product

The scalar product of two vectors yields a scalar quantity and is defined as:

• Formula:

• Component Form:

• Properties:

  • Positive if

  • Zero if (vectors are perpendicular)

  • Negative if

• Example: , ;

Vector (Cross) Product

The vector product of two vectors yields a vector perpendicular to both, with magnitude and direction given by the right-hand rule.

• Formula:

• Direction: Determined by the right-hand rule.

• Anticommutative Property:

• Example: , ;

Summary Table: SI Base Units and Prefixes

Additional info: These notes expand on the brief points in the slides and text, providing definitions, formulas, and examples for each concept. The table summarizes SI base units and common prefixes for clarity.