Chapter-0-Units-Physical-Quantities-and-Vectors-1
Chapter 1: Units, Physical Quantities, and Vectors
Page 1: Introduction
Covers fundamental concepts in physics related to quantitative measurements and vector analysis.
Page 2: Goals for Chapter 1
Learn three fundamental quantities of physics and their measurement units.
Track significant figures in calculations.
Understand the distinction between vectors and scalars; learn to add vectors graphically.
Determine vector components and their use in calculations.
Understand unit vectors and their application with vector components.
Learn two methods for multiplying vectors.
Page 3: Nature of Physics
Physics is an experimental science aimed at uncovering patterns in nature.
Established patterns are termed physical theories.
Widely accepted theories are known as physical laws or principles.
Page 4: Standards and Units
SI Units: International System (Système International), also known as MKS (Meter-Kilogram-Second) system.
Length: meter (m)
Mass: kilogram (kg)
Time: second (s)
US Customary Units:
Length: feet
Time: seconds
Mass: slugs (often pounds instead of mass).
Page 5: Length
SI Unit: Meter (m)
Definition: The distance traveled by light in a vacuum during a duration of approximately 3.33 nanoseconds.
Page 6: Mass
SI Unit: Kilogram (kg)
Definition: Mass of a standard cylinder of platinum-iridium at the International Bureau of Standards in France.
Page 7: Time
Unit: Seconds (s)
Definition: The time of oscillation of radiation from a cesium atom (measured by atomic clocks).
Page 8: Unit Prefixes
Various prefixes represent larger and smaller units for fundamental quantities.
Refer to Table 1.1 for commonly used prefixes (e.g., mega-, kilo-, milli-).
Page 9: Unit Consistency and Conversions
Equations must be dimensionally consistent; all terms must share the same units.
Carry units through all calculations and convert as necessary.
Examples:
Multiplication: 50 cm x 150 cm = 7500 cm²
Division: 20 m / 5 s = 4 m/s
Addition: 50 cm + 150 cm = 200 cm
Page 10: Uncertainty and Significant Figures
Every measurement has uncertainty.
Significant figures indicate the precision of a measurement.
For example, a length measurement can be expressed in various significant figures (e.g., 2.8 cm).
Page 11: Rules for Significant Figures
Nonzero integers: Always count as significant figures.
Zeros:
Leading zeros: Never count.
Captive zeros: Always count.
Trailing zeros: Count only if there is a decimal point.
Page 12: Operations with Significant Figures
Multiplication/Division: Result has no more significant figures than the measurement with the fewest significant figures.
Addition/Subtraction: Result's significant figures determined by the value with the largest uncertainty (fewest digits after the decimal).
Page 13: Vectors and Scalars
Scalar Quantities: Defined by a single number (e.g., time, mass).
Vector Quantities: Defined by both a magnitude and a direction (e.g., displacement, velocity).
Vectors are represented in bold italic with an arrow.
Page 14: Drawing Vectors
Visual representation of vectors includes a line with an arrow indicating direction and magnitude.
Page 15: Adding Two Vectors Graphically
Use the Parallelogram Method or Head-to-Tail Method for graphical vector addition.
Page 16: Subtracting Vectors
Subtraction is performed by adding the negative of the vector (A + (-B) = A - B).
Page 17: Multiplying a Vector by a Scalar
If c is a scalar, the resultant vector cA alters the magnitude of A by |c|.
Page 18: Addition of Two Vectors at Right Angles
Example: A skier skis 1 km north and 2 km east; the resultant displacement is 2.24 km at 63.4° east of north.
Page 19: Components of a Vector
Any vector can be decomposed into x-component (Ax) and y-component (Ay).
Use trigonometric functions: Ax = A cos θ, Ay = A sin θ.
Page 20: Positive and Negative Components
Vector components can be either positive or negative based on directional context.
Page 21: Calculations Using Components
Vector magnitude and direction can be calculated using their components:
Example involves inspecting diagrams for angle determination.
Page 22: Finding Components
Example exercise involves calculating x- and y-components of given vectors based on magnitude and angle.
Page 23: Calculations Using Components (Continued)
Components of a set of vectors can be added to determine overall component sums.
Page 24: Problem-solving Strategy for Adding Vectors
Identify relevant concepts; set up a schematic; execute the solution by calculating components; evaluate results.
Page 25: Adding Vectors Using Their Components
Example scenario of three displacements directing to treasure location determined through vector addition.
Page 26: Unit Vectors
Unit vectors have a magnitude of 1. They indicate direction along the x, y, and z axes. Any vector can be expressed in terms of its components using unit vectors.
Page 27: The Scalar Product (1 of 2)
Definition: The scalar (dot) product of two vectors relates to their magnitudes and the cosine of the angle between them.
Page 28: The Scalar Product (2 of 2)
The scalar product's magnitude can be positive, negative, or zero, based on the angle formed between the vectors.
Page 29: Calculating a Scalar Product Using Components
Formula including component sums to find the scalar product of two vectors.
Page 30: Finding an Angle Using the Scalar Product
Example provided, demonstrating how to find the angle between two vectors using their components.
Page 31: The Vector Product
Definition: The cross product of two vectors, yielding a vector that is perpendicular to the plane formed by the two original vectors.
Page 32: The Vector Product is Anticommutative
The vector product follows an anticommutative property, where A x B = -B x A.
Page 33: Calculating the Vector Product
Utilize both the sine function for magnitude and the right-hand rule for direction determination.