Fundamental Quantities, Units, and Vectors in Mechanics

Preliminaries

Course Introduction

This course, University Physics I - Mechanics, introduces the foundational concepts of classical mechanics, focusing on motion and the forces that cause or change motion. The course is taught by Assoc. Prof. Nicholas Matlis, Department of Physics.

• Attendance: Mandatory; arriving late may result in being marked absent.

• Participation: Mandatory; students are encouraged to take notes and ask questions.

• Cheating: Strictly prohibited on homework and exams; may result in failing the class or expulsion.

What is Mechanics?

Definition and Key Concepts

Mechanics is a branch of physics that studies motion and the forces that cause or change motion. It is divided into several key areas:

• Motion (Kinematics): Describes how objects move in 1D and 2D (e.g., free fall, projectile motion).

  • Position, displacement, velocity, and speed are fundamental terms.

• Mass and Inertia: Explains why objects resist changes in motion.

• Forces and Interactions (Dynamics): Explains why motion changes, including Newton's Laws, friction, and resistance.

• Energy & Momentum: Powerful tools for analyzing motion, especially in interactions or collisions.

  • Types: Kinetic, potential, gravitational, elastic, work.

• Rotational Motion: Extends mechanics to spinning objects, including angular position, velocity, and acceleration.

• Equilibrium and Statics: Deals with objects at rest or moving at constant velocity, focusing on stability and balance.

Fundamental Quantities and Units

SI Units and Measurement

Physical quantities are measured using standardized units. The International System of Units (SI) is used in physics:

Fundamental Quantity	Base unit (SI)	Unit abbreviation
length	meter	m
time	second	s
mass	kilogram	kg

• Quantities are described by a value and a unit of measurement, e.g., 2 m, 7.14 s, 3.5 kg.

Scalar & Vector Quantities

Definitions and Examples

• Scalar quantity: Has only magnitude (numerical value with units).

  • Examples: Time (s), mass (kg), energy (J).

• Vector quantity: Has both magnitude and direction in space.

  • Examples: Velocity (m/s, direction), acceleration (m/s2, direction), force (N, direction).

Vector Representation

Graphical Representation

• A vector is represented by an arrow.

• Magnitude: Length of the arrow (always positive).

• Direction: Counterclockwise angle, $\theta$, from the positive x-axis.

Vectors in Everyday Life

Applications

• Vectors are used to describe directions and distances, such as giving directions for a flying car from an airport to a university.

• Example: From PHX to ASU, distance = 2.5 miles, direction = E by SE.

Adding and Subtracting Vectors

Graphical Methods

• Adding vectors: Place vectors head-to-tail or use the parallelogram method.

  • $\vec{C} = \vec{A} + \vec{B}$

• Subtracting vectors: Equivalent to adding the negative of a vector.

  • $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Multiplying a Vector by a Scalar

Effects of Scalar Multiplication

• Positive scalar: Preserves direction, changes magnitude.

  • $2\vec{A}$ is twice as long as $\vec{A}$.

• Negative scalar: Changes direction, changes magnitude.

  • $-3\vec{A}$ is three times as long as $\vec{A}$ and points in the opposite direction.

Unit Vectors

Definition and Use

• A unit vector has magnitude 1 and no units, pointing in the positive direction of a given axis.

• In Cartesian coordinates:

  • $\hat{i}$ points in the +x direction

  • $\hat{j}$ points in the +y direction

  • $\hat{k}$ points in the +z direction

Components of Vectors

Vector and Scalar Components

• Vector component: Projection of a vector along a chosen axis.

• $\vec{A}_x$ and $\vec{A}_y$ are the vector components of $\vec{A}$.

• Expressed in terms of unit vectors:

  • $\vec{A}_x = A_x \hat{i}$

  • $\vec{A}_y = A_y \hat{j}$

• Scalar components: $A_x$ and $A_y$ (just the magnitudes).

• Unit vector notation: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$

Vector Components in Everyday Life

Practical Example

• Describing a route using vector components, e.g., from PHX to ASU: $\vec{v} = 5$ blocks $\hat{j} - 1$ block $\hat{i}$.

Components can be Positive or Negative

Quadrant Analysis

• Depending on the direction, vector components can be positive or negative.

• For example, in different quadrants, $A_x$ and $A_y$ may change sign.

Vector Addition by Components

Mathematical Approach

• Vectors can be added by summing their components:

  • $R_x = A_x + B_x$

  • $R_y = A_y + B_y$

• The magnitude and direction of the resultant vector $\vec{R}$ can be found using:

  • $R = \sqrt{R_x^2 + R_y^2}$

  • $\theta = \cos^{-1}\left(\frac{R_x}{R}\right)$

Vector-Component Relationship

Right Triangle Relationship

• In the xy-coordinate system, vector $\vec{A}$ and its components form a right triangle.

• $A = \sqrt{A_x^2 + A_y^2}$

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

Worked Example: Vector Addition

Hiker Problem

• A hiker walks 2.50 km at 45.0° southeast, then 4.00 km at 60.0° north of east. Find the total displacement from the car.

• Solution Steps:

  1. Draw and find the x and y components of vector $\vec{A}$.

  2. Draw and find the x and y components of vector $\vec{B}$.

  3. Find the x and y components of $\vec{R} = \vec{A} + \vec{B}$.

  4. Draw the $\vec{R}$ vector in the coordinate system based on the components found in step 3.

  5. Use Pythagorean theorem and trigonometry to find the magnitude and direction of $\vec{R}$.

Summary

This guide covers the foundational concepts of units, physical quantities, and vectors in mechanics, including their representation, addition, subtraction, and practical applications. These principles are essential for understanding motion and forces in physics.