Fundamental Concepts in College Physics: Motion, Forces, Energy, and Momentum

CHAPTER 2: Straight Line Motion

2.1 - Avg & Instantaneous Velocity & Acceleration

This section introduces the basic concepts of motion in a straight line, focusing on velocity and acceleration.

• Vector Addition: Displacement, time, and average velocity are vector quantities.

• Instantaneous Velocity: The rate of change of displacement at a specific instant.

• Instantaneous Acceleration: The rate of change of velocity at a specific instant.

• Key Equations:

  • $v = \frac{dx}{dt}$

  • $a = \frac{dv}{dt}$

• Example: A car accelerating from rest along a straight road.

2.2 - Motion with Constant Acceleration

Describes motion where acceleration remains constant, allowing the use of kinematic equations.

• Kinematic Equations: Used to solve problems involving constant acceleration.

  • $v = v_0 + at$

  • $x = x_0 + v_0 t + \frac{1}{2} a t^2$

  • $v^2 = v_0^2 + 2a(x - x_0)$

• Free Falling Objects: Objects in free fall experience constant acceleration due to gravity ($g$).

• Example: Dropping a ball from a height and calculating its velocity after a certain time.

2.3 - Quiz & Problem Overview

Practice problems to reinforce concepts of straight-line motion and acceleration.

CHAPTER 3: Motion in 2 & 3 Dimensions

3.1 - Position, Velocity & Acceleration

Extends motion analysis to two and three dimensions, using vector components.

• Vector Components: Position, velocity, and acceleration are analyzed in $x$, $y$, and $z$ directions.

• Equations:

  • $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$

  • $\vec{v} = \frac{d\vec{r}}{dt}$

  • $\vec{a} = \frac{d\vec{v}}{dt}$

3.2 - Projectile Motion & Motion in a Circle

Analyzes motion under gravity and circular motion.

• Projectile Motion: Objects given initial velocity and then follow a parabolic path under gravity.

  • Horizontal motion: $x = v_{0x} t$

  • Vertical motion: $y = v_{0y} t - \frac{1}{2} g t^2$

• Uniform Circular Motion: Constant speed around a fixed point.

  • Average acceleration: $a = \frac{v^2}{r}$

  • Period $T$ is the time for one revolution.

• Example: Calculating the time for a satellite to orbit Earth.

CHAPTER 4: Forces

4.1 - Newton's Laws of Motion - I

Introduces the fundamental laws governing forces and motion.

• Four Common Forces:

  • Normal (N) - contact force

  • Tension (T) - contact force

  • Friction (f) - contact force

  • Weight (W) - long-range force

• Newton's First Law: An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net external force.

• Newton's Second Law: $\vec{F}_{net} = m\vec{a}$

• Newton's Third Law: "Every action has an equal and opposite reaction."

• Free Body Diagrams (FBDs): Visual representation of forces acting on an object.

• Example: Analyzing forces on a block sliding down an inclined plane.

CHAPTER 5: Newton's Laws of Motion - II

5.1 - Applying Newton's Laws

Application of Newton's laws to equilibrium and dynamics.

• Equilibrium: Sum of forces in any direction is zero ($\sum F = 0$).

• Dynamics: When $\sum F \neq 0$, the object accelerates.

• Apparent Weight: The normal force experienced in an accelerating frame (e.g., elevator).

• Example: Calculating apparent weight in an elevator moving upward.

5.2 - Frictional Forces

Explores the nature of friction between surfaces.

• Kinetic Friction: Acts when an object is in motion. $f_k = \mu_k N$

• Static Friction: Acts when the object is not moving but could move. $f_s \leq \mu_s N$

• Example: Pushing a box across the floor and determining the force needed to start motion.

CHAPTER 6: Work & Kinetic Energy

6.1 - Work & Energy (conservation) Introduction

Defines work and energy, and introduces the concept of conservation.

• Work (scalar): $W = \vec{F} \cdot \vec{d}$

• Work on a Constant Force: $W = Fd \cos \theta$

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Work-Energy Theorem: $W_{net} = \Delta KE = KE_{final} - KE_{initial}$

• Example: Calculating work done by a force moving a box across a surface.

6.2 - Work, Energy & Power

Explores the relationship between work, energy, and power, including springs.

• Springs: $F = kx$; Work done $W = \frac{1}{2}kx^2$

• Power: $P = \frac{dW}{dt}$; Average power $P_{avg} = \frac{W}{\Delta t}$

• Example: Calculating the power output of a car engine.

CHAPTER 7: Potential & Mechanical Energy (conservation)

7.1 - Conservation of Mechanical Energy

Discusses the principle of conservation of energy in mechanical systems.

• Conservation of Mechanical Energy: $KE + GPE = KE_{i} + GPE_{i}$ (when only conservative forces act)

• Gravitational Potential Energy: $U_{gravity} = mgh$

• Elastic Potential Energy: $U_{spring} = \frac{1}{2}kx^2$

• Example: A pendulum swinging without air resistance.

7.2 - Potential & Mechanical Energy (conservation) - II

Further explores potential energy, conservative and nonconservative forces.

• Conservative Forces: Allow conversion between kinetic and potential energy.

• Nonconservative Forces: (e.g., friction) dissipate energy as heat.

• Law of Conservation of Energy: $\Delta KE + \Delta PE + \Delta E_{thermal} = 0$

• Example: A block sliding down a ramp with friction.

CHAPTER 8: Momentum

8.1 - Momentum, Impulse & Collisions (+ conservation)

Introduces momentum, impulse, and their conservation in collisions.

• Momentum: $\vec{p} = m\vec{v}$

• Impulse: $I = \sum F \Delta t$

• Impulse-Momentum Theorem: $\Delta p = I$

• Example: Calculating the change in momentum when a baseball is hit.

8.2 - Momentum, Impulse & Collisions (+ elasticity & center of mass)

Explores elastic and inelastic collisions, and the concept of center of mass.

• Elastic Collisions: Total kinetic energy is conserved.

  • $E_{K,1} + E_{K,2} = E_{K,1}' + E_{K,2}'$

  • $p_1 + p_2 = p_1' + p_2'$

• Inelastic Collisions: Kinetic energy is not conserved.

• Center of Mass: The point representing the average position of mass in a system.

  • $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

• Example: Two cars colliding and moving together after impact.

CHAPTER 9: Rotation of Rigid Bodies

9.1 - Angular Velocity & Angular Acceleration

Describes rotational motion and its kinematic quantities.

• Angular Coordinates: Angle in radians $\theta = \frac{s}{r}$

• Angular Velocity: $\omega = \frac{d\theta}{dt}$

• Angular Acceleration: $\alpha = \frac{d\omega}{dt}$

• Rotation with Constant Angular Acceleration: Kinematic equations for rotation.

  • $\omega = \omega_0 + \alpha t$

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

• Example: Calculating the angular velocity of a spinning wheel after a given time.

9.2 - Relating Linear & Angular Kinematics

Shows the relationship between linear and angular motion.

• Relation between $v$ & $\omega$: $v = r\omega$

• Relation between $a$ & $\alpha$: $a = r\alpha$

• Example: The speed at the rim of a rotating disk.

Table: Comparison of Elastic and Inelastic Collisions

Type of Collision	Kinetic Energy Conserved?	Momentum Conserved?	Example
Elastic	Yes	Yes	Billiard balls
Inelastic	No	Yes	Car crash (cars stick together)
Completely Inelastic	No	Yes	Clay balls sticking together

Table: Common Forces in Mechanics

Force	Description	Example
Normal (N)	Perpendicular contact force	Book on a table
Tension (T)	Pulling force via a string/rope	Hanging mass
Friction (f)	Resists motion between surfaces	Sliding box
Weight (W)	Gravitational force	Falling object

Additional info: Some equations and context have been expanded for clarity and completeness. The notes cover introductory college-level physics topics, including kinematics, dynamics, energy, momentum, and rotation, suitable for exam preparation.