PHYS 180 Study Guide: Chapters 1–4 (Units, Kinematics, Vectors, and Circular Motion)

Chapter 1: Introduction to Physics

1.1 Units, Dimensional Analysis, and Significant Figures

Understanding units and dimensional analysis is fundamental in physics, ensuring that equations and calculations are consistent and meaningful. Significant figures reflect the precision of measurements, and coordinate systems provide a framework for describing physical quantities.

• Units and Dimensional Analysis: Every physical quantity has units (e.g., meters, seconds, kilograms). Dimensional analysis checks the consistency of equations and helps with unit conversions.

• Unit Conversion: To convert between units, multiply by conversion factors. For example, to convert 5 km to meters: $5\ \text{km} \times \frac{1000\ \text{m}}{1\ \text{km}} = 5000\ \text{m}$

• Significant Figures: The number of meaningful digits in a measurement. Rules for significant figures ensure proper reporting of precision.

• Coordinate System: Establishing an origin and positive directions (x, y) is essential for describing positions and motions.

• Example: Converting 2.5 hours to seconds: $2.5\ \text{hr} \times \frac{3600\ \text{s}}{1\ \text{hr}} = 9000\ \text{s}$

Chapter 2: One-Dimensional Kinematics

2.1 Definitions and Visualization of Motion in 1D

Kinematics describes the motion of objects without considering the forces causing the motion. Key quantities include displacement, velocity, speed, and acceleration.

• Displacement: The change in position of an object; a vector quantity. $\Delta x = x_{\text{final}} - x_{\text{initial}}$

• Velocity: The rate of change of displacement. Average velocity: $v_{\text{avg}} = \frac{\Delta x}{\Delta t}$; Instantaneous velocity: $v = \frac{dx}{dt}$

• Speed: The rate of change of distance; a scalar quantity. $\text{Speed} = \frac{\text{distance}}{\text{time}}$

• Acceleration: The rate of change of velocity. Average acceleration: $a_{\text{avg}} = \frac{\Delta v}{\Delta t}$; Instantaneous acceleration: $a = \frac{dv}{dt}$

• Example: A car moves from 0 m to 100 m in 10 s. Average velocity: $v_{\text{avg}} = \frac{100\ \text{m}}{10\ \text{s}} = 10\ \text{m/s}$

• Comparison: Speed is always positive; velocity can be positive or negative depending on direction.

2.2 Kinematic Equations and Free Fall

Kinematic equations describe motion with constant acceleration. Free fall refers to motion under gravity alone.

• Kinematic Equations (for constant acceleration):

  • $v = v_0 + a t$

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

  • $v^2 = v_0^2 + 2 a (x - x_0)$

• Variables: $x$ = position, $v$ = velocity, $a$ = acceleration, $t$ = time, subscript 0 = initial value.

• Application: Use these equations when acceleration is constant.

• Free Fall: On Earth, free fall means acceleration $a = -g$ (where $g \approx 9.8\ \text{m/s}^2$ downward).

• Example: Dropping a ball from rest: $v = 0$, $a = -g$, $x = x_0 - \frac{1}{2} g t^2$

Chapter 3: Scalars, Vectors, and Trigonometry

3.1 Vector Representation and Arithmetic

Vectors are quantities with both magnitude and direction. They can be represented in multiple ways and manipulated mathematically.

• Vector Representation:

  • (a) Magnitude and Direction: $\vec{A}$ has length $A$ and angle $\theta$.

  • (b) Unit Vector Form: $\vec{A} = A_x \hat{i} + A_y \hat{j}$, where $A_x$ and $A_y$ are components along x and y axes.

• Conversion: $A_x = A \cos \theta$, $A_y = A \sin \theta$; $A = \sqrt{A_x^2 + A_y^2}$, $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

• Vector Arithmetic: Add/subtract vectors by components: $\vec{C} = \vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

• Pictorial Representation: Vectors can be drawn as arrows; addition uses the "tip-to-tail" method.

• Example: $\vec{A} = 3 \hat{i} + 4 \hat{j}$; magnitude $A = 5$, direction $\theta = \tan^{-1}(4/3)$

Chapter 4: Two-Dimensional Kinematics and Circular Motion

4.1 Two-Dimensional Motion and Projectile Motion

Motion in two dimensions involves vectors for displacement, velocity, and acceleration. Projectile motion is a classic example of 2D kinematics.

• Displacement, Velocity, Acceleration as Vectors: $\vec{r}$, $\vec{v}$, $\vec{a}$ each have x and y components.

• 2D Kinematic Equations: Apply 1D equations to each axis separately (assuming constant acceleration).

  • $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$

  • $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$

• Projectile Motion: An object launched with initial velocity $v_0$ at angle $\theta$ follows a parabolic path.

  • Horizontal motion: $a_x = 0$, $v_x = v_0 \cos \theta$

  • Vertical motion: $a_y = -g$, $v_y = v_0 \sin \theta - g t$

• Frame of Reference: Choose axes so that one is along the direction of launch.

• Always True: Gravity acts downward; horizontal velocity is constant; vertical velocity changes due to gravity.

• Example: A ball launched at $30^\circ$ with $v_0 = 10\ \text{m/s}$: $v_{0x} = 10 \cos 30^\circ$, $v_{0y} = 10 \sin 30^\circ$

4.2 Uniform Circular Motion and Acceleration

Circular motion involves objects moving in a circle at constant speed. Acceleration in circular motion has radial (centripetal) and tangential components.

• Uniform Circular Motion: Motion in a circle with constant speed.

• Centripetal Acceleration: Points toward the center of the circle; magnitude $a_c = \frac{v^2}{r}$

• Direction: Always directed radially inward.

• Radial vs. Tangential Acceleration:

  • Radial (centripetal): Changes direction of velocity, not magnitude.

  • Tangential: Changes magnitude of velocity (speed).

• Period (T): Time for one complete revolution. $T = \frac{2\pi r}{v}$

• Angular Speed ($\omega$): $\omega = \frac{2\pi}{T} = \frac{v}{r}$

• Example: A car travels in a circle of radius 20 m at 5 m/s: $a_c = \frac{(5)^2}{20} = 1.25\ \text{m/s}^2$

Table: Radial vs. Tangential Acceleration

Type	Direction	Effect	Formula
Radial (Centripetal)	Toward center	Changes direction	$a_c = \frac{v^2}{r}$
Tangential	Along tangent	Changes speed	$a_t = \frac{dv}{dt}$

Additional info: The above notes expand brief learning objectives into full academic explanations, including formulas, examples, and a comparison table for acceleration types in circular motion.