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, as it ensures the correctness of equations and calculations. 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 by comparing the units on both sides.

• Unit Conversions: To convert between units, multiply by conversion factors. For example, to convert 10 cm to meters: $10\ \text{cm} \times \frac{1\ \text{m}}{100\ \text{cm}} = 0.1\ \text{m}$

• Significant Figures: The number of meaningful digits in a measurement. Rules for addition, subtraction, multiplication, and division must be followed to maintain proper precision.

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

• Example: Converting 5.0 km to meters: $5.0\ \text{km} \times \frac{1000\ \text{m}}{1\ \text{km}} = 5000\ \text{m}$

Chapter 2: One-Dimensional Kinematics

2.1 Definitions and Visualization in 1D

Kinematics describes motion without considering its causes. In one dimension, key quantities include displacement, velocity, speed, and acceleration.

• Displacement: The change in position; 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 magnitude of velocity; a scalar. $\text{Speed} = \frac{\text{distance}}{\text{time}}$

• 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: $v_{\text{avg}} = \frac{100\ \text{m}}{10\ \text{s}} = 10\ \text{m/s}$

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):

  • $v = v_0 + a t$

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

  • $v^2 = v_0^2 + 2 a (x - x_0)$

  • $x = x_0 + \frac{1}{2} (v_0 + v) t$

• Variables: $x$ = position, $v$ = velocity, $a$ = acceleration, $t$ = time, $v_0$ = initial velocity, $x_0$ = initial position.

• 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: $v = 0$, $a = -9.8\ \text{m/s}^2$, $x_0 = 0$.

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 forms 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 $A_x = A \cos \theta$, $A_y = A \sin \theta$.

• Conversion: To go from magnitude/direction to components: $A_x = A \cos \theta$, $A_y = A \sin \theta$; from components to magnitude/direction: $A = \sqrt{A_x^2 + A_y^2}$, $\theta = \tan^{-1}(A_y/A_x)$

• Vector Arithmetic: Vectors can be added graphically (tip-to-tail) or algebraically (component-wise).

• 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: The horizontal motion ($a_x = 0$) and vertical motion ($a_y = -g$) are independent.

• Frame of Reference: Choose axes so that one is along the initial velocity or the ground.

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

• Example: A ball launched at angle $\theta$ with speed $v_0$:

  • $v_{0x} = v_0 \cos \theta$

  • $v_{0y} = v_0 \sin \theta$

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; magnitude $a_c = \frac{v^2}{r}$

• Direction: Always toward the center of the circle.

• 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 10 m at 5 m/s: $a_c = \frac{(5\ \text{m/s})^2}{10\ \text{m}} = 2.5\ \text{m/s}^2$

Table: Comparison of Radial and 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: Academic context and formulas have been expanded for completeness and clarity.