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: Units are standard quantities used to specify measurements. Dimensional analysis checks the consistency of equations by comparing the units on both sides.

• Unit Conversions: To convert between units, multiply by conversion factors that relate the original unit to the desired unit.

• Significant Figures: The number of meaningful digits in a measurement, indicating its precision. Calculations should preserve the correct number of significant figures.

• Coordinate System: A system (usually Cartesian) with an origin and defined positive directions for axes (x, y) is essential for describing positions and motions.

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

Chapter 2: One-Dimensional Kinematics

2.1 Displacement, Velocity, Speed, and Acceleration

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):

  • $v = v_0 + a t$

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

  • $v^2 = v_0^2 + 2 a (x - x_0)$

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

• Application: These equations apply when acceleration is constant.

• Free Fall: On Earth, free fall means acceleration due to gravity ($g \approx 9.8\ \text{m/s}^2$ downward). In free fall, $a = -g$ (if upward is positive).

• Example: Dropping a ball from rest: $v = 0 + (-9.8) t$

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 Representations:

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

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

• Conversion: $A_x = A \cos \theta$, $A_y = A \sin \theta$

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

• Example: $\vec{A} = 3 \hat{i} + 4 \hat{j}$; magnitude $A = \sqrt{3^2 + 4^2} = 5$

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, combining horizontal and vertical motion.

• Displacement, Velocity, Acceleration as Vectors: Each has x and y components.

• Kinematic Equations in 2D: Apply 1D equations separately to x and y directions (for constant acceleration).

• Projectile Motion: The horizontal motion has constant velocity; vertical motion has constant acceleration (gravity).

• Frame of Reference: Establish axes, origin, and positive directions for problem solving.

• Always True: On Earth, vertical acceleration is $-g$; horizontal acceleration is zero (neglecting air resistance).

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

  • Horizontal: $x = v_0 \cos \theta \cdot t$

  • Vertical: $y = v_0 \sin \theta \cdot t - \frac{1}{2} g t^2$

4.2 Uniform Circular Motion and Acceleration

Circular motion involves objects moving in a circle at constant speed. Acceleration arises from changing direction, not speed.

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

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

• Radial vs. Tangential Acceleration:

  • Radial (centripetal): Directed toward center, changes direction.

  • Tangential: Directed along the tangent, changes 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: $a_c = \frac{5^2}{10} = 2.5\ \text{m/s}^2$

Table: Types of Acceleration in Circular Motion

Type	Direction	Effect
Radial (Centripetal)	Toward center	Changes direction
Tangential	Along tangent	Changes speed

Additional info: Academic context and examples were added to clarify definitions and applications for each learning objective.