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 accuracy and consistency of calculations. Significant figures reflect the precision of measurements, and coordinate systems provide a framework for describing physical quantities.

• Units and Dimensional Analysis: Physical quantities are always expressed with units (e.g., meters, seconds). 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 significant digits in a measurement indicates its precision. When performing calculations, the result should reflect the least precise measurement.

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

• Example: Converting 2.5 hours to seconds: $2.5\ \text{hr} \times \frac{60\ \text{min}}{1\ \text{hr}} \times \frac{60\ \text{s}}{1\ \text{min}} = 9000\ \text{s}$

Chapter 2: One-Dimensional Kinematics

2.1 Fundamental Quantities in 1D Motion

One-dimensional kinematics describes motion along a straight line using 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: $v = \frac{d x}{d t}$

• 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: $a = \frac{d v}{d t}$

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

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 $a$:

  • $v = v_0 + a t$

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

  • $v^2 = v_0^2 + 2 a (x - x_0)$

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

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

• 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 $A_x$ 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: 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: All are vectors with x and y components.

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

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

  • Frame of Reference: Choose origin and axes; typically, x is horizontal, y is vertical.

  • Always True: Horizontal acceleration is zero; vertical acceleration is $-g$.

• Example: A ball thrown at $v_0$ and $\theta$; $v_{0x} = v_0 \cos \theta$, $v_{0y} = v_0 \sin \theta$

4.2 Uniform Circular Motion and Acceleration

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

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

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

• Direction: Always directed radially inward.

• Radial vs. Tangential Acceleration:

  • Radial (centripetal): Changes direction of velocity.

  • Tangential: Changes speed along the path.

• 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	Formula	Direction
Radial (Centripetal)	$a_c = \frac{v^2}{r}$	Toward center
Tangential	$a_t = \frac{d v}{d t}$	Along tangent

Additional info: Academic context and formulas have been expanded for clarity and completeness.