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: Every physical quantity has associated 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 so that unwanted units cancel out. For example, converting 10 km to meters: $10\ \text{km} \times \frac{1000\ \text{m}}{1\ \text{km}} = 10,000\ \text{m}$

• 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 miles to kilometers using significant figures and conversion factors.

Chapter 2: One-Dimensional Kinematics

2.1 Definitions and Visualization of Motion in 1D

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: $v = \frac{dx}{dt}$

• 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. Its average velocity is $10\ \text{m/s}$.

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, subscript 0 = initial value.

• 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: $v = -g t$, $x = x_0 - \frac{1}{2} g t^2$

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

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

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

Additional info: Unit vectors $\hat{i}$ and $\hat{j}$ point in the x and y directions, respectively.

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 motion under constant acceleration.

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

• Kinematic Equations in 2D: Apply 1D equations to each component:

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

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

• Projectile Motion: Horizontal acceleration $a_x = 0$, vertical acceleration $a_y = -g$.

• Frame of Reference: Choose axes so that one is horizontal and one is vertical.

• 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 is directed toward the center (centripetal) and can also have tangential components if speed changes.

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

• Centripetal Acceleration: Magnitude $a_c = \frac{v^2}{r}$; direction is toward the center of the circle.

• Radial vs. Tangential Acceleration:

  • Radial (centripetal): Points toward center; responsible for changing direction.

  • Tangential: Points along the tangent; responsible for changing 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. $a_c = \frac{25}{10} = 2.5\ \text{m/s}^2$

Table: Types of Acceleration in Circular Motion

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

Additional info: If speed is constant, tangential acceleration is zero; only centripetal acceleration is present.