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, kilograms). Dimensional analysis checks the consistency of equations and helps with unit conversions.

• Unit Conversion: To convert between units, multiply by conversion factors that relate the old unit to the new unit. 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 significant digits in a measurement indicates its precision. When performing calculations, the result should reflect the least precise measurement.

• Coordinate System: A coordinate system consists of an origin and axes (typically x and y). The positive direction for each axis must be established to describe positions and motions.

Chapter 2: One-Dimensional Kinematics

2.1 Displacement, Velocity, Speed, and Acceleration in 1D

Kinematics describes the motion of objects without considering the forces causing the motion. In one dimension, 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 magnitude of velocity; a scalar quantity. Speed is always positive and does not indicate direction.

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

• Example: If a car moves from 0 m to 10 m in 2 s, its average velocity is $\frac{10\ \text{m}}{2\ \text{s}} = 5\ \text{m/s}$.

2.2 Kinematic Equations and Free Fall

Kinematic equations describe motion with constant acceleration. Free fall refers to motion under gravity alone, with acceleration $g \approx 9.8\ \text{m/s}^2$ downward.

• 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.

• Application: These equations apply when acceleration is constant, such as in free fall.

• Free Fall: In free fall, $a = -g$ (downward). For 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 ways and manipulated mathematically.

• Vector Representation:

  • (a) Magnitude and Direction: A vector is described by its length and angle relative to a reference axis. Example: $\vec{A} = 5\ \text{m}$ at $30^\circ$ above the x-axis.

  • (b) Unit Vector Form: A vector is written as the sum of perpendicular components: $\vec{A} = A_x \hat{i} + A_y \hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in the x and y directions.

• Conversion: To convert between forms: $A_x = A \cos \theta$, $A_y = A \sin \theta$; $A = \sqrt{A_x^2 + A_y^2}$, $\theta = \tan^{-1}(A_y/A_x)$

• Vector Arithmetic: Vectors can be added graphically (tip-to-tail) or algebraically (component-wise). Example: $\vec{C} = \vec{A} + \vec{B}$ means $C_x = A_x + B_x$, $C_y = A_y + B_y$.

• Example: If $\vec{A} = 3 \hat{i} + 4 \hat{j}$, its magnitude is $5$and direction is$\tan^{-1}(4/3)$.

Chapter 4: Two-Dimensional Kinematics and Circular Motion

4.1 Displacement, Velocity, and Acceleration as Vectors

In two dimensions, displacement, velocity, and acceleration are vector quantities, each with x and y components.

• Displacement: $\vec{r} = x \hat{i} + y \hat{j}$

• Velocity: $\vec{v} = v_x \hat{i} + v_y \hat{j}$

• Acceleration: $\vec{a} = a_x \hat{i} + a_y \hat{j}$

• Example: If an object moves from $(0,0)$ to $(3,4)$, its displacement is $3 \hat{i} + 4 \hat{j}$.

4.1 Kinematic Equations in Two Dimensions

The kinematic equations apply to each component separately when acceleration is constant.

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

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

• Application: Used for projectile motion and other 2D problems.

4.1 Projectile Motion

Projectile motion is a classic example of two-dimensional kinematics, where an object moves under the influence of gravity.

• Frame of Reference: Establish x (horizontal) and y (vertical) axes.

• Horizontal Motion: $a_x = 0$, $v_x$ is constant.

• Vertical Motion: $a_y = -g$, $v_y$ changes due to gravity.

• Key Equations:

  • $x = x_0 + v_{0x} t$

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

• Always True: Gravity acts downward; horizontal and vertical motions are independent.

• Example: A ball thrown horizontally from a height follows a parabolic path.

4.2 Uniform Circular Motion and Acceleration

Uniform circular motion occurs when an object moves in a circle at constant speed. Acceleration is present due to the changing direction of velocity.

• 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 speed.

  • Tangential: Changes speed along the circle.

• 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 driving around a circular track experiences centripetal acceleration toward the center.

Quantity	Definition	Equation
Centripetal Acceleration	Acceleration toward center of circle	$a_c = \frac{v^2}{r}$
Period (T)	Time for one revolution	$T = \frac{2\pi r}{v}$
Angular Speed ($\omega$)	Rate of change of angle	$\omega = \frac{2\pi}{T}$, $\omega = \frac{v}{r}$
Radial Acceleration	Directed toward center	$a_r = \frac{v^2}{r}$
Tangential Acceleration	Directed along tangent	$a_t = \frac{d|v|}{dt}$