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 involves checking equations for consistency in units.

• Unit Conversions: 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. Rules for significant figures must be followed in calculations.

• Coordinate Systems: A coordinate system consists of an origin and axes (e.g., x and y directions). Choosing a coordinate system is essential for describing positions and directions.

• Example: When measuring the length of a table as 1.23 m, the measurement has three significant figures.

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 does not include 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.

• Kinematic Equations (for constant acceleration):

  • $v = v_0 + a t$

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

  • $v^2 = v_0^2 + 2 a (x - x_0)$

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

• Application: These equations apply when acceleration is constant.

• Free Fall: On Earth, free fall means acceleration due to gravity ($a = -g$, $g \approx 9.8\ \text{m/s}^2$ 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 ways and manipulated mathematically.

• Vector Representation:

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

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

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

• Example: Adding $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 1 \hat{i} + 2 \hat{j}$: $\vec{A} + \vec{B} = (3+1) \hat{i} + (4+2) \hat{j} = 4 \hat{i} + 6 \hat{j}$

Chapter 4: Two-Dimensional Kinematics and Circular Motion

4.1 Two-Dimensional Motion: Displacement, Velocity, and Acceleration

In two dimensions, displacement, velocity, and acceleration are vectors. Kinematic equations can be extended to describe motion in both x and y directions.

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

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

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

• Kinematic Equations: Apply separately to x and y components (for constant acceleration).

• Example: Projectile motion: $x = x_0 + v_{0x} t$, $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

4.1 Projectile Motion

Projectile motion is a classic example of two-dimensional motion under constant acceleration (gravity).

• Frame of Reference: Choose axes (usually x horizontal, y vertical).

• Key Properties: Horizontal motion is constant velocity; vertical motion is constant acceleration ($-g$).

• Always True: Gravity acts downward; air resistance is often neglected.

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

4.2 Uniform Circular Motion and Acceleration

Uniform circular motion describes an object moving at constant speed in a circle. Acceleration is always present due to changing direction.

• Uniform Circular Motion: Motion with constant speed along a circular path.

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

• Direction: Always radially inward.

• Radial vs. Tangential Acceleration:

  • Radial (centripetal): Changes direction of velocity.

  • Tangential: Changes speed (if present).

• 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 in a circle at constant speed experiences centripetal acceleration toward the center.

Table: Circular Motion Quantities

Quantity	Symbol	Formula	Description
Centripetal Acceleration	$a_c$	$a_c = \frac{v^2}{r}$	Acceleration toward center
Period	$T$	$T = \frac{2\pi r}{v}$	Time for one revolution
Angular Speed	$\omega$	$\omega = \frac{2\pi}{T}$	Rate of rotation
Radial Acceleration	$a_r$	$a_r = a_c$	Same as centripetal
Tangential Acceleration	$a_t$	$a_t = \frac{dv}{dt}$	Changes speed

Additional info: Table entries for tangential acceleration and radial acceleration are inferred for completeness.