PHYS 201 Final Exam Study Guide: Key Concepts and Applications

Chapter 1: Models, Measurements, and Vectors

SI Units, Prefixes, and Conversion Factors

Physics relies on standardized units for measurement, known as the International System of Units (SI). Prefixes are used to denote multiples or fractions of units, and conversion factors allow for changing between units.

• SI Units: Fundamental units include meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).

• Prefixes: Examples include kilo- (103), milli- (10-3), micro- (10-6).

• Conversion Factors: Used to convert between units, e.g., 1 km = 1000 m.

• Example: Converting 5 km to meters: $5 \times 10^3 = 5000$ m.

Scalars vs. Vectors

Physical quantities are classified as scalars or vectors.

• Scalar: Has magnitude only (e.g., mass, temperature).

• Vector: Has both magnitude and direction (e.g., displacement, velocity).

• Example: Velocity is a vector; speed is a scalar.

Vector Addition, Subtraction, and Component Conversion

Vectors can be added or subtracted graphically or algebraically. They can be expressed in component form or magnitude-angle form.

• Component Form: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Magnitude-Angle Form: $A = \sqrt{A_x^2 + A_y^2}$, $\theta = \tan^{-1}(A_y/A_x)$

• Vector Addition: Add corresponding components.

• Example: $\vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

Chapter 2: Motion Along a Straight Line

Position, Displacement, and Distance

Motion is described by position, displacement, and distance traveled.

• Position: Location of an object at a given time.

• Displacement: Change in position; vector quantity.

• Distance: Total path length traveled; scalar quantity.

• Example: If an object moves from 0 m to 5 m and back to 2 m, displacement is 2 m, distance is 7 m.

Velocity vs. Speed

Velocity is a vector; speed is a scalar.

• Velocity: $v = \frac{\Delta x}{\Delta t}$

• Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Instantaneous Velocity and Acceleration

Instantaneous values are limits as time intervals approach zero.

• Instantaneous Velocity: $v = \frac{d x}{d t}$

• Instantaneous Acceleration: $a = \frac{d v}{d t}$

Constant Acceleration Kinematics Equations

For constant acceleration, use the following equations:

• $v = v_0 + a t$

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

• $v^2 = v_0^2 + 2 a (x - x_0)$

Free Fall and Highest Point

In free fall, the highest point is where velocity is zero.

• At the highest point: $v = 0$

• Acceleration: $a = -g$ (downward)

Chapter 3: Motion in a Plane

2D Vectors: Displacement, Velocity, Acceleration

Motion in two dimensions uses vector components.

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

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

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

Breaking Motion into Components

Analyze x- and y-components separately.

• Projectile Motion: Horizontal motion is constant velocity; vertical motion is affected by gravity.

• Equations: $x = v_{0x} t$, $y = v_{0y} t - \frac{1}{2} g t^2$

Uniform Circular Motion and Radial Acceleration

In uniform circular motion, speed is constant but direction changes, so acceleration is not zero.

• Radial (Centripetal) Acceleration: $a_r = \frac{v^2}{r}$

Chapter 4: Newton’s Laws of Motion

Force and Net Force

A force is a push or pull; the net force determines motion.

• Net Force: Sum of all forces acting on an object.

Newton’s Laws

• First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net force.

• Second Law: $\vec{F}_{\text{net}} = m \vec{a}$

• Third Law: For every action, there is an equal and opposite reaction.

Free-Body Diagrams

Draw all forces acting on an object to analyze motion.

Chapter 5: Applications of Newton’s Laws

Axes Choice and Equilibrium

Choose axes to simplify force analysis, especially on inclines.

• Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$

Dynamics

• $\sum F_x = m a_x$, $\sum F_y = m a_y$

Friction and Hooke’s Law

• Static Friction: Prevents motion; $f_s \leq \mu_s N$

• Kinetic Friction: Opposes motion; $f_k = \mu_k N$

• Hooke’s Law: $F = -k x$

Chapter 6: Circular Motion and Gravitation

Curves and Maximum Speed

On flat curves, friction limits maximum speed before sliding.

• $v_{\text{max}} = \sqrt{\mu_s g r}$

Banked Curves

Banking angle affects design speed.

• $v = \sqrt{g r \tan \theta}$

Newton’s Law of Gravitation

• $F = G \frac{m_1 m_2}{r^2}$

Circular Orbits

• Orbital Speed: $v = \sqrt{\frac{G M}{r}}$

• Orbital Period: $T = 2 \pi \sqrt{\frac{r^3}{G M}}$

Chapter 7: Work and Energy

Work and Its Sign

Work is the product of force and displacement; the angle affects its sign.

• $W = F d \cos \theta$

Zero Work

Forces perpendicular to displacement do zero work.

Kinetic Energy and Work-Energy Theorem

• Kinetic Energy: $K = \frac{1}{2} m v^2$

• Work-Energy Theorem: $W = \Delta K$

Potential Energy

• Gravitational: $U_g = m g h$

• Elastic: $U_e = \frac{1}{2} k x^2$

Conservation of Mechanical Energy

• $E = K + U$ is constant if only conservative forces act.

Chapter 8: Momentum

Momentum and Conservation

• Momentum: $\vec{p} = m \vec{v}$

• Conservation: Total momentum is conserved in isolated systems.

Collisions

• Elastic: Both momentum and kinetic energy conserved.

• Inelastic: Only momentum conserved.

• Completely Inelastic: Objects stick together.

Impulse

• $\vec{J} = \Delta \vec{p} = \int \vec{F} dt$

Chapter 9: Rotational Motion

Angular Quantities

• Angular Displacement: $\theta$ (radians)

• Angular Velocity: $\omega = \frac{d \theta}{dt}$

• Angular Acceleration: $\alpha = \frac{d \omega}{dt}$

Linear and Angular Relations

• $v = r \omega$

• $a = r \alpha$

Rotational Kinematics

• $\omega = \omega_0 + \alpha t$

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)$

Moment of Inertia and Rotational Kinetic Energy

• Moment of Inertia: $I = \sum m_i r_i^2$

• Rotational Kinetic Energy: $K_{\text{rot}} = \frac{1}{2} I \omega^2$

Total Kinetic Energy

• $K_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$

Rolling Without Slipping

• $v = r \omega$

Chapter 10: Dynamics of Rotational Motion

Torque and Lever Arm

• $\tau = r F \sin \theta$

Rotational Analog of Newton’s 2nd Law

• $\sum \tau = I \alpha$

Angular Momentum and Conservation

• $L = I \omega$

• Conservation: $\sum \tau = 0 \Rightarrow L = \text{constant}$

Rotational Work and Power

• $W = \tau \theta$

• $P = \tau \omega$

Rigid Body Equilibrium

• $\sum F = 0$, $\sum \tau = 0$

Chapter 11: Elasticity and Periodic Motion

Period, Frequency, and Angular Frequency

• Period (T): Time for one cycle.

• Frequency (f): Cycles per second; $f = \frac{1}{T}$

• Angular Frequency (\omega): $\omega = 2 \pi f$

Simple Harmonic Motion (SHM)

• Restoring Force: $F = -k x$

• Position: $x(t) = A \cos(\omega t + \phi)$

• Velocity: $v(t) = -A \omega \sin(\omega t + \phi)$

• Acceleration: $a(t) = -A \omega^2 \cos(\omega t + \phi)$

Mass-Spring Oscillator

• $T = 2 \pi \sqrt{\frac{m}{k}}$

• $f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}$

• $\omega = \sqrt{\frac{k}{m}}$

Simple Pendulum (Small Angle Approximation)

• $T = 2 \pi \sqrt{\frac{L}{g}}$

• Period depends on length (L) and gravity (g), not mass.

Chapter 12: Mechanical Waves and Sound

Wave Properties

• Amplitude: Maximum displacement.

• Wavelength (\lambda): Distance between successive crests.

• Period (T): Time for one cycle.

• Frequency (f): $f = \frac{1}{T}$

• Wave Speed (v): $v = f \lambda$

Wave Speed on a Rope

• $v = \sqrt{\frac{T}{\mu}}$ (T = tension, \mu = mass per unit length)

Interference Conditions

• Constructive: Path difference = integer multiples of wavelength.

• Destructive: Path difference = half-integer multiples of wavelength.

Chapter 13: Fluid Mechanics

Density and Pressure

• Density: $\rho = \frac{m}{V}$

• Pressure: $P = \frac{F}{A}$

Pressure Variation with Depth

• $P = P_0 + \rho g h$

Pascals Law and Hydraulic Lift

• Pressure applied to a fluid is transmitted undiminished.

• $F_2 = F_1 \frac{A_2}{A_1}$

Buoyant Force

• $F_b = \rho_{\text{fluid}} V_{\text{displaced}} g$

Floating and Sinking

• If object density < fluid density, it floats; otherwise, it sinks.

Chapter 14: Temperature and Heat

Temperature vs. Heat

• Temperature: Measure of average kinetic energy.

• Heat: Energy transferred due to temperature difference.

Temperature Conversions

• $T_{\text{K}} = T_{\text{C}} + 273.15$

• $T_{\text{F}} = \frac{9}{5} T_{\text{C}} + 32$

Specific Heat and Heat Transfer

• $Q = m c \Delta T$

Latent Heat and Phase Change

• $Q = m L$

• Phase change involves energy transfer without temperature change.

Chapter 15: Thermal Properties of Matter

Mole, Avogadro’s Number, and Molar Mass

• Mole: Amount of substance containing $6.022 \times 10^{23}$ particles.

• Molar Mass: Mass of one mole of a substance.

Ideal Gas Law

• $P V = n R T$

Average Translational Kinetic Energy

• $\langle K \rangle = \frac{3}{2} k_B T$

RMS Speed

• $v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}$

Molar Heat Capacities

• $C_V$ (constant volume), $C_P$ (constant pressure)

Work on a pV Diagram

• $W = \int P dV$

Internal Energy and First Law of Thermodynamics

• Internal Energy: For ideal gas, depends only on temperature.

• First Law: $\Delta U = Q - W$

Thermodynamic Processes

• Isochoric: Constant volume

• Isobaric: Constant pressure

• Isothermal: Constant temperature

• Adiabatic: No heat exchange