Comprehensive Study Notes: Foundations of College Physics

Physical Quantities, Units, and Measurement

SI Units and Scientific Notation

Physics relies on standardized units for measurement, primarily the International System of Units (SI). Scientific notation is used to express very large or small numbers efficiently.

• Base SI Units: meter (m) for length, kilogram (kg) for mass, second (s) for time.

• Prefixes: kilo (103), mega (106), giga (109), centi (10-2), milli (10-3), micro (10-6), nano (10-9).

• Scientific Notation: Numbers are written as a product of a coefficient and a power of ten, e.g., $3.16 \times 10^7$.

Example: 1 kilogram (kg) = 1000 grams (g); 1 centimeter (cm) = 0.01 meters (m).

Dimensional Analysis

Dimensional analysis checks the consistency of equations and converts between units.

• Key Quantities: Length (L), Mass (M), Time (T).

• Example: Velocity has dimensions of $[L][T]^{-1}$.

Motion in One and Two Dimensions

Displacement, Velocity, and Acceleration

Motion is described using displacement, velocity, and acceleration.

• Displacement ($\Delta x$): Change in position, $\Delta x = x_f - x_i$.

• Average Velocity ($\bar{v}$): $\bar{v} = \frac{\Delta x}{\Delta t}$.

• Average Acceleration ($\bar{a}$): $\bar{a} = \frac{\Delta v}{\Delta t}$.

Kinematic Equations (Constant Acceleration)

• $v = v_0 + at$

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

• $v^2 = v_0^2 + 2a(x - x_0)$

Example: A car accelerates from rest ($v_0 = 0$) at $2.8\,\mathrm{m/s^2}$; time to reach $7\,\mathrm{m/s}$ is $t = \frac{7}{2.8} = 2.5\,\mathrm{s}$.

Free Fall

• Objects in free fall experience constant acceleration due to gravity ($g = 9.8\,\mathrm{m/s^2}$ downward).

• Equations of motion apply with $a = -g$ (upward positive).

Vectors and Two-Dimensional Motion

• Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$.

• Projectile Motion: Horizontal and vertical motions are independent.

• Equations:

  • Horizontal: $x = v_{0x} t$

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

Example: A stone thrown horizontally from a $58.1\,\mathrm{m}$ cliff with $v_{0x} = 24.6\,\mathrm{m/s}$ takes $t = \sqrt{\frac{2 \times 58.1}{9.8}} \approx 3.45\,\mathrm{s}$ to hit the ground.

Forces and Newton's Laws

Types of Forces

• Contact Forces: Result from physical contact (e.g., friction, tension).

• Field Forces: Act at a distance (e.g., gravity, electromagnetic).

Newton's Laws of Motion

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

• Second Law: $\sum \vec{F} = m \vec{a}$

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

Friction

• Static Friction: Prevents motion up to a maximum value $f_s \leq \mu_s N$.

• Kinetic Friction: $f_k = \mu_k N$

Gravity and Weight

• Weight: $W = mg$

• Newton's Law of Universal Gravitation: $F_g = G \frac{Mm}{r^2}$, $G = 6.67 \times 10^{-11}\,\mathrm{N\,m^2/kg^2}$

Work, Energy, and Power

Work and Kinetic Energy

• Work: $W = Fd \cos \theta$

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

• Work-Energy Theorem: $W_{net} = \Delta KE$

Potential Energy

• Gravitational: $PE_g = mgh$ (near Earth's surface)

• Elastic (Spring): $PE_s = \frac{1}{2} k x^2$

Conservation of Mechanical Energy

• $KE_i + PE_i = KE_f + PE_f$ (if only conservative forces act)

Power

• Average Power: $P = \frac{W}{t}$

• Unit: Watt (W) = Joule/second

Momentum and Collisions

Linear Momentum and Impulse

• Momentum: $\vec{p} = m \vec{v}$

• Impulse: $\vec{J} = \Delta \vec{p} = F_{avg} \Delta t$

Conservation of Momentum

• In a closed system, total momentum is conserved: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$

Types of Collisions

• Elastic: Both momentum and kinetic energy are conserved.

• Inelastic: Momentum conserved, kinetic energy not conserved.

Rotational Motion and Equilibrium

Rotational Kinematics

• Angular Displacement: $\Delta \theta$ (radians)

• Angular Velocity: $\omega = \frac{\Delta \theta}{\Delta t}$

• Angular Acceleration: $\alpha = \frac{\Delta \omega}{\Delta t}$

• Relationship to Linear Quantities: $v = r \omega$, $a_t = r \alpha$

Moment of Inertia and Torque

• Moment of Inertia (I): $I = \sum m r^2$ (depends on mass distribution)

• Torque: $\tau = r F \sin \theta$

• Rotational Dynamics: $\sum \tau = I \alpha$

Rotational Kinetic Energy

• $KE_{rot} = \frac{1}{2} I \omega^2$

Angular Momentum

• $L = I \omega$

• Conservation: $L_{initial} = L_{final}$ if net external torque is zero.

Equilibrium

• Translational: $\sum F = 0$

• Rotational: $\sum \tau = 0$

Solids and Fluids

Density and Pressure

• Density: $\rho = \frac{m}{V}$

• Pressure: $P = \frac{F}{A}$ (SI unit: Pascal, Pa)

Elasticity

• Young's Modulus (Y): $Y = \frac{\text{stress}}{\text{strain}} = \frac{F/A}{\Delta L/L}$

• Shear Modulus (S): $S = \frac{F/A}{\Delta x/h}$

• Bulk Modulus (B): $B = -\frac{\Delta P}{\Delta V/V}$

Fluids in Static Equilibrium

• Hydrostatic Pressure: $P = P_0 + \rho g h$

• Archimedes' Principle: Buoyant force equals the weight of displaced fluid.

• Equation of Continuity: $A_1 v_1 = A_2 v_2$ (incompressible fluids)

• Bernoulli's Equation: $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

Oscillations and Waves

Simple Harmonic Motion (SHM)

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

• Period: $T = 2\pi \sqrt{\frac{m}{k}}$ (mass-spring), $T = 2\pi \sqrt{\frac{L}{g}}$ (pendulum)

• Frequency: $f = \frac{1}{T}$

• Maximum Speed: $v_{max} = \omega A$

• Maximum Acceleration: $a_{max} = \omega^2 A$

Waves

• Wave Speed: $v = f \lambda$

• Types: Mechanical (require medium), Electromagnetic (do not require medium)

• Sound: Audible range: 20 Hz to 20,000 Hz; speed in air $\approx 330\,\mathrm{m/s}$

Interference and Doppler Effect

• Constructive Interference: Path difference $n\lambda$

• Destructive Interference: Path difference $(n + 1/2)\lambda$

• Doppler Effect: $f' = f \frac{v + v_o}{v - v_s}$ (for moving source/observer)

Thermal Physics

Temperature and Thermal Expansion

• Temperature Scales: Celsius, Kelvin ($T_K = T_C + 273.15$), Fahrenheit

• Thermal Expansion:

  • Linear: $\Delta L = \alpha L_0 \Delta T$

  • Area: $\Delta A = \gamma A_0 \Delta T$

  • Volume: $\Delta V = \beta V_0 \Delta T$

Ideal Gas Law

• $PV = nRT$

• $R = 8.314\,\mathrm{J/(mol\,K)}$

• Number of molecules: $N = n N_A$, $N_A = 6.02 \times 10^{23}\,\mathrm{mol^{-1}}$

Kinetic Theory of Gases

• Average kinetic energy per molecule: $\langle KE \rangle = \frac{3}{2} k_B T$

• Total internal energy (monatomic ideal gas): $U = \frac{3}{2} nRT$

• Boltzmann constant: $k_B = 1.38 \times 10^{-23}\,\mathrm{J/K}$

Heat and Thermodynamics

• Specific Heat: $Q = mc\Delta T$

• Heat Conduction: $P = kA \frac{T_h - T_c}{L}$

• First Law of Thermodynamics: $\Delta U = Q - W$

• Work by Gas: $W = P \Delta V$ (isobaric process)

Thermodynamic Processes

• Isobaric: $P$ constant

• Isochoric: $V$ constant, $W = 0$

• Isothermal: $T$ constant, $\Delta U = 0$

• Adiabatic: $Q = 0$

Heat Engines and Entropy

• Efficiency: $e = \frac{W_{eng}}{Q_h} = 1 - \frac{Q_c}{Q_h}$

• Second Law: No engine can convert all heat into work; some is always expelled to a cold reservoir.

• Entropy Change: $\Delta S = \frac{Q_{rev}}{T}$

Sample Table: SI Prefixes

Prefix	Symbol	Factor
kilo	k	$10^3$
mega	M	$10^6$
giga	G	$10^9$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	μ	$10^{-6}$
nano	n	$10^{-9}$

Summary

• Physics uses SI units and scientific notation for clarity and precision.

• Motion, forces, energy, and momentum are foundational concepts, described mathematically.

• Rotational motion, equilibrium, and fluid mechanics extend these ideas to more complex systems.

• Oscillations, waves, and thermal physics connect microscopic and macroscopic phenomena.

Additional info: Some explanations and formulas were expanded for clarity and completeness based on standard college physics curricula.