IMAT Physics Master Handbook & Compendium

2D Vector Decomposition & Reconstitution

$$A_x = |A| \cos\theta, \quad A_y = |A| \sin\theta, \quad |A| = \sqrt{A_x^2 + A_y^2}, \quad \theta = \arctan\left(\frac{A_y}{A_x}\right)$$

📌 Concept & Application: Any 2D vector A can be broken into independent perpendicular components along the x and y axes. Angle θ is measured counterclockwise from the positive x-axis.

Dot Product (Scalar Product) vs Cross Product (Vector Product)

$$\vec{A} \cdot \vec{B} = |A||B| \cos\theta = A_x B_x + A_y B_y, \qquad |\vec{A} \times \vec{B}| = |A||B| \sin\theta$$

📌 Concept & Application: Dot product yields a scalar (max when vectors are parallel θ=0°). Cross product yields a vector perpendicular to both (max when vectors are orthogonal θ=90°).

SUVAT Equations Summary

$$v = u + at, \qquad s = \frac{u + v}{2} \cdot t, \qquad s = ut + \frac{1}{2}at^2, \qquad v^2 = u^2 + 2as$$

2D Projectile Motion Decomposition

$$x(t) = (v_0 \cos\theta) t, \qquad y(t) = (v_0 \sin\theta) t - \frac{1}{2}gt^2$$

📌 Concept: Horizontal velocity remains constant (a_x = 0). Vertical motion undergoes free fall acceleration (a_y = -g).

Key Trajectory Formulas (Flight Time, Max Height, Range)

$$t_{flight} = \frac{2 v_0 \sin\theta}{g}, \qquad H_{max} = \frac{v_0^2 \sin^2\theta}{2g}, \qquad R = \frac{v_0^2 \sin(2\theta)}{g}$$

📌 Concept: Max range R occurs at θ = 45°. Complementary launch angles (e.g., 30° and 60°) yield identical horizontal ranges.

Uniform Circular Motion Relations

$$v = r \omega = \frac{2\pi r}{T}, \qquad \omega = 2\pi f = \frac{2\pi}{T}, \qquad a_c = \frac{v^2}{r} = r \omega^2$$

📌 Concept: Angular velocity ω (rad/s), period T (seconds per revolution), frequency f (revolutions per second in Hz).

SHM Displacement, Velocity, Acceleration & Periods

$$x(t) = A \sin(\omega t), \quad v(t) = A \omega \cos(\omega t), \quad a(t) = -A \omega^2 \sin(\omega t) = -\omega^2 x$$

📌 Concept: Maximum speed v_max = Aω occurs at equilibrium (x=0). Maximum acceleration a_max = Aω² occurs at extreme amplitudes (x=±A).

Friction Forces (Static vs Kinetic)

$$f_{s, max} = \mu_s N, \qquad f_k = \mu_k N \quad (\text{where } \mu_k < \mu_s)$$

📌 Concept: Static friction f_s balances applied pushing force up to a maximum limit μ_s N. Once sliding begins, kinetic friction f_k remains constant.

Conditions for Mechanical Equilibrium & Torque Formula

$$\sum \vec{F} = 0 \quad (\text{Translational}), \qquad \sum \vec{\tau} = 0 \quad (\text{Rotational}), \qquad \tau = r F \sin\theta$$

📌 Concept: Torque τ is the turning effect of a force. It equals force magnitude F times perpendicular distance from pivot (moment arm r sin θ).

Work Definition & Work-Kinetic Energy Theorem

$$W = \vec{F} \cdot \vec{d} = F d \cos\theta, \qquad W_{net} = \Delta K = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$$

📌 Concept: Work is a scalar (Joule). Work is positive if θ < 90°, negative if θ > 90° (e.g. friction), and zero if force is perpendicular (θ = 90°).

Potential Energies & Conservation of Mechanical Energy

$$U_{grav} = m g h, \qquad U_{elastic} = \frac{1}{2}k x^2, \qquad E_{mech} = K + U = \text{constant (if } W_{nc} = 0)$$

📌 Concept: If only conservative forces (gravity, springs) do work, total mechanical energy E = K + U is conserved.

Mechanical Power & Efficiency

$$P = \frac{W}{t} = \vec{F} \cdot \vec{v}, \qquad \eta = \frac{P_{useful}}{P_{input}} \times 100\%$$

📌 Concept: Power is the rate of doing work (1 Watt = 1 J/s). Power can also be written as Force × Velocity.

Linear Momentum, Impulse & Collisions

$$\vec{p} = m \vec{v}, \qquad \vec{I} = \vec{F}_{net} \Delta t = \Delta \vec{p}, \qquad e = -\frac{v_{1}' - v_{2}'}{v_{1} - v_{2}}$$

📌 Concept: Total momentum is conserved in all isolated collisions. Elastic collisions conserve kinetic energy (e=1). Completely inelastic collisions stick together (e=0).

Stevin's Law (Hydrostatic Law)

$$P = P_0 + \rho g h$$

📌 Concept: Hydrostatic pressure at depth h in a liquid equals surface atmospheric pressure P_0 plus column pressure ρgh. Pressure depends only on depth h, NOT container shape!

Pascal's Principle (Hydraulics)

$$\frac{F_1}{A_1} = \frac{F_2}{A_2} \implies F_2 = F_1 \left(\frac{A_2}{A_1}\right), \qquad W_1 = W_2 \implies F_1 d_1 = F_2 d_2$$

📌 Concept: Pressure applied to an enclosed fluid is transmitted undiminished throughout. Mechanical advantage multiplies force by area ratio, while conserving total work done.

Archimedes' Buoyancy Law

$$F_b = \rho_{fluid} \cdot V_{submerged} \cdot g$$

📌 Concept: Any object completely or partially submerged experiences an upward buoyant force F_b equal to the weight of the fluid it displaces.

Continuity Equation (Volume Flow Rate)

$$Q = A_1 v_1 = A_2 v_2 = \text{constant} \quad (\text{m}^3/\text{s})$$

📌 Concept: Mass flow rate is conserved. When pipe cross-section constricts (A↓), fluid speed must increase proportionally (v↑).

Bernoulli's Equation (Energy Conservation in Fluids)

$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$

📌 Concept: Sum of static pressure P, dynamic pressure (0.5ρv²), and hydrostatic energy density (ρgh) is constant along any streamline. High velocity implies lower pressure!

Torricelli's Law (Efflux Speed from Tank)

$$v_{efflux} = \sqrt{2 g h}$$

📌 Concept: The speed of liquid flowing out of an orifice at depth h below an open surface equals free-fall speed from height h.

Sensible Heat & Latent Heat Transformations

$$Q = m c \Delta T = C \Delta T, \qquad Q_{phase} = m L$$

📌 Concept: Specific heat c (J/(kg·K)) governs temperature changes. Latent heat L (J/kg) governs phase transitions (fusion L_f or vaporization L_v) at CONSTANT temperature.

Calorimetry Thermal Equilibrium Principle

$$Q_{gained} + Q_{lost} = 0 \implies m_1 c_1 (T_f - T_1) + m_2 c_2 (T_f - T_2) = 0$$

📌 Concept: In an insulated calorimeter, total heat lost by hotter substances equals total heat gained by cooler substances until final equilibrium temperature T_f is reached.

Ideal Gas Law & Microscopic Kinetic Energy

$$P V = n R T = N k_B T, \qquad K_{avg} = \frac{3}{2} k_B T, \qquad U = \frac{3}{2} n R T$$

📌 Concept: Universal gas constant R = 8.314 J/(mol·K). Boltzmann constant k_B = R/N_A = 1.38 × 10⁻²³ J/K. Internal energy U of a monatomic ideal gas depends ONLY on temperature T!

First Law of Thermodynamics

$$\Delta U = Q - W$$

📌 Concept: Energy conservation: change in internal energy ΔU equals heat Q added to gas minus work W done BY gas on surroundings.

Thermal Engine Efficiency & Carnot Limit

$$\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}, \qquad \eta_{Carnot} = 1 - \frac{T_C}{T_H}$$

📌 Concept: Carnot efficiency is the theoretical upper limit for any heat engine operating between hot reservoir T_H and cold reservoir T_C (Temperatures MUST be in Kelvin!).

Fundamental Wave Equation & Frequency

$$v = f \lambda = \frac{\lambda}{T}, \qquad f = \frac{1}{T}$$

📌 Concept: Wave speed v depends entirely on the medium properties (e.g., tension and density for strings).

General Doppler Effect Formula

$$f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$$

📌 Concept: Use top signs (+ in numerator, - in denominator) when observer and source approach each other (higher apparent pitch f').

Snell's Law of Refraction & Critical Angle

$$n_1 \sin\theta_1 = n_2 \sin\theta_2, \qquad n = \frac{c}{v}, \qquad \sin\theta_c = \frac{n_2}{n_1} \quad (\text{for } n_1 > n_2)$$

📌 Concept: Light refracts toward normal when entering denser medium (n2 > n1). Total internal reflection occurs when incident angle exceeds critical angle θ_c.

Thin Lens & Spherical Mirror Equations

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}, \qquad m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}, \qquad P = \frac{1}{f} \quad (\text{Diopters m}^{-1})$$

📌 Concept: Sign convention: Convex lens has positive focal length ($f>0$). Real image has positive image distance ($d_i>0$). Negative $m$ indicates an inverted image.

Resonant Frequencies of Strings & Pipes

$$f_n = \frac{n v}{2L} \; (\text{string / open pipe}), \qquad f_n = \frac{n v}{4L} \; (\text{pipe closed at one end, } n = 1,3,5,\dots)$$

📌 Concept: $L$ is length and $v$ is wave speed. $n=1$ is fundamental frequency; higher $n$ give harmonics.

Coulomb's Law & Electric Field Intensity

$$F = k \frac{|q_1 q_2|}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{r^2}, \qquad \vec{E} = \frac{\vec{F}}{q_0} = k \frac{|Q|}{r^2} \hat{r}$$

📌 Coulomb constant $k \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2 \approx 9 \times 10^9$, permittivity $\epsilon_0 \approx 8.85 \times 10^{-12} \text{ F/m}$.

Gauss's Law & Electric Flux Applications

$$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}$$

📌 Key Applications: Infinite charged plate field $E = \frac{\sigma}{2\epsilon_0}$; Parallel plate field $E = \frac{\sigma}{\epsilon_0}$; Outside conducting sphere $E = k \frac{Q}{r^2}$; Inside conductor $E_{inside} = 0$.

Electric Potential, Potential Energy & Field Gradient

$$V = k \frac{Q}{r}, \qquad U_E = k \frac{q_1 q_2}{r} = q V, \qquad E = -\frac{\Delta V}{\Delta r}$$ $$W_{field} = -\Delta U_E = -q \Delta V = q (V_i - V_f)$$

Parallel Plate Capacitance & Dielectric Effect

$$C = \frac{Q}{V} = \epsilon \frac{A}{d} = \kappa \epsilon_0 \frac{A}{d}, \qquad U_C = \frac{1}{2} C V^2 = \frac{1}{2} Q V = \frac{Q^2}{2C}$$

📌 Dielectric constant $\kappa > 1$ increases capacitance. Field energy density $u_E = \frac{1}{2}\epsilon_0 E^2 \quad (\text{J/m}^3)$.

Ohm's Law, Resistivity & Joule Power Dissipation

$$V = I R, \qquad R = \rho \frac{L}{A}, \qquad P = V I = I^2 R = \frac{V^2}{R}, \qquad V_{term} = \mathcal{E} - I r$$

Kirchhoff's Rules (KCL & KVL)

$$\text{Junction Rule (KCL): } \sum I_{in} = \sum I_{out}, \qquad \text{Loop Rule (KVL): } \sum \Delta V = 0$$

RC Charging & Time Constant Formula

$$q(t) = Q_{max} \left(1 - e^{-t / \tau}\right), \qquad I(t) = I_0 e^{-t / \tau}, \qquad \tau = R C \quad (\text{seconds})$$

Lorentz Force & Cyclotron Motion

$$F_B = q v B \sin\theta, \qquad r = \frac{m v}{q B}, \qquad T = \frac{2\pi m}{q B}, \qquad F_{wire} = I L B \sin\theta$$

Faraday-Lenz Law & Motional EMF

$$\Phi_B = B A \cos\theta, \qquad \mathcal{E} = -N \frac{d\Phi_B}{dt}, \qquad \mathcal{E}_{motional} = B L v, \qquad U_L = \frac{1}{2} L I^2$$

Ideal Transformer Formula

$$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s} \implies P_p = P_s$$

Photon Energy & Photoelectric Effect Equation

$$E = h f = \frac{h c}{\lambda}, \qquad h f = W_0 + K_{max} = W_0 + e V_{stop}$$

📌 Concept: Planck constant h = 6.63 × 10⁻³⁴ J·s. Incident photon energy hf must exceed work function W_0 to eject electrons with max kinetic energy K_max.

De Broglie Matter Wavelength

$$\lambda = \frac{h}{p} = \frac{h}{m v}$$

📌 Concept: Wave-particle duality: every particle with momentum $p$ exhibits an associated matter wavelength $\lambda$. This is why electrons diffract like waves.

Bohr Model: Energy Levels & Atomic Spectra

$$E_n = -\frac{13.6}{n^2} \text{ eV}, \qquad \Delta E = E_{high} - E_{low} = h f = \frac{hc}{\lambda}$$

📌 Concept: Electrons occupy quantised energy levels ($n = 1, 2, 3, \dots$). A photon is absorbed when an electron jumps up and emitted when it falls down, producing the discrete lines of an atomic spectrum. ($1 \text{ eV} = 1.6\times10^{-19}$ J.)

Mass Defect & Nuclear Binding Energy

$$E = m c^2, \qquad \Delta E = \Delta m \cdot c^2$$

📌 Concept: Mass defect Δm during nuclear fusion or fission releases nuclear binding energy ΔE.

Radioactive Decay Law & Half-Life Formula

$$N(t) = N_0 e^{-\lambda t} = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}, \qquad T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$$

📌 Concept: Half-life T_1/2 is the time taken for half of the radioactive parent nuclei N_0 to decay.

Fission vs. Fusion (Energy from Mass Defect)

$$\text{Fission: heavy nucleus} \to \text{2 lighter nuclei} + \text{neutrons} + \text{energy}$$ $$\text{Fusion: 2 light nuclei} \to \text{heavier nucleus} + \text{energy} \quad (E = \Delta m\, c^2)$$

📌 Concept: Both release energy because the products have a higher binding energy per nucleon (peak near iron, Fe-56). Fission powers reactors/bombs; fusion powers stars and the Sun.