Unit I : Kinematics
11th Class
→Frame of Reference & Motion in a Straight Line
A frame of reference is a coordinate system used to describe the position of objects.
Uniform motion: equal displacement in equal intervals of time → velocity = constant.
Non-uniform motion: velocity changes with time (acceleration ≠ 0).
v = u + at | s = ut + ½at² | v² = u² + 2as
v-t graph: slope = acceleration; area under curve = displacement.
x-t graph: slope = velocity; straight line → uniform motion.
Equations of motion valid only for constant acceleration.
→Vectors & Scalars
Unit vector: magnitude = 1
î, ĵ, k̂ along x, y, z axes
â = A⃗ / |A|
î, ĵ, k̂ along x, y, z axes
â = A⃗ / |A|
Scalar: magnitude only (mass, speed, temperature, energy).
Vector: magnitude + direction (displacement, velocity, force, acceleration).
Relative velocity: V⃗_AB = V⃗_A − V⃗_B
Resolution: Aₓ = A cosθ, Aᵧ = A sinθ
Triangle law / Parallelogram law for vector addition.
Dot product A⃗·B⃗ = AB cosθ (scalar); Cross product A⃗×B⃗ = AB sinθ n̂ (vector)
→Projectile Motion & Uniform Circular Motion
Projectile: horizontal → uniform motion; vertical → uniformly accelerated.
Range R = u²sin2θ/g | Max height H = u²sin²θ/2g | T = 2u sinθ/g
Maximum range at θ = 45°.
UCM: speed constant, velocity changes direction → centripetal acceleration.
a_c = v²/r = ω²r | F_c = mv²/r
Centripetal force is directed towards centre; it is NOT a new force but provided by friction/tension/gravity.
Unit II : Laws of Motion 11th Class
→Newton's Laws
1st Law (Inertia): Body remains at rest or uniform motion unless external force acts.
2nd Law: F⃗ = ma⃗ = dp⃗/dt (net force = rate of change of momentum).
3rd Law: Every action has equal and opposite reaction (different bodies).
Impulse J = FΔt = Δp = m(v−u)
Conservation of Linear Momentum: If ΣF_ext = 0, total momentum conserved.
→Friction
Static friction (f_s): acts on stationary body; f_s ≤ μ_s N
Kinetic friction (f_k): acts on moving body; f_k = μ_k N (μ_k < μ_s)
Rolling friction << kinetic friction (that's why wheels are used!)
Angle of friction: tanλ = μ | Angle of repose: tanθ = μ
Friction is self-adjusting up to limiting value (static friction).
→Dynamics of Circular Motion
Banking of roads: tanθ = v²/rg (no friction needed at this speed).
Conical pendulum: T cosθ = mg; T sinθ = mω²r
v_max (banked) = √(rg·tan(θ+λ)) where λ = friction angle
At top of vertical circle: v_min = √(gr) (minimum for contact)
Unit III : Work, Energy and Power 11th Class
Work W = F⃗·d⃗ = Fd cosθ (scalar); negative when θ > 90°
Kinetic Energy KE = ½mv² | Work-Energy theorem: W_net = ΔKE
Conservative force: work done is path-independent (gravity, spring force).
Non-conservative: work is path-dependent (friction, air resistance).
PE of spring = ½kx² | Conservation: KE + PE = constant (conservative forces)
Power P = W/t = Fv cosθ; SI unit: Watt (1 hp = 746 W)
Elastic collision: KE conserved. Inelastic: KE not conserved. Perfectly inelastic: bodies stick together.
e = (v₂−v₁)/(u₁−u₂) — coefficient of restitution (0 ≤ e ≤ 1)
Unit IV : Classical Mechanics UG–PG Level
→D'Alembert's Principle & Lagrangian Mechanics
D'Alembert: (F − ma)·δr = 0 → transforms dynamics into statics using virtual work.
Lagrangian L = T − V; Euler-Lagrange: d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0
Generalised coordinates qᵢ describe system with constraints.
Cyclic coordinate: ∂L/∂qᵢ = 0 → corresponding canonical momentum pᵢ is conserved.
Canonical momentum: pᵢ = ∂L/∂q̇ᵢ
→Hamiltonian Mechanics & Poisson Brackets
H = Σpᵢq̇ᵢ − L = T + V (for conservative systems)
Hamilton's equations: q̇ᵢ = ∂H/∂pᵢ; ṗᵢ = −∂H/∂qᵢ
Poisson Bracket: {f,g} = Σ(∂f/∂qᵢ·∂g/∂pᵢ − ∂f/∂pᵢ·∂g/∂qᵢ)
Fundamental PBs: {qᵢ,pⱼ} = δᵢⱼ; {qᵢ,qⱼ} = 0; {pᵢ,pⱼ} = 0
Hamilton-Jacobi equation: H(qᵢ, ∂S/∂qᵢ, t) + ∂S/∂t = 0
→Central Force & Kepler's Laws
Central force: F = f(r) r̂ — always directed along line joining bodies.
Angular momentum L conserved → motion in a plane.
Kepler's Laws: (1) Elliptical orbits; (2) Equal areas in equal times (L conserved); (3) T² ∝ a³
Kepler's 3rd: T² = (4π²/GM) a³
→Special Theory of Relativity
Postulates: (1) Laws of physics same in all inertial frames; (2) Speed of light c is constant.
Lorentz factor: γ = 1/√(1−v²/c²)
Time dilation: Δt = γΔt₀ (moving clock runs slow)
Length contraction: L = L₀/γ (moving rod is shorter)
Mass-energy equivalence: E = mc² | E² = (pc)² + (m₀c²)²
Relativistic momentum: p = γm₀v
Relativistic velocity addition: u' = (u−v)/(1−uv/c²)
Unit V : Oscillations and Waves 11th / 12th
→Simple Harmonic Motion (SHM)
SHM: restoring force ∝ −displacement; F = −kx → a = −ω²x
x = A sin(ωt + φ) | ω = √(k/m) | T = 2π/ω = 2π√(m/k)
Energy in SHM: KE = ½mω²(A²−x²); PE = ½mω²x²; Total E = ½mω²A² (constant)
Simple pendulum: T = 2π√(L/g) (valid for small angles)
Spring-mass: Series: 1/k_eff = 1/k₁ + 1/k₂; Parallel: k_eff = k₁ + k₂
→Damped, Forced Oscillations & Resonance
Damped: d²x/dt² + 2β dx/dt + ω₀²x = 0; x = Ae^(−βt)cos(ω't+φ)
Underdamped (β < ω₀): oscillates with decreasing amplitude.
Overdamped: no oscillation, slow return. Critically damped: fastest return without oscillation.
Resonance: when driving frequency = natural frequency → maximum amplitude.
Quality factor Q = ω₀/2β (sharpness of resonance)
→Waves
Transverse waves: displacement ⊥ propagation (light, string). Longitudinal: displacement ∥ propagation (sound).
v = fλ | y = A sin(kx − ωt) | k = 2π/λ, ω = 2πf
Standing waves: y = 2A sin(kx) cos(ωt) — nodes (zero amplitude), antinodes (max amplitude).
String: f_n = n/2L · √(T/μ); Organ pipe (closed): f_n = (2n−1)v/4L
Doppler: f' = f(v ± v_o)/(v ∓ v_s)
Beats: f_beat = |f₁ − f₂| (heard as periodic variation in loudness)
Superposition principle: resultant = algebraic sum of individual displacements.
Unit VI : Gravitation 11th Class
Newton's Law: F = GMm/r² (attractive, acts along line joining masses)
g = GM/R² (surface); g_h = g(1 − 2h/R) at height h; g_d = g(1 − d/R) at depth d
Gravitational PE = −GMm/r | Escape velocity v_e = √(2GM/R) = √(2gR)
v_e at Earth = 11.2 km/s
Orbital velocity: v_o = √(GM/r) = √(gR²/(R+h))
Geo-stationary orbit: T = 24 h, h ≈ 36,000 km above equator, v_o ≈ 3.07 km/s
Kepler's 3rd: T² ∝ r³ → T² = (4π²/GM)r³
Gravitational potential V = −GM/r (negative, zero at infinity)
Unit VII : Electrostatics 12th Class
→Coulomb's Law & Electric Field
F = kq₁q₂/r² | k = 1/4πε₀ = 9×10⁹ Nm²/C²
Electric field E = F/q₀ = kq/r² (due to point charge)
Electric flux Φ = E⃗·A⃗ = EA cosθ
Gauss's Law: Φ = Σq_enc/ε₀
Applications: infinite sheet → E = σ/2ε₀; sphere → E = kQ/r² (outside)
Dipole: p = qd; E along axis = 2kp/r³; E along equator = kp/r³
→Electric Potential & Capacitance
V = kq/r | E = −dV/dr | W = q(V_A − V_B)
Equipotential surfaces: perpendicular to field lines, no work done moving along them.
Capacitance C = Q/V; Parallel plate: C = ε₀A/d; with dielectric: C = κε₀A/d
Energy in capacitor: U = Q²/2C = ½CV² = QV/2
Series: 1/C_eq = Σ1/Cᵢ; Parallel: C_eq = ΣCᵢ
Van de Graaff generator: builds high static potential using conveyor belt.
Unit VIII : Electric Conduction 12th Class
Drift velocity v_d = eE τ/m | I = nev_dA | J = σE = E/ρ
Mobility μ = v_d/E = eτ/m
Resistivity ρ = ρ₀(1 + αΔT) — increases with temperature for metals
Ohm's Law: V = IR; Power P = VI = I²R = V²/R
Kirchhoff's: Junction rule ΣI = 0; Loop rule ΣV = 0
Wiedemann-Franz Law: K/σT = L (Lorenz number, constant for metals)
Colour code for resistors: B B ROY Great Britain Very Good Wife (Black, Brown, Red, Orange, Yellow, Green, Blue, Violet, Grey, White → 0–9)
Unit IX : Magnetic Effects of Current & Magnetism 12th Class
Biot-Savart: dB = μ₀/4π · Idl sinθ/r²
Ampere's Law: ∮B⃗·dl⃗ = μ₀I_enc
Force on moving charge: F = qv×B (Lorentz force)
Cyclotron: r = mv/qB; frequency f = qB/2πm (independent of speed)
Force between parallel wires: F/L = μ₀I₁I₂/2πd (attractive if same direction)
Torque on loop: τ = NIAB sinθ = M×B
Moving coil galvanometer: θ = NIAB/k; converted to ammeter by low shunt, voltmeter by high series R.
Para (χ > 0, weak), Dia (χ < 0), Ferro (χ >> 0) — know examples!
Unit X : EM Induction & Alternating Currents 12th Class
Faraday: ε = −dΦ/dt | Φ = BA cosθ
Lenz's law: induced current opposes cause (conservation of energy).
Self inductance: ε = −L dI/dt; energy in inductor U = ½LI²
Mutual inductance M: ε₂ = −M dI₁/dt; M = k√(L₁L₂)
AC: I = I₀ sinωt; V = V₀ sinωt; I_rms = I₀/√2; V_rms = V₀/√2
X_L = ωL (inductive); X_C = 1/ωC (capacitive); Z = √(R² + (X_L−X_C)²)
Resonance in LCR: ω₀ = 1/√(LC); Q = ω₀L/R
Power factor: cosφ = R/Z; Wattless current: I sinφ (no power dissipation)
Transformer: V₂/V₁ = N₂/N₁ = I₁/I₂; Eddy currents — losses (laminated core reduces them)
Unit XI : Mathematical Methods of Physics UG–PG Level
Vector Calculus: Gradient ∇f, Divergence ∇·F⃗, Curl ∇×F⃗
Gauss's theorem: ∮F⃗·dA⃗ = ∫∇·F⃗ dV | Stokes: ∮F⃗·dl⃗ = ∫(∇×F⃗)·dA⃗
Matrices: Orthogonal (AᵀA = I), Unitary (A†A = I), Hermitian (A† = A)
Eigenvalue equation: Ax = λx; det(A−λI) = 0
Special functions: Legendre (P_n), Bessel (J_n), Hermite (H_n), Laguerre (L_n) — arise in wave equations
Tensors: covariant (lower index), contravariant (upper index), mixed. Metric tensor g_μν
Christoffel symbol Γᵅ_μν — connection coefficients; Ricci tensor R_μν — curvature
Epsilon tensor ε_ijk: completely antisymmetric, ε₁₂₃ = 1; used in cross products