1. Simple Harmonic Motion (SHM) Kinematics

Differential Equation of linear SHM: d²x/dt² + ω² x = 0, where angular frequency ω = 2π/T = √(k/m). Kinematic equations: Displacement: x(t) = A sin(ωt + φ). Velocity: v = dx/dt = A ω cos(ωt + φ) = ±ω√(A² - x²). Maximum velocity at mean position (x = 0): v_max = A ω. Acceleration: a = dv/dt = -A ω² sin(ωt + φ) = -ω² x. Maximum acceleration at extremes (x = ±A): a_max = ω² A. Acceleration is always directly proportional to displacement and directed towards the mean equilibrium position.

2. Energy in Simple Harmonic Motion

Kinetic Energy at displacement x: K = (1/2) m v² = (1/2) m ω² (A² - x²). Maximum at mean position (K_max = 1/2 m ω² A²), zero at extreme amplitudes. Potential Energy: U = (1/2) k x² = (1/2) m ω² x². Zero at mean position, maximum at extremes (U_max = 1/2 m ω² A²). Total Mechanical Energy: E = K + U = (1/2) m ω² A² = constant (conserved at every point of oscillation). Average kinetic energy over full cycle equals average potential energy: <K> = <U> = (1/4) m ω² A² = E / 2.

3. Pendulums & Spring-Mass Oscillation Systems

Simple Pendulum of length l: Time period for small angular amplitude: T = 2π√(l / g). Frequency f = (1 / 2π)√(g / l). Seconds pendulum (T = 2s) has length l ≈ 0.993 m ≈ 1 m on Earth. Spring-Mass System (mass m, spring constant k): T = 2π√(m / k). Combinations of springs: (1) Series combination: 1/k_eq = 1/k1 + 1/k2 (forces are equal, displacements add); (2) Parallel combination: k_eq = k1 + k2 (displacements are equal, restoring forces add).

4. Progressive Wave Equation & Speed of Sound

Equation of 1D sinusoidal wave traveling in positive x-direction: y(x, t) = A sin(k x - ω t + φ) = A sin[2π (x/λ - t/T)]. Wave number (propagation constant): k = 2π / λ. Angular frequency: ω = 2π f. Wave speed: v = ω / k = f λ. Speed of transverse wave on stretched string of mass per unit length μ under tension T: v = √(T / μ). Speed of longitudinal sound wave in fluid of bulk modulus B and density ρ: Newton's formula with Laplace correction: v = √(γ P / ρ) = √(γ R T / M). Sound speed increases with temperature: v_T ≈ v_0 + 0.61 t(°C).

5. Standing Waves in Organ Pipes & Stretched Strings

Superposition of two identical waves traveling in opposite directions produces standing waves with nodes (zero displacement) and antinodes (maximum displacement). Distance between two consecutive nodes or antinodes is λ/2; between adjacent node and antinode is λ/4. (1) Open Organ Pipe (open at both ends, length L): Fundamental frequency f1 = v / (2L); Harmonics: fn = n (v / 2L), where n = 1, 2, 3... (ALL integer harmonics present). (2) Closed Organ Pipe (closed at one end, length L): Fundamental f1 = v / (4L); Harmonics: fn = (2n - 1) (v / 4L), where n = 1, 2, 3... (ONLY ODD harmonics present).

6. The Doppler Effect for Sound

Apparent frequency f' heard by observer when source (speed vs) and observer (speed vo) move along line of sight with sound speed v: General formula: f' = f · [(v ± vo) / (v ∓ vs)]. Sign rule: If motion tends to REDUCE the distance between source and observer (approaching), apparent frequency INCREASES (use + in numerator, - in denominator). If motion tends to INCREASE the distance (receding), apparent frequency DECREASES (use - in numerator, + in denominator). Note: Doppler effect depends on relative motion to medium; no frequency shift occurs if wind blows while source and observer are stationary.