Chapter 1: Simple Harmonic Motion

SHM is defined by a = −ω^{2}x. The sinusoidal displacement x = A sin(ωt + φ) produces velocity v = ω√($A^{2}$ − $x^{2}$) and acceleration a = −ω^{2}x. The velocity is maximum (Aω) at the equilibrium position and zero at the extremes; acceleration is the reverse. Time period T = 2π/ω.

Energy in SHM: total energy E = ½mω^{2}$A^{2}$ is constant. KE = ½mω^{2}($A^{2}$ − $x^{2}$); PE = ½mω^{2}$x^{2}$. These are equal at x = A/√2 (not A/2 — the single most tested NEET trap in this chapter). At x = A/2, KE = 3E/4 and PE = E/4; they are unequal.

Chapter 2: Oscillating Systems

Spring-mass system: T = 2π√(m/k). Completely independent of amplitude and gravity. Series combination: 1/k_eff = 1/k_{1} + 1/k_{2} (lower effective k, longer T). Parallel: k_eff = k_{1} + k_{2} (higher k, shorter T).

Simple pendulum: T = 2π√(L/g). Independent of mass and amplitude (for θ < 15°). Effect of gravity changes: lift up → g_eff = g + a → T decreases; lift down → g_eff = g − a → T increases; free fall → g_eff = 0 → T = ∞.

Chapter 3: Wave Motion

Progressive wave: y = A sin(kx − ωt), wave number k = 2π/λ. Wave speed v = fλ = ω/k. Transverse wave in string: v = √(T/μ). Sound in gas: v = √(γRT/M) ∝ √T_K. Speed order: v_solid > v_liquid > v_gas.

Chapter 4: Standing Waves & Pipes

Standing wave: y = 2A sin(kx) cos(ωt). Nodes at x = nλ/2; antinodes at x = (2n+1)λ/4; node-to-node spacing = λ/2.

Closed pipe fundamental = ½ open pipe fundamental (same L). Closed pipe first overtone = 3rd harmonic.

Chapter 5: Beats and Doppler Effect

Beats: f_beat = |f_{1} − f_{2}|. When two tuning forks of 440 Hz and 444 Hz are sounded together, 4 beats per second are heard.

Doppler: f' = f(v ± v_O)/(v ∓ v_S). Use + numerator and − denominator for toward motion; reverse for away. Source approaching: f' > f. Source receding: f' < f. Applies to sound; analogous (relativistic) formula for light explains cosmological redshift.