SHM

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Simple Harmonic Motion — Key Facts SHM definition: acceleration a = -ω²x; restoring force proportional to displacement Equation: x = A cos(ωt + φ); v = -Aω sin(ωt + φ); a = -Aω² cos(ωt + φ) Time period: T = 2π√(m/k) for spring-mass; T = 2π√(L/g) for simple pendulum (small amplitude) Energy: PE = ½kx², KE = ½k(A² - x²); total E = ½kA² (constant) ⚡ Exam tip: At extreme (x=±A): v=0, a=max; At centre (x=0): v=max, a=0

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Simple Harmonic Motion — NEET/JEE Study Guide Spring-mass: T = 2π√(m/k); series spring: 1/k_eq = Σ(1/k_i); parallel springs: k_eq = Σk_i Damped SHM: amplitude decreases as A = A₀e^(-bt/2m); logarithmic decrement δ = ln(A_i/A_f) Forced oscillations: resonance when driving frequency = natural frequency; amplitude peaks at ω_r = √(ω₀² - 2β²) Simple pendulum: T = 2π√(L/g); valid for small θ₀; length L measured to centre of bob

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Simple Harmonic Motion — Comprehensive Notes Differential equation: d²x/dt² + (k/m)x = 0; solution x = A cos(ωt + φ) Quality factor Q = ω₀/(2β) for damped oscillator; Q = 2π × (energy stored)/(energy lost per period) Lissajous figures: superposition of two perpendicular SHMs; closed curves when frequency ratio is rational Physical pendulum: T = 2π√(I/mgd) where d = distance from pivot to COM; used to find g experimentally Oscillations in LC circuit: q = Q₀ cos(ωt), I = -I₀ sin(ωt); ω = 1/√(LC); analogous to spring-mass

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