Simple Harmonic Motion

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Definition: A particle executes SHM if it moves to and fro about a fixed mean position under a restoring force that is directly proportional to its displacement from the mean position and is always di…

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Periodic Motion: Any motion that repeats itself after a fixed interval of time (time period T). → Example: Revolution of Earth around the Sun (repeats every 365 days) — Non-oscillatory periodic motion…

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General Equations of SHM: y = a sin(ωt + φ₀) OR y = a cos(ωt + φ₀) Where: • y = displacement from mean position at time t (m) • a = amplitude (maximum displacement) (m) • ω = angular frequency…

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Derivation Using Circle of Reference: Step 1: Consider a point M moving with constant speed v in a circle of radius a (amplitude). At t=0, M is at X. The phasor OM rotates with angular velocity ω = v…

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Given: • Mass of tray, m₁ = 9 kg, T₁ = 1.0 s • Mass of tray + block, m₂ = 9 + M, T₂ = 2.0 s Formula: T = 2π√(m/k) → T² = 4π²m/k → m = kT²/4π² Step 1: For empty tray: 9 = k(1)²/4π² → k = …

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Given: • k = 1600 N/m • m = 4.0 kg • Amplitude a = 4.0 cm = 0.04 m Step 1 – Angular Frequency: ω = √(k/m) = √(1600/4) = √400 = 20 rad/s Step 2 – Frequency: ν = ω/2π = 20/2π = 20/6.28 ≈ 3.18 Hz Step…

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Derivation: Setup: A bob of mass m hangs from a string of length l. It is displaced by a small angle θ from vertical. Step 1: Forces on bob: • Weight mg (downward) • Tension T (along string, upward)…

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Why independent of mass: Restoring force F = −mg sinθ ≈ −(mg/l)x Acceleration a = F/m = −(g/l)x The mass m cancels out. So acceleration (and hence ω and T) does not depend on mass. T = 2π√(l/g) Fac…