Oscillations

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Simple Harmonic Motion is the periodic motion of a body in which the restoring force is directly proportional to the displacement from the mean position and always directed towards the mean position.

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Differential equation of SHM: d²x/dt² + ω²x = 0 Where ω² = k/m (k = force constant, m = mass) Standard solution: x = A sin(ωt + φ) Where: - A = amplitude - ω = angular frequency - φ = initial phase

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Given: x = A sin(ωt + φ) Velocity: v = dx/dt = d/dt[A sin(ωt + φ)] v = Aω cos(ωt + φ) Acceleration: a = dv/dt = d/dt[Aω cos(ωt + φ)] a = -Aω² sin(ωt + φ) a = -ω²x This confirms that acceleration is

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Given: A = 4 cm = 0.04 m, T = 2 s Step 1: Find angular frequency ω = 2π/T = 2π/2 = π rad/s Step 2: Maximum velocity vₘₐₓ = ωA = π × 0.04 = 0.04π m/s ≈ 0.126 m/s Step 3: Maximum acceleration aₘₐₓ =

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Period of simple pendulum: T = 2π√(L/g) Where L = length, g = acceleration due to gravity Laws of simple pendulum: 1. Period is directly proportional to √L 2. Period is inversely proportional to √g

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Given: L = 1 m, g_Earth = 9.8 m/s², g_Moon = 1.6 m/s² Using T = 2π√(L/g) On Earth: T_Earth = 2π√(1/9.8) = 2π√(0.102) = 2π × 0.319 = 2.006 s On Moon: T_Moon = 2π√(1/1.6) = 2π√(0.625) = 2π × 0.791 =

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A second's pendulum is a simple pendulum whose time period is exactly 2 seconds. For second's pendulum: T = 2 s Using T = 2π√(L/g) 2 = 2π√(L/9.8) Squaring both sides: 4 = 4π²(L/9.8) L = 4 × 9.8/(4π

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For SHM: x = A sin(ωt + φ), v = Aω cos(ωt + φ) Kinetic Energy: KE = ½mv² = ½mA²ω² cos²(ωt + φ) Potential Energy: PE = ½kx² = ½kA² sin²(ωt + φ) = ½mω²A² sin²(ωt + φ) Total Energy: E = KE + PE = ½mω²