Simple Harmonic Motion

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Periodic motion is motion that repeats itself at equal intervals of time. The smallest interval of time after which the motion repeats is called the time period (T). Examples: 1. Motion of planets ar…

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Periodic Motion: Any motion that repeats after equal intervals of time (includes rotational motion like planets around sun) Oscillatory Motion: To and fro motion about a mean position (like pendulum,…

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Simple Harmonic Motion: A particle executes SHM if its acceleration is directly proportional to its displacement from mean position and is always directed towards the mean position. Mathematical Cond…

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Differential Equation of SHM: d²x/dt² + ω²x = 0 General Solutions: x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ) Where: - A = amplitude - ω = angular frequency - φ = initial phase - t = time…

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Amplitude (A): Maximum displacement from mean position (units: m) Time Period (T): Time taken to complete one oscillation (units: s) T = 2π/ω Frequency (f): Number of oscillations per second (units:…

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Given: x(t) = A sin(ωt + φ) Velocity: v = dx/dt = d/dt[A sin(ωt + φ)] v = Aω cos(ωt + φ) Maximum velocity: vₘₐₓ = Aω (at mean position) Acceleration: a = dv/dt = d/dt[Aω cos(ωt + φ)] a = -Aω² sin(ω…

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SHM can be represented as the projection of uniform circular motion on a diameter. Consider a particle moving in a circle of radius A with angular velocity ω: - Position: y = A sin(ωt) - The projecti…

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For horizontal spring-mass system: Restoring force: F = -kx (Hooke's law) From Newton's second law: F = ma Therefore: ma = -kx a = -(k/m)x Comparing with a = -ω²x: ω² = k/m ω = √(k/m) Time period: …