Simple Harmonic Motion — Equation and Key Formulas

Displacement: x = A sin(ωt + φ)

Or equivalently: x = A cos(ωt + φ)

Where:

• x = displacement from mean position (metres)
• A = amplitude (maximum displacement)
• ω = angular frequency = 2π/T = 2πf (rad/s)
• t = time (seconds)
• φ = initial phase angle (radians)
• T = time period (seconds)
• f = frequency (Hz)

By differentiating x = A sin(ωt + φ):

Velocity: v = dx/dt = Aω cos(ωt + φ)

Maximum velocity (at mean position): v_max = Aω

Acceleration: a = dv/dt = −Aω² sin(ωt + φ) = −ω²x

Maximum acceleration (at extreme position): a_max = Aω²

Key relationship: a = −ω²x (acceleration is proportional to displacement and opposite in direction — this IS the definition of SHM)

For a simple pendulum: T = 2π√(L/g) Where L = length, g = 9.8 m/s²

For a spring-mass system: T = 2π√(m/k) Where m = mass, k = spring constant

For any SHM: T = 2π/ω = 1/f

Kinetic Energy: KE = ½m(Aω)²cos²(ωt) = ½mω²(A²−x²) Potential Energy: PE = ½mω²x² Total Energy: E = KE + PE = ½mω²A² (constant!)

At mean position (x=0): KE is maximum, PE = 0 At extreme position (x=A): KE = 0, PE is maximum Total energy is conserved throughout SHM.