Simple Harmonic Motion

F = −kx

y = a sin(ωt + φ₀) OR y = a cos(ωt + φ₀)

v = ωa cos(ωt)

v = ω√(a² − y²)

Acceleration: A = −ω²y

T = 2π√(m/k) [Spring-Mass — Horizontal]

T = 2π√(m/k) [Spring-Mass — Vertical]

T = 2π√(l/g) [Simple Pendulum]

ω = √(g/l) [Simple Pendulum Angular Frequency]

k = mg/l [Equivalent force constant for pendulum]

KE = (1/2)mω²a²cos²(ωt) = (1/2)mω²(a²−y²)

PE = (1/2)ky² = (1/2)mω²y² = (1/2)mω²a²sin²(ωt)

Total Energy E = (1/2)mω²a² = (1/2)ka²

E = KE + PE = constant

At y = a/√2: KE = PE = E/2