Simple Harmonic Motion — Formula Sheet

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√(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

For a simple pendulum

T = 2π√(l/g). Note that the time period depends only on length l and acceleration due to gravity g — NOT on mass of the bob or amplitude (for small oscillations).

For vertical oscillations of a spring-mass system

T = 2π√(m/k). Gravity does NOT affect the time period of vertical spring oscillations because the equilibrium position shifts but the restoring force

The total mechanical energy of a simple harmonic oscillator is constant (conserved) and equals E = ½mω²a² = ½ka².

Kinetic Energy (K) at displacement y

K = ½mω²(a² – y²). Maximum at y = 0 (mean position) and zero at y = ±a (extreme positions).

Potential Energy (U) at displacement y

U = ½mω²y² = ½ky². Minimum (zero) at mean position and maximum at extreme positions.