Simple Harmonic Motion (SHM)

SHM is periodic motion where the restoring force is proportional to displacement:

• F = -kx

Position as a function of time:

• x(t) = A cos(ωt + φ)

where A is amplitude, ω is angular frequency, φ is phase constant.

Period and Frequency:

• T = 2π/ω = 2π√(m/k)
• f = 1/T = ω/2π

Energy in Simple Harmonic Motion

In SHM, energy oscillates between kinetic and potential:

• KE = ½mv² = ½mω²A² sin²(ωt + φ)
• PE = ½kx² = ½kA² cos²(ωt + φ)
• Etotal = ½kA² = constant

SHM vs Circular Motion

SHM is the projection of uniform circular motion onto a diameter. An object moving in a circle at constant speed projects SHM onto any diameter.

Pendulums

For small angles (< 15°), a simple pendulum exhibits SHM:

• T = 2π√(L/g)

where L is length and g is gravitational acceleration.