C.1 Simple harmonic motion

Definition

Simple harmonic motion (SHM) occurs when the acceleration is proportional to the displacement from the equilibrium position and always directed towards it: a = −ω²x.

Quantities

• Period T and frequency f = 1 / T. Angular frequency ω = 2π / T = 2πf.
• Displacement x = x₀ sin ωt (starting at equilibrium) or x = x₀ cos ωt (starting at maximum displacement).
• Velocity v = ±ω√(x₀² − x²). Maximum speed v_max = ωx₀ at the equilibrium position.
• Maximum acceleration a_max = ω²x₀ at the extremes of the motion.

Two standard systems

• Mass on a spring: T = 2π√(m / k).
• Simple pendulum (small angles): T = 2π√(L / g).

Energy

• Total energy E = ½mω²x₀² is constant when there is no damping.
• Kinetic energy Ek = ½mω²(x₀² − x²) and potential energy Ep = ½mω²x².
• At the equilibrium position Ek is a maximum and Ep is zero; at the extremes Ep is a maximum and Ek is zero.