Simple Harmonic Motion

Ideal Springs F Applied =kx k = spring constant x = displacement of the spring +x  pulled displacement -x  compressed displacement

Hooke’s Law F = -kx Describes the restoring force of an ideal spring Negative sign indicates that this force always goes in the opposite direction of displacement

Hooke’s Law This type of restoring force will create a back and forth or up and down type of motion This type of friction-free motion is designated simple harmonic motion The maximum excursion from equilibrium is the amplitude  A

Period Mass-Spring Complex T = 2π√(m/k) Pendulum T = 2π√(L/g)

The Reference Circle Simply  a ball moving in uniform circular motion The shadow cast by the ball on a film creates the same type of sinusoidal pattern It makes another model of simple harmonic motion

Displacement x = A cos q = A cos wt w = 2p/ T f = 1 / T w = 2pf ( w is often called angular frequency) X = A cos 2πft

Energy & Simple Harmonic Motion Welastic = ½ kxo2 – ½ kxf2 Us = ½ kx2 ETotal = ½ mv2 + ½ Iw2 + mgh + ½ kx2