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 Describes the restoring force of an ideal spring F = -kx 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  = A cos  t  = 2  / T f = 1 / T  = 2  f (  is often called angular frequency)

Velocity & Acceleration v = -A  sin  = -A  sin  t v max = A  a = -A  2 cos  = -  2 cos  t a max = A  2

Frequency of Vibration  = (k / m) 1/2  must be in radians per second k  spring constant m  mass

Energy & Simple Harmonic Motion W elastic = ½ kx o 2 – ½ kx f 2 PE elastic = ½ kx 2 E Total = ½ mv 2 + ½  2 + mgh + ½ kx 2