Simple Harmonic Motion

SHM Position, Velocity, and Acceleration

Springs and Simple Harmonic Motion

Equations of Motion

Conservation of Energy allows a calculation of the velocity of the object at any position in its motion…

Energy in SHM

Energy-time graphs KE PE Total Note:

For a spring-mass system:

KE = ½ mv 2  KE is zero when v = 0 PE = ½ kx 2  PE is zero when x = 0 (i.e. at v max )

Energy –displacement graphs

energy KE PE Total -x o displacement +x o Note:

For a spring-mass system:

KE = ½ mv 2  KE is zero when v = 0 (i.e. at x o ) PE = ½ kx 2  PE is zero when x = 0

Conservation of Energy For A Spring in Horizontal Motion E = Kinetic + Elastic Potential E = ½ mv 2 + ½ kx 2 = Constant

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At maximum displacement, velocity is zero and all energy is elastic potential, so total energy is equal to ½ kx o 2

Simple Harmonic Motion

Energy E k (max) = 1 / 2 mv o 2 E p (max) = 1 / 2 kx o 2 Where they happen •E k : 0 max 0 •E p : max 0 max

Potential energy in SHM

If a = ω 2 x then the average force applied trying to pull the object back to the equilibrium position as it moves away from the equilibrium position is… F = ½ mω 2 x Work done by this force must equal the PE it gains (e.g in the springs being stretched). Thus..

E p (max) = ½ mω 2 x o 2