Oscillations and Waves

Oscillations and Waves Energy Changes During Simple Harmonic Motion

velocity KE energy PE Total Energy in SHM Energy-time graphs Note:For a spring-mass system: KE = ½ mv2  KE is zero when v = 0 PE = ½ kx2  PE is zero when x = 0 (i.e. at vmax)

energy -xo +xo displacement KE PE Total Energy–displacement graphs Note:For a spring-mass system: KE = ½ mv2  KE is zero when v = 0 (i.e. at xo) PE = ½ kx2  PE is zero when x = 0

Kinetic energy in SHM We know that the velocity at any time is given by… v = ω √ (xo2 – x2) So if Ek = ½ mv2 then kinetic energy at an instant is given by… Ek = ½ mω2 (xo2 – x2)

Potential energy in SHM If a = - ω2x 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 mω2x Work done by this force must equal the PE it gains (e.g in the springs being stretched). Thus.. Ep = ½ mω2x2

Total Energy in SHM Clearly if we add the formulae for KE and PE in SHM we arrive at a formula for total energy in SHM: ET = ½ mω2xo2 Summary: Ek = ½ mω2 (xo2 – x2) Ep = ½ mω2x2 ET = ½ mω2xo2

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Oscillations and Waves