Oscillations and Waves Energy Changes During Simple Harmonic Motion

Energy in SHM Energy-time graphs velocity KE PE Total energy 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 displacement +x o -x o KE PE Total 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

Kinetic energy in SHM We know that the velocity at any time is given by… v = ω (x o 2 – x 2 ) So if E k = ½ mv 2 then kinetic energy at an instant is given by… E k = ½ mω 2 (x o 2 – x 2 )

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 = ½ mω 2 x 2

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: E T = ½ mω 2 x o 2 Summary: E k = ½ mω 2 (x o 2 – x 2 ) E p = ½ mω 2 x 2 E T = ½ mω 2 x o 2

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