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Oscillations and Waves

This resource explores the energy transformations occurring during simple harmonic motion (SHM) in spring-mass systems. It covers key concepts including kinetic energy (KE) expressed as KE = ½ mv², potential energy (PE) as PE = ½ kx², and total energy in SHM, given by ET = ½ mω²xo². Detailed energy-time graphs illustrate the variations in KE and PE as displacement changes, highlighting that KE is zero at maximum displacement (x0) and PE is zero at equilibrium. This comprehensive guide aids in understanding these fundamental physics principles.

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Oscillations and Waves

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  1. Oscillations and Waves Energy Changes During Simple Harmonic Motion

  2. 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)

  3. 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

  4. 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)

  5. 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

  6. 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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