  SHM17

### 1.3 a)    Variation with time of energy in simple harmonic motion If the variation with time of displacement is as shown, then the energies should be drawn as shown.

recalling Energy formula

KE = ½ m v2

PE =  ½ k x2

in terms of time t,

x = x0 sin(ωt)

differentiating with t gives

v = v0 cos (ωt)

therefore, KE = ½ m v2= ½ m (v0 cos (ωt))2= ½ m (x02ω2)cos (ωt))2

similarly

PE = ½ k x2= ½ (mω2 )(x0 sin (ωt))2= ½ m (x02ω2 )sin (ωt))2

therefore total energy is a constant value in the absence of energy loss due to drag (resistance)

TE = KE + PE = ½ m (x02ω2 )[cos2(ωt) + sin2(ωt))] = ½ m (x02ω2) this is how the x vs t looks together of the energy vs t graphs ### 1.3.1 Summary

the table shows some of the common values
 general energy formula SHM energy formula when t = 0 when t = T/4 when t = T/2 when t = 3T/4 when t = T KE = ½ m v2 ½ m (x02ω2)cos (ωt))2 ½ m (x02ω2) 0 ½ m (x02ω2) 0 ½ m (x02ω2) PE =  ½ k x2 ½ m (x02ω2)cos (ωt))2 0 ½ m (x02ω2) 0 ½ m (x02ω2) 0 TE = KE + PE TE = ½ m (x02ω2) ½ m (x02ω2) ½ m (x02ω2) ½ m (x02ω2) ½ m (x02ω2) ½ m (x02ω2)

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