### About

### 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

^{2}

PE = ½ k x^{2}

in terms of time t,

x = x_{0} sin(ωt)

differentiating with t gives

v = v_{0} cos (ωt)

^{2}= ½ m (v

_{0}cos (ωt))

^{2}= ½ m (x

_{0}

^{2}ω

^{2})cos (ωt))

^{2}

similarly

PE = ½ k x^{2}= ½ (mω^{2} )(x_{0} sin (ωt))^{2}= ½ m (x_{0}^{2}ω^{2} )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 (x_{0}^{2}ω^{2} )[cos^{2}(ωt)
+ sin^{2}(ωt))] = ½ m (x_{0}^{2}ω^{2})

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 valuesgeneral 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 v^{2} | ½ m (x_{0}^{2}ω^{2})cos
(ωt))^{2} | ½ m (x_{0}^{2}ω^{2}) | 0 | ½ m (x_{0}^{2}ω^{2}) | 0 | ½ m (x_{0}^{2}ω^{2}) |

PE = ½ k x^{2} | ½ m (x_{0}^{2}ω^{2})cos
(ωt))^{2} | 0 | ½ m (x_{0}^{2}ω^{2}) | 0 | ½ m (x_{0}^{2}ω^{2}) | 0 |

TE = KE + PE | TE = ½ m (x_{0}^{2}ω^{2}) | ½ m (x_{0}^{2}ω^{2}) | ½ m (x_{0}^{2}ω^{2}) | ½ m (x | ½ m (x_{0}^{2}ω^{2}) | ½ m (x_{0}^{2}ω^{2}) |

### 1.3.2 Model:

### Translations

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### end faq

## Apps

https://play.google.com/store/apps/details?id=com.ionicframework.shm17app307408&hl=en

Other resources

### end faq

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