Energy in a horizontal vibrating spring

Energy transformation always takes place between elastic potential energy and kinetic energy

The maximum of the kinetic energy  occurs when the object is passing through the central position.

The maximum in potential energy occurs in the extreme position

In an undamped vibration :  Emech =  Ek + Ep  = constant.

E mech = Ek + Ep = Ek,max = Ep,max

Emech = ½ m vmax2 = ½ m (2 π A/T)2 = ½ m (4 π2 A2 / T2) = (2 π2 A2) /T2 = 2 m π2 A2 f2

Emech = ½ C A2      (Ep in extreme position)

 T = 2 π √m/C     so     C = 4 π2 m/T2     

This gives :

Emech = ½    (4 π2 m/T2) A2 = (2 m π2 A2) T2 =      2 m π2 A2 f2

  • Emech  in J
  • m   mass in kg
  • A    amplitude in m
  • f     frequency in Hz
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