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Energy in SHM

A particle executing SHM possesses two types of energy: potential energy and kinetic energy.

Potential energy

Potential energy is an account of the displacement of the particle from its mean position. The restoring force F = –ky against which work has to be done.
 
So 60273.png
 
 
Important Points
  • Potential energy maximum and equal to total energy at extreme positions
     
    60705.png
     
    when y = ±aωt = π/2; t = T/4
  • Potential energy is minimum at mean position,
     
    Umin = 0, when y = 0; ωt = 0; t = 0

Kinetic energy

Kinetic energy is because of the velocity of the particle.
 
Kinetic energy,
 
60230.png
  • Kinetic energy is maximum at mean position and equal to total energy at mean position.
60206.png when y = 0; t = 0; ω t = 0
  • Kinetic energy is minimum at extreme position.
Kmin = 0, when y = at = T/4, ωt = π/2;

Total energy

Total mechanical energy = Kinetic energy + Potential energy
 
E = 60200.png
 
Total energy is not a position function, i.e., it always remains constant.

Energy position graph

Kinetic energy (K60194.png
 
Potential Energy (U) = 60187.png
 
Total energy (E) = 62250.png
 
It is clear from Fig. 2 that
 
60181.png
Fig. 2
  • Kinetic energy is maximum at mean position and minimum at extreme position.
  • Potential energy is maximum at extreme position and minimum at mean position.
  • Total energy always remains constant.
Kinetic energy
 
60169.png
60162.png60156.png
 
Potential energy
 
60150.png60144.png
 
where ω = 2ω and 60136.png, i.e., in SHM, kinetic energy and potential energy vary periodically with double the frequency of SHM (i.e., with time period T = T/2).
 
From Fig. 3, we note that potential energy or kinetic energy completes two vibrations in a time during which SHM completes one vibration. Thus, the frequency of potential energy or kinetic energy is double than that of SHM.
 
60130.png
Fig. 3




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