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Lecture 2 [10 Feb, 2020] PHY 115

 PHY 115

Wave and Oscillation




The solution of the differential equation of SHM: 
We know that, the differential equation of simple harmonic motion is,
  x=Asin(ωt+ϕ)  


    ♦ The Velocity:  v=dxdt=acos(ωt+ϕ)v=Aω1sin2(ωt+ϕ)=±Aω1(xA)2=±AωA2x2A2v=±ωA2x2  

  vmin=0,   when  x=avmax=ωA,   when x=0  

    ♦ The Acceleration:  a=dvdt=aω2sin(ωt+ϕ)=ω2x  | [x=Asin(ωt+ϕ)]ax  




    ♦ Total Energy: Total Energy = Kinetic energy + Potential energy

Total energy of a body executing SHM:
    • Kinetic Energy: 
  v=ωA2x2=Aωcos(ωt+ϕ)K.E=12mv2=12mω2(A2x2)K.E=12mv2=12mA2ω2cos2(ωt+ϕ)  

    • Potential Energy:
  F=ma=mω2xF=mω2asin(ωt+ϕ)P.E=x0F dx=x0mω2x dx=x0mω2x dx=mω2x0x dx12mω2x2  


  P.E=12mω2A2sin2(ωt+ϕ)  Total Energy = (5) + (7)=12mω2(A2x2)+12mω2x2=12mω2A2=12kA2             |ω2=km=12m(2πT)2A2=12m(2πn)2A2=2π2n2A2m  


    Assignment: Find out the average value of K.E and P.E of harmonic oscillator.

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