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=Aω√1−sin2(ωt+ϕ)=±Aω√1−(xA)2=±Aω√A2−x2A2∴v=±ω√A2−x2
vmin=0, when x=avmax=ωA, when x=0
♦ The Acceleration:
a=dvdt=−aω2sin(ωt+ϕ)=−ω2x | [x=Asin(ωt+ϕ)]∴a∝−x
♦ Total Energy: Total Energy = Kinetic energy + Potential energy
Total energy of a body executing SHM:
• Kinetic Energy:
v=ω√A2−x2=Aωcos(ωt+ϕ)K.E=12mv2=12mω2(A2−x2)K.E=12mv2=12mA2ω2cos2(ωt+ϕ)
• Potential Energy:
F=ma=−mω2xF=−mω2asin(ωt+ϕ)P.E=−∫x0F dx=−∫x0−mω2x dx=∫x0mω2x dx=mω2∫x0x dx12mω2x2
∴P.E=12mω2A2sin2(ωt+ϕ) Total Energy = (5) + (7)=12mω2(A2−x2)+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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