Question #154508

Show the energy consideration of harmonic oscillator at any time t and show that the energy decreases exponentially at a faster rate , E(t)= E0e-2βt  where E0= 1/2KA2

1
Expert's answer
2021-01-11T11:42:47-0500

Answer

harmonic oscillator differential equation is given by

md2xdt2+kx=0\frac{md^2x}{dt^2}+kx=0

Solution of above euation can be calculated by differential equation

Which is found

x=Aeβtx=Ae^{-\beta t}

Differentiate with respect to t

v=dxdt=Aβeβtv=\frac{dx}{dt}=-A\beta e ^{-\beta t}

the energy consideration of harmonic oscillator at any time t is given by

E=mv22E=\frac{mv^2}{2}

Putting value of velocity

E=e2βt2KA2=E0e2βtE=\frac{e^{-2\beta t}}{2KA^2}=E_0e^{-2\beta t}

Where

E0=1/2KA2E_0= 1/2KA^2


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