Question

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

Answer 1

Answer

harmonic oscillator differential equation is given by

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

Solution of above euation can be calculated by differential equation

Which is found

$x=Ae^{-\beta t}$

Differentiate with respect to t

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

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

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

Putting value of velocity

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

Where

$E_0= 1/2KA^2$