Write the expression of mechanical energy of damped oscillator.

Explanation

The mechanical energy of SHO `E= (1)/(2) kA^(2)`.
If the amplitude of oscillation as `Ae^(-(bt)/(2m))`, then mechanical energy of oscillator obtain.
Mechanical energy of SHO at time t,
`E(t) = (1)/(2)kA^(2) e^(-(bt)/(2m))" ""………"(1)`
Hence, mechanical energy is not constant but decreases exponentially with time.
This equation is ture for small damped oscillation for small damping `b lt lt sqrt(km)`, where `(b)/(sqrt(km)) lt lt 1`, means the ratio is much less than 1 and it is dimensionless.
If b=0, all equations of a damped oscillator reduced to the corresponding equation of an undamped oscillator.
For example : Equation of damped motion,
`x(t) = Ae^(-(bt)/(2m)) cos (omega.t + phi)`
and `omega. = sqrt((k)/(m)-(b^2)/(4m^2))" let us put " b=0, =sqrt((k)/(m))`
`therefore omega. = omega`
and `x(t) = Ae^(0) cos (omega t + phi)`
`=A cos (omega t+phi) " "[therefore e^(0) =1]` and mechanical energy `E(t) = (1)/(2)kA^(2)e^(-(bt)/(2m))," if "b=0`
`E(t) = (1)/(2) kA^(2)` and in amplitude `=Ae^(-(bt)/(2m)), b=0 " "= A`
and the equation of simple harmonic oscillator `m(d^(2)x)/(dt^2)+b(dx)/(dt)+kx =0," if "b=0`
`m(d^(2)x)/(dt)+kx=0`
This becomes the differential equation of SHM.