Write the expression for the mechanical energy of a damped oscillator.

(N/A) The mechanical energy of a simple harmonic oscillator $(SHO)$ is given by $E = \frac{1}{2} k A^{2}$.
For a damped oscillator,the amplitude at time $t$ is given by $A(t) = A e^{-\frac{b t}{2 m}}$.
Substituting this amplitude into the energy expression,the mechanical energy $E(t)$ of the damped oscillator at time $t$ is:
$E(t) = \frac{1}{2} k \left(A e^{-\frac{b t}{2 m}}\right)^{2} = \frac{1}{2} k A^{2} e^{-\frac{b t}{m}}$.
Thus,the mechanical energy is not constant but decreases exponentially with time.
This equation is valid for small damping where $b \ll \sqrt{k m}$,implying that the dimensionless ratio $\frac{b}{\sqrt{k m}} \ll 1$.
If $b = 0$,the expression reduces to the energy of an undamped oscillator,$E = \frac{1}{2} k A^{2}$.