Question

Derive the expression of the mechanical energy for under damped system

Answer 1

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Answer 2

Answer:

The expression of the mechanical energy for under damped system is $\bold{E(t)=\frac{1}{2} kA^{2} e^\frac{bt}{2m} }$

Explanation:

The mechanical energy of simple harmonic motion is

$E=\frac{1}{2} kA^2$

Now if the amplitude of oscillation is   $Ae^-\frac{bt}{2m}$, then the mechanical energy of the oscillator obtain

The mechanical energy of simple harmonic motion at a time $t$ is

$E(t)=\frac{1}{2} kA^2e^-\frac{bt}{2m}$

Hence, the expression of the mechanical energy for under damped system is $\bold{E(t)=\frac{1}{2} kA^{2} e^\frac{bt}{2m} }$

Note: mechanical energy is not constant but decreases exponentially with time.