Energy in Simple Harmonic Motion Video Lessons

Concept

# Problem: The amplitude of a lightly damped oscillator decreases by 3.0 % during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?

###### FREE Expert Solution

When there is no damping, the mechanical energy of the oscillator is expressed as:

$\overline{){\mathbf{E}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{k}}{{\mathbf{A}}}^{{\mathbf{2}}}}$

For the damped oscillator, the amplitude is decreased by 3.0% per cycle.

The mechanical energy is now expressed as:

$\begin{array}{rcl}{\mathbf{E}}_{\mathbf{d}}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}\mathbf{k}{\mathbf{\left(}\mathbf{A}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{A}\mathbf{\right)}}^{\mathbf{2}}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}\mathbf{k}{\mathbf{\left(}\mathbf{0}\mathbf{.}\mathbf{97}\mathbf{A}\mathbf{\right)}}^{\mathbf{2}}\\ & \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}{\mathbf{A}}^{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{.}\mathbf{9409}\mathbf{\right)}\end{array}$

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###### Problem Details

The amplitude of a lightly damped oscillator decreases by 3.0 % during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?