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## Homework Statement

Find the Laplace transform of the pendulum equation.

## Homework Equations

The pendulum equation:

[tex]

\frac{d^{2}\theta}{dt^{2}} + \alpha \frac{d \theta}{dt} + g \sin(\theta) = 0

[/tex]

[tex]

s = \sigma + i \omega

[/tex]

## The Attempt at a Solution

Taking the laplace transform I get

[tex]

s^{2} + \alpha s + g \frac{\omega}{s^{2}+\omega^{2}} = 0

[/tex]

I usually drop the [tex] \sigma [/tex] term in [tex]s = \sigma + i \omega[/tex], which leaves

[tex]

s^{2} + \alpha s + g \frac{\omega}{-\omega^{2}+\omega^{2}} = 0

[/tex]

The [tex]\frac{\omega}{-\omega^{2}+\omega^{2}}[/tex] is problematic, but I'm not sure what to do. I don't see the usefulness of the [tex] \sigma [/tex] variable, and I'm not sure that I'm doing this correctly to begin with