# This should've been mechanics!

Calculus Level 5

Consider a mathematical pendulum. It is held at a position as shown above at $$t=0$$, before it is released. Find the time taken by it to complete one oscillation, that is the time taken by it to make a complete round from its initial position to back to the initial position. If the time taken can be represented as:

$\sqrt{\dfrac{L^{\alpha}}{\pi^{\beta}g^{\gamma}}}\Gamma^{\delta}\left(\dfrac{1}{\eta}\right)$

where $$\Gamma$$ is the gamma function and $$\alpha$$, $$\beta$$, $$\gamma$$, $$\delta$$, and $$\eta$$ are all positive integers. Find $$\alpha +\beta +\gamma +\eta^{\delta}$$.

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