# Average Period

An ideal pulley (massless, frictionless) is set up, as shown in the diagram. Before the system is released, mass $$M$$ is set into motion as a pendulum. From this initial point in time until the mass $$2M$$ hits the ground, the average period of the oscillation of $$M$$ is given by

$a{\pi} \left( \sqrt {b \left(\frac{l_{o} -d}{g}\right)} + \sqrt{\frac{c}{gd}}\, l_{o} \sin^{-1} \sqrt{\frac{d}{l_{o}}} \right),$

where $$a$$ , $$b$$, and $$c$$ are constants. Find $$a*(b+c)$$.

Note: $$l_{o} \geq d$$.

Temporary Edit: Ignore effects due to centripetal acceleration.

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