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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