Question $03-$

$($a$)$ For the physical pendulum $($of mass, $m,$ and moment of inertia, $1,$ about the indicated pivot point, pre derive the period of oscillation for small angle amplitudes. Take as given the $d$ and the center of mass $($labeled com$).(10$ points$)$

$($b$)$ A uniform disk of mass, $M,$ and radius, $R,$ is pivoted at its rim. Find its period for small oscillations. You may use the result from the first part of this question $($which should have resulted in a derived period of $T=2\sqrt[\phantom{2}]{\frac{l}{mgd}}.$ You may use this result even if you did not complete the first part of the question$).$ Furthermore, you may take as given that the moment of inertia for a uniform dis$/$ rotated about an axis passing through its center of mass is $I_{\text{c}\text{o}\text{m}}=(\frac{1}{2})M{R}^2.(5$ points$)$

At Fauilihrium

Rontated Throunh a Small Anole About Pivot

$($c$)$ For the physical pendulum described in the second part of this problem, what would be the length of an ideal pendulum of mass, $2$ M $,$ that has an identical oscillation period through small angles? $(5$ points$)$