The derivation is similar to that of a simple pendulum since one can consider all the mass M to be located at the body's center of mass. Then a physical pendulum looks like a simple pendulum except that its moment of inertia is found using the parallel axis theorem.
Icm = the body's moment of inertia about its center of mass.
h = Distance from pivot point to the center of mass.
M = Mass of the body.
If the pivot joint is frictionless then the net torque acting on the planar object is given by the force of gravity perpendicular to lever arm, Mg sin(q), times the length of the lever arm, h:
When the angle of oscillation is small then the value of sin(q) and q or nearly the same provided q is measured in radians. Using this approximation, the above torque equation can be solved for the angular acceleration,
Since a =d2q/dt2, this equation is structurally similar to the differential equation for any type of SHM,
Matching terms with SHM equations,
Here w' is instantaneous angular velocity of the body while w is angular frequency of the body's SHM. They are not the same. Moreover, w is constant while w' varies as the body oscillates back and forth.
Bar Physical Pendulum Simulation
A 2.00 meter bar is pivoted about some point on the bar. The location of the pivot point and the initial angle of the bar from vertical can be varied by clicking and dragging. Displayed is the graph of the angle of oscillation as function of time. Also show are the resulting values of small angle period, momentum of inertia about the pivot point, and the distance of the pivot point from the center of mass of the bar.