If the angle of swing is kept to not more than about 30⁰, then θ and sin θ will differ by not more that about 4%. This equation, therefore, allow us to determine the moment of inertia of any object through any specific pivot point on that object! Note the assumption that θ is assumed to be small. The acceleration of gravity g is taken as a known quantity, and T, M, and d are all measurable. Solving this equation for the moment of inertia I, the equation in the box at the bottom of the figure is obtained. By considering the dynamics involved, the figure shows the derivation of an equation for the period T of the physical pendulum. The distance between the pivot point and the center of mass is d. The equilibrium position is when the center of mass C of the object is directly below the pivot point. The object is displaced from its equilibrium position by an angle θ. Figure 1 shows an irregular shaped object that pivots about a frictionless axis perpendicular to the plane of the figure at point P.
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