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Compound pendulums

A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency (ω n {\displaystyle \omega _{\text{n}}}) of a compound pendulum depends on its moment of inertia, I P {\displaystyle I_{P}}, ω n = m g r I P, {\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},} where m {\displaystyle m} is the mass of the object, g {\displaystyle g} is local acceleration of gravity, and r {\displaystyle r} is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.

Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point P {\displaystyle P} so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (t {\displaystyle t}), to obtain I P = m g r ω n 2 = m g r t 2 4 π 2, {\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},} where t {\displaystyle t} is the period (duration) of oscillation (usually averaged over multiple periods).