Compound Pendulum Period — the NEET Physics formula with its derivation, variables, validity constraints and worked solver.
Compound Pendulum Period This formula provides the period of oscillation for a rigid body pivoted at a point, assuming small angular displacements and negligible damping. Apply conservation of energy to find the potential energy change (PE) and kinetic energy (KE). The restoring torque = -mgd , leading to the equation of motion: I d 2 dt 2 = -mgd . For small angles, approximate , yielding the Simple Harmonic Motion (SHM) equation: d 2 dt 2 = mgd I . For small angles, the angular frequency is = mgd I , which gives the period T = 2 = 2 I mgd . This standard result is valid when I is the moment of inertia about the pivot point. The period T is derived from the angular frequency = mgd I (if the formula was T = 2 I mgd ). Let's use the standard derivation steps for the period: = mgd I leads to T = 2 I mgd only if I is defined differently. We will list the steps leading to the final form. The period T is found by solving the differential equation of motion for small angles, resulting in the formula T = 2 I mgd . (small angle approximation) d > 0 I > 0 Confusing the compound pendulum period with the simple pendulum period (where I is replaced by md ). Forgetting the small angle approximation ( ), which is necessary for this formula to be valid. Assuming the period depends only on the length of the pendulum and ignoring the mass distribution (Moment of Inertia, I).