Summary
Highlights
The video begins by recalling previous derivations of simple harmonic motion equations for mass-spring systems and simple pendulums. The condition for simple harmonic motion is stated as the second derivative of position with respect to time equaling the negative of the square of angular frequency times position. A simple pendulum is defined as an idealized pendulum with negligible string mass, a point mass bob, no friction, and an inextensible string. A physical pendulum, however, is introduced as a rigid body suspended at a point, oscillating in simple harmonic motion without friction or a string, potentially having any shape.
The derivation of the physical pendulum's angular frequency and period begins by considering the net torque about the axis of rotation. The only force causing torque is gravity acting on the center of mass. The net torque is set equal to the rotational inertia times angular acceleration. Through substitution and rearrangement, the equation for the second derivative of angular position with respect to time is found to be based on the mass, gravitational field strength, distance to the center of mass, and rotational inertia, multiplied by the sine of the angular displacement.
To fit the condition for simple harmonic motion, the small angle approximation is applied, where the sine of the angle is approximately equal to the angle itself (in radians). This allows the equation to match the simple harmonic motion condition. From this, the square of the angular frequency is determined, and subsequently, the angular frequency is found by taking the square root. The period of the physical pendulum is then derived using the relationship between period and angular frequency, resulting in the period equaling two pi times the square root of (rotational inertia divided by mass times gravitational field strength times the length to the center of mass).
The derived equation for the physical pendulum's period is then applied to a simple pendulum scenario, demonstrating that it reduces to the known simple pendulum period equation. The video emphasizes that while the angular frequencies differ, the fundamental equations of motion for simple harmonic motion remain consistent. An experiment is then conducted using a uniform, long, thin rod as a physical pendulum. By measuring the time for 10 oscillations, the observed period is calculated. Using the known formula for the rotational inertia of such a rod and the derived period equation, both the accepted and observed values for rotational inertia are determined. The close match between these values, initially appearing as zero percent error (later corrected to a small percentage), confirms the validity of the derived equations.