Describe the properties of a physical pendulum.

A physical pendulum is a rigid body that undergoes oscillations about a fixed axis.

For small amplitudes of motion, the period of a physical pendulum is derived from the application of Newton’s second law in rotational form. \[T = 2\pi \sqrt{\dfrac{I}{mgd}}\]

When displaced from equilibrium, the gravitational force exerted on a physical pendulum’s center of mass provides a restoring torque. \[\tau = -mgd \sin{\theta}\]

For small amplitudes of motion, the small angle approximation can be applied to the restoring torque. \[\sin{\theta} \approx \theta\] \[\tau \approx -mgd\theta = I \alpha\]

The small-angle approximation and Newton’s second law in rotational form yield a second order differential equation that describes SHM: \[\dfrac{d^2\theta}{dt^2} = -\omega^2 \theta\]

A simple pendulum is a special case of physical pendulums in which the hanging object can be modeled as a point mass at a distance, l, from the pivot point. \[T = 2\pi \sqrt{\dfrac{l}{g}}\]

A torsion pendulum is a case of SHM where the restoring torque is proportional to the angular displacement of a rotating system. For example, a horizontal disk that is suspended from a wire attached to its center of mass may undergo rotational oscillations about the wire in the horizontal plane. \[I \alpha = - k \Delta\theta\]