A grandfather clock uses a physical pendulum to keep time....

A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass $M$ and length $L$ that is pivoted freely about one end. with a solid sphere of the same mass, $M,$ and a radius of $L / 2$ centered about the free end of the rod. a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of $M$ and $L$. b) Obtain an expression for the period of the pendulum for small oscillations.

A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass $M$ and length $L$ that is pivoted freely about one end. with a solid sphere of the same mass, $M,$ and a radius of $L / 2$ centered about the free end of the rod.
a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of $M$ and $L$.
b) Obtain an expression for the period of the pendulum for small oscillations.

Key Concepts

Physical Pendulum

A physical pendulum is a rigid body that swings about a pivot point under the influence of gravity. Unlike the simple pendulum, which treats the mass as a point, the mass distribution in a physical pendulum contributes to its behavior. The period of small oscillations depends on the moment of inertia of the object about the pivot and the distance from the pivot to its center of mass.

Moment of Inertia

The moment of inertia measures the resistance of a body to angular acceleration about a specified axis. It is calculated by integrating the mass distribution with respect to the distance from the axis. For systems comprising multiple parts, such as a rod and a sphere, the total moment of inertia is the sum of the individual moments, each calculated about the same axis.

Parallel-Axis Theorem

The parallel-axis theorem is used to find the moment of inertia of an object about any axis parallel to an axis through its center of mass. It states that the moment of inertia about a new axis is equal to the moment of inertia about the center of mass plus the mass of the body times the square of the distance between the two axes.

Simple Harmonic Motion

Simple harmonic motion describes oscillatory motion where the restoring force (or torque) is directly proportional to the displacement from equilibrium, and acts in the opposite direction. In the context of a physical pendulum undergoing small oscillations, this relationship allows the derivation of the period, which depends on the moment of inertia and the gravitational torque acting on the pendulum.

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