The pendulum consists of two slender rods A B and O C which...

The pendulum consists of two slender rods $A B$ and $O C$ which have a mass of $3 \mathrm{kg} / \mathrm{m}$. The thin plate has a mass of $12 \mathrm{kg} / \mathrm{m}^{2} .$ Determine the location $\bar{y}$ of the center of mass $G$ of the pendulum, then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through $G$

Key Concepts

Center of Mass of Composite Systems

This concept involves determining the overall balance point of a system by calculating the weighted average of the positions of its constituent parts. For a composite body, each component's center of mass is multiplied by its mass, and these products are divided by the total mass, providing the effective point where gravity can be considered to act.

Moment of Inertia

The moment of inertia quantifies the resistance of a body to angular acceleration about a given axis. It is fundamentally defined as the integral of the mass distribution times the square of the distance from the axis of rotation. When dealing with composite bodies, the total moment of inertia is the sum of the contributions from each individual component.

Parallel-Axis Theorem

This theorem allows for the determination of the moment of inertia about any axis parallel to one passing through an object’s center of mass. By adding the product of the mass of the component and the square of the distance between the two axes to the moment of inertia about the center of mass, one can shift the axis of rotation effectively in the composite setup.

Integration in Continuous Mass Distributions

For bodies with continuous mass distributions, such as rods or plates, integration is used to sum the contributions of all infinitesimally small mass elements. This approach is critical for calculating properties like the center of mass and the moment of inertia, ensuring an accurate analysis of parts with non-discrete masses.

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