From a uniform disc of radius R, a circular hole of radius R...

From a uniform disc of radius $R$, a circular hole of radius $R / 2$ is cut. The centre of the hole is at $R / 2$ from the centre of the original disc. Locate the centre of mass of the resulting flat body.

Key Concepts

Center of Mass

The center of mass of an object is a point that represents the average location of the mass distribution within the body. This concept is crucial for analyzing the motion and stability of objects, and it plays an important role in mechanics, especially when dealing with composite or irregularly shaped bodies.

Composite Body Method

The composite body method involves breaking down a complex shape into simpler parts where the center of mass is easier to compute. For problems involving holes or removed sections, one typically treats the removed part as having -1 times its mass and computes the overall center of mass as a weighted average of the parts.

Uniform Density

Uniform density implies that mass is distributed evenly throughout the object, making the mass of any section proportional to its area. This simplification is essential for calculating the center of mass because it ensures that the center of mass corresponds directly to geometric centroids for symmetric parts of the object.

Superposition Principle in Mass Distribution

The superposition principle in the context of mass distributions allows one to add and subtract mass contributions from different parts of an object to find the net effect. This principle is fundamental when dealing with composite bodies, such as objects with cutouts, because it enables using negative mass for the removed portions.

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