Mass of bigger disc having radius 2 R is M. A disc of radius...

Mass of bigger disc having radius $2 R$ is $M$. A disc of radius $R$ is cut from bigger disc as shown in figure. Moment of inertia of disc about an axis passing through periphery and perpendicular to plane (shown in figure) is: (a) $\frac{27 M R^{2}}{8}$ (b) $\frac{29 M R^{2}}{8}$ (c) $3.5 \mathrm{MR}$ (d) $2 M R^{2}$

Mass of bigger disc having radius $2 R$ is $M$. A disc of radius $R$ is cut from bigger disc as shown in figure. Moment of inertia of disc about an axis passing through periphery and perpendicular to plane (shown in figure) is:
(a) $\frac{27 M R^{2}}{8}$
(b) $\frac{29 M R^{2}}{8}$
(c) $3.5 \mathrm{MR}$
(d) $2 M R^{2}$

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Key Concepts

Moment of Inertia

This fundamental concept in rotational dynamics quantifies an object’s resistance to angular acceleration about a specific axis. It is computed based on the mass distribution relative to the axis of rotation, and for standard geometries, formulas exist that relate the mass and shape to the moment of inertia.

Parallel Axis Theorem

This theorem allows for the calculation of the moment of inertia about any axis parallel to one through the 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 axis plus the product of the mass and the square of the distance between the two axes.

Composite Body Analysis

In cases where a body is modified by adding or removing parts, the overall moment of inertia can be determined by considering the body as a combination of simpler sub?bodies. The contributions (or subtractions) of these parts, calculated using their own inertias and relative positions, are then combined to find the total moment of inertia.

Uniform Mass Distribution

When dealing with bodies of uniform density, the mass is evenly spread over the area or volume. This uniformity simplifies calculations, as the mass per unit area (or volume) remains constant, making it easier to relate geometric properties to the moment of inertia.

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