Two thin dises each of mass M and radius R are placed at...

Two thin dises each of mass $M$ and radius $R$ are placed at either end of a rod of mass $m$, length $l$ and radius $r$. Moment of inertia of the system about an axis passing through the centre of rod and perpendicular to its length is (a) $\frac{m L^{2}}{12}+\frac{1}{4} M R^{2}+\frac{1}{4} M L^{2}$ (b) $\frac{M^{2}}{12}+\frac{1}{2} m R^{2}+\frac{1}{2} m \mathcal{L}^{2}$ (c) $\frac{1}{2} m L^{2}+\frac{m R^{2}}{2}+\frac{M L^{2}}{12}$ (d) $\frac{m L^{2}}{12}+M R^{2}+\frac{1}{2} M E^{2}$

Two thin dises each of mass $M$ and radius $R$ are placed at either end of a rod of mass $m$, length $l$ and radius $r$. Moment of inertia of the system about an axis passing through the centre of rod and perpendicular to its length is
(a) $\frac{m L^{2}}{12}+\frac{1}{4} M R^{2}+\frac{1}{4} M L^{2}$
(b) $\frac{M^{2}}{12}+\frac{1}{2} m R^{2}+\frac{1}{2} m \mathcal{L}^{2}$
(c) $\frac{1}{2} m L^{2}+\frac{m R^{2}}{2}+\frac{M L^{2}}{12}$
(d) $\frac{m L^{2}}{12}+M R^{2}+\frac{1}{2} M E^{2}$

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Key Concepts

Parallel Axis Theorem

This theorem is used to relate the moment of inertia of an object about an axis through its center of mass to its moment of inertia about any parallel axis. It is essential when the components of a composite system rotate about an axis that does not pass through their individual centers of mass.

Standard Formulas for Rigid Bodies

Knowledge of the standard formulas for the moments of inertia of common shapes, such as rods, disks, and spheres, is crucial. These formulas provide the base values that can be adjusted using the parallel axis theorem when the axis of rotation does not coincide with the symmetry axis of the object.

Moment of Inertia

This is a physical quantity representing the resistance of a body to angular acceleration about a chosen axis. It depends on the mass distribution relative to the axis of rotation and is the rotational analogue of mass in linear motion.

Composite Systems

When dealing with a body composed of several parts, the moment of inertia of the overall system is found by summing the individual moments of inertia of each part. This approach requires attention to the location of each component’s center of mass relative to the rotation axis.

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