Two spheres each of mass M and radius R / 2 are connected...

Two spheres each of mass $M$ and radius $R / 2$ are connected with a massless rod of length $2 R$ as shown in the figure. The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod is (a) $\frac{21}{5} M R^{2}$ (b) $\frac{2}{5} M R^{2}$ (c) $\frac{5}{2} M R^{2}$ (d) $\frac{5}{21} M R^{2}$

Two spheres each of mass $M$ and radius $R / 2$ are connected with a massless rod of length $2 R$ as shown in the figure.

The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod is
(a) $\frac{21}{5} M R^{2}$
(b) $\frac{2}{5} M R^{2}$
(c) $\frac{5}{2} M R^{2}$
(d) $\frac{5}{21} M R^{2}$

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Key Concepts

Parallel Axis Theorem

This theorem allows one to calculate the moment of inertia of an object about any axis parallel to an axis through its center of mass. By adding the moment of inertia about the center-of-mass axis to the product of the mass and the square of the distance between the two axes, it facilitates the computation for composite or shifted rotational systems.

Moment of Inertia

This is a measure of an object’s resistance to changes in its rotational motion about an axis. It quantifies how the mass of an object is distributed relative to the chosen axis, affecting the angular acceleration induced by a given torque.

Composite Body Analysis

When dealing with systems made up of multiple bodies, the total moment of inertia is obtained by summing the individual contributions of each component. This involves calculating or knowing the moment of inertia for each part about the chosen axis, possibly using the parallel axis theorem if the component’s center of mass does not lie on the axis.

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