Four spheres, each of mass M and radius r are situated at...

Four spheres, each of mass $M$ and radius $r$ are situated at the four corners of square of side $R$. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its centre will be (a) $\frac{5}{2} M\left(4 r^{2}+5 R^{2}\right)$ (b) $\frac{2}{5} M\left(4 r^{2}+5 R^{2}\right)$ (c) $\frac{2}{5} M\left(4 r^{2}+5 r^{2}\right)$ (d) $\frac{5}{2} M\left(4 r^{2}+5 r^{2}\right)$

Four spheres, each of mass $M$ and radius $r$ are situated at the four corners of square of side $R$. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its centre will be
(a) $\frac{5}{2} M\left(4 r^{2}+5 R^{2}\right)$
(b) $\frac{2}{5} M\left(4 r^{2}+5 R^{2}\right)$
(c) $\frac{2}{5} M\left(4 r^{2}+5 r^{2}\right)$
(d) $\frac{5}{2} M\left(4 r^{2}+5 r^{2}\right)$

Key Concepts

Moment of Inertia

This is a measure of an object's resistance to changes in its rotation and is defined by summing over all mass elements multiplied by the square of their distances from the axis of rotation. In composite systems, the overall moment of inertia is found by summing the individual contributions of each part, accounting for both their intrinsic rotational inertia and the distribution of their mass relative to the rotation axis.

Parallel Axis Theorem

The Parallel Axis Theorem allows the calculation of the moment of inertia about any axis parallel to an axis through the object's center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes. This theorem is crucial when the object is not rotating about its own center of mass but about an external axis.

Intrinsic Moment of Inertia of a Sphere

For a solid sphere, its intrinsic moment of inertia about an axis through its center is given by (2/5)MR², where M is the mass and r is the radius of the sphere. This value is used as the base inertia before applying any shifts in the axis of rotation using the Parallel Axis Theorem.

Geometry of the System

The arrangement of bodies in space directly influences the overall moment of inertia. In the context of the problem, the spheres are positioned at the corners of a square, making the distance from each sphere to the center of the square (the axis of rotation) vital for calculating the additional moment of inertia contributed by each sphere. Understanding this geometry leads to correctly determining the displacement distance in the parallel axis calculation.

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