The moment of inertia of a sphere of mass M and radius R...

The moment of inertia of a sphere of mass $M$ and radius $R$ about an axis passing through its centre is $\frac{2}{5} M R^{2}$. The radius of gyration of the sphere about a parallel axis to the above and tangent to the sphere is (a) $\frac{7}{5} R$ (b) $\frac{3}{5} R$ (c) $\left(\sqrt{\frac{7}{5}}\right) R$ (d) $\left(\sqrt{\frac{3}{5}}\right) R$

The moment of inertia of a sphere of mass $M$ and radius $R$ about an axis passing through its centre is $\frac{2}{5} M R^{2}$. The radius of gyration of the sphere about a parallel axis to the above and tangent to the sphere is
(a) $\frac{7}{5} R$
(b) $\frac{3}{5} R$
(c) $\left(\sqrt{\frac{7}{5}}\right) R$
(d) $\left(\sqrt{\frac{3}{5}}\right) R$

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Key Concepts

Radius of Gyration

The radius of gyration is defined as the distance from the axis of rotation at which the total mass of the body can be assumed to be concentrated for the purposes of analyzing rotational motion. It is calculated by taking the square root of the moment of inertia divided by the mass of the object, providing a useful parameter for comparing the distribution of mass in different shapes.

Parallel Axis Theorem

The parallel axis theorem is used to determine the moment of inertia of a body about any axis parallel to an axis through its center of mass. It states that this moment of inertia equals the moment of inertia about the centroidal axis plus the product of the mass and the square of the distance between the two axes. This theorem simplifies calculations when shifting the axis of rotation from the center of mass.

Moment of Inertia

The moment of inertia is a measure of how resistant a body is to angular acceleration about an axis. It depends on the mass of the object and its distribution relative to the axis of rotation. A key point is that for different shapes and orientations, the moment of inertia can be calculated using integrals or standard formulas that take into account the geometry of the object.

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