A ball rolls without slipping. The radius of gyration of the...

A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is $K$. If radius of the ball be $R$, then the fraction of total energy associated with its rotational energy will be: (a) $\frac{K^{2}}{R^{2}}$ (b) $\frac{K^{2}}{K^{2}+R^{2}}$ (c) $\frac{R^{2}}{K^{2}+R^{2}}$ (d) $\frac{K^{2}+R^{2}}{R^{2}}$

A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is $K$. If radius of the ball be $R$, then the fraction of total energy associated with its rotational energy will be:
(a) $\frac{K^{2}}{R^{2}}$
(b) $\frac{K^{2}}{K^{2}+R^{2}}$
(c) $\frac{R^{2}}{K^{2}+R^{2}}$
(d) $\frac{K^{2}+R^{2}}{R^{2}}$

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Key Concepts

Rolling without slipping

Rolling without slipping is a condition where the point at which a rolling object contacts the surface is momentarily at rest relative to the surface. This condition creates a fixed relationship between the translational velocity of the center of mass and the rotational speed, allowing one to relate angular velocity to linear velocity through the object’s radius.

Kinetic energy decomposition in rolling motion

The kinetic energy of a rolling object can be decomposed into translational kinetic energy, associated with the movement of the center of mass, and rotational kinetic energy, associated with the rotation about the center of mass. This decomposition is essential for understanding how energy is partitioned during the motion of rolling bodies.

Moment of inertia and radius of gyration

The moment of inertia describes how the mass of an object is distributed relative to an axis of rotation and is a measure of an object's resistance to rotational acceleration. The radius of gyration is defined as the distance from the axis at which the entire mass of the object could be concentrated to yield the same moment of inertia. This concept simplifies the analysis of rotational dynamics in physical systems.

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