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 fractio of total energy associated with its rotation will be (a) $\frac{k^{2}+R^{2}}{R^{2}}$ (b) $\frac{k^{2}}{R^{2}}$ (c) $\frac{k^{2}}{1 \cdot 2+R^{2}}$ (d) $\frac{R^{2}}{k^{2}+R^{2}}$

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 fractio of total energy associated with its rotation will be
(a) $\frac{k^{2}+R^{2}}{R^{2}}$
(b) $\frac{k^{2}}{R^{2}}$
(c) $\frac{k^{2}}{1 \cdot 2+R^{2}}$
(d) $\frac{R^{2}}{k^{2}+R^{2}}$

Key Concepts

Rolling Without Slipping

This concept refers to the condition where a rolling object, such as a ball or wheel, moves in such a way that the point of contact with the surface has zero velocity relative to the surface. This introduces a geometric constraint linking the translational speed of the object’s center of mass to its angular velocity, which is critical in analyzing both the linear and rotational aspects of its motion.

Radius of Gyration

The radius of gyration is a measure that describes the distribution of an object's mass around an axis of rotation. It is defined so that the moment of inertia can be expressed as the product of the total mass and the square of the radius of gyration. This concept simplifies the analysis of rotational dynamics by providing a single characteristic distance that encapsulates the mass distribution.

Kinetic Energy Components

In the context of a rolling object, the total kinetic energy is composed of two parts: translational kinetic energy (due to the motion of the center of mass) and rotational kinetic energy (due to the object’s rotation about its center of mass). Understanding the relationship and conversion between these two types of energy is essential for analyzing dynamics of rolling motion.

Energy Distribution in Rolling Motion

When an object rolls without slipping, its energy is partitioned between translational and rotational kinetic energy. The fraction of the total energy associated with rotation depends on its moment of inertia (or radius of gyration) relative to its size. This distribution is derived from combining the rolling constraint with the expressions for translational and rotational kinetic energies, highlighting how the object's mass distribution influences its overall dynamics.

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