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

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

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

Key Concepts

Translational and Rotational Kinetic Energy

In a rolling object, the total kinetic energy is composed of both translational kinetic energy, due to the motion of the center of mass, and rotational kinetic energy, due to the body's spinning motion. These two forms of energy are interconnected through the rolling-at-no-slipping condition.

Energy Partition in Rolling Motion

Analyzing the fraction of total energy that is rotational involves expressing the rotational kinetic energy in terms of the moment of inertia and angular velocity, and relating it to the total energy that also includes translational kinetic energy. This partitioning is critical in studies of rolling motion and dynamics, illustrating how the distribution of energy depends on the mass distribution (through the radius of gyration) and the geometry of the rolling body.

Rolling Without Slipping

This concept refers to the condition where an object rolls in such a way that its point of contact with the surface has zero velocity relative to that surface. This condition creates a fixed relationship between linear and angular velocities, which is fundamental in analyzing the dynamics of rolling bodies.

Moment of Inertia and Radius of Gyration

The moment of inertia quantifies an object's resistance to changes in its rotational motion and depends on its mass distribution around the axis of rotation. The radius of gyration is a derived value that represents the distance from the axis at which the entire mass of the object could be concentrated to yield the same moment of inertia. Together, these concepts are essential for understanding rotational dynamics.

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