A sphere of radius R is rolling on a rough horizontal...

A sphere of radius $R$ is rolling on a rough horizontal surface. The magnitude of velocity of $A$ with respect to ground will be :(a) $\sqrt{2} v_{\mathrm{cm}}$
(b) $2 v_{\mathrm{cm}} \sin \theta$
(c) $\sqrt{2} v_{\mathrm{cm}} \sqrt{1+\sin \theta}$
(d) no sufficient information

Key Concepts

Rolling Without Slipping

This concept refers to a situation where the point of contact of a rolling object with the surface has zero relative velocity with respect to the surface at any instant. It establishes a fixed relationship between the translational velocity of the object's center of mass and its angular velocity, which is crucial in problems involving rolling motion.

Rigid Body Kinematics

Rigid body kinematics involves analyzing the motion of all points on an object by breaking it down into translational motion of the center of mass and a rotational component about the center of mass. This decomposition allows one to study the velocity of any point on the object by vectorially adding the translational and rotational contributions.

Angular Velocity

Angular velocity is a measure of the rate of rotation of an object around an axis. In the context of rolling motion, it directly relates to the rotational component of the velocity of points on the object and is tied to the translational velocity in cases of no slip, providing a means to calculate the overall motion of specific points on the rolling body.

Velocity Composition in Rolling Motion

The velocity of any point on a rolling object is obtained by the vector sum of the translational velocity of the center of mass and the tangential velocity produced by the object's rotation about the center of mass. Understanding this principle is essential for determining the resultant velocities at different points on the object.

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