A sphere is rolling without slipping on a fixed horizontal...

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, $A$ is the point of contact, $B$ is the centre of the sphere and $C$ is its topmost point. Then, (a) $\mathrm{v}_{C}-\mathrm{v}_{A}=2\left(\mathrm{v}_{A}-v_{C}\right)$ (b) $v_{C}-v_{B}=v_{B}-v_{A}$ (c) $\left|v_{C}-v_{A}\right|=2\left|v_{B}-v_{C}\right|$ (d) $\left|v_{C}-v_{A}\right|=4 v_{B}$

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, $A$ is the point of contact, $B$ is the centre of the sphere and $C$ is its topmost point. Then,
(a) $\mathrm{v}_{C}-\mathrm{v}_{A}=2\left(\mathrm{v}_{A}-v_{C}\right)$
(b) $v_{C}-v_{B}=v_{B}-v_{A}$
(c) $\left|v_{C}-v_{A}\right|=2\left|v_{B}-v_{C}\right|$
(d) $\left|v_{C}-v_{A}\right|=4 v_{B}$

Key Concepts

Rolling Without Slipping

This concept refers to the condition in rolling motion where the point of contact between the rolling object and the surface has zero instantaneous velocity relative to the surface. It implies a specific relationship between the translational velocity of the center of mass and the angular velocity of the object, ensuring that there is no relative sliding at the contact point.

Instantaneous Center of Rotation

In a body undergoing rolling without slipping, there exists a point (often the point of contact with the surface) that is momentarily at rest. This point acts as the instantaneous center of rotation, allowing for simplified analysis of the velocities of different points on the object by treating the motion as pure rotation about this center.

Relative Velocity in Rigid Body Motion

This concept involves understanding that the velocity of any point on a rigid body can be expressed as the velocity of a reference point plus the contribution from the body's rotation. The relative velocity between two points is found by subtracting their individual velocities, which helps in understanding the kinematic relationships among different points during the motion.

Vector Subtraction and Magnitude Relationships

In the study of rigid body kinematics, vector subtraction is used to determine the relative motion between different points. By comparing these vector differences and their magnitudes, one can establish proportional relationships between the speeds of various points on a rolling object, which is crucial for solving problems involving motion analysis.

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