A circular disc of radius R rolls without slipping along the...

A circular disc of radius $R$ rolls without slipping along the horizontal surface with constant velocity $v_{0}$. We consider a point $A$ on the surface of the disc. Then the acceleration of the point $A$ is IUP SEE 2005] (a) constant in magnitude as well as in direction (b) constant in direction (c) constant in magnitude (d) constant

A circular disc of radius $R$ rolls without slipping along the horizontal surface with constant velocity $v_{0}$. We consider a point $A$ on the surface of the disc. Then the acceleration of the point $A$ is IUP SEE 2005]
(a) constant in magnitude as well as in direction
(b) constant in direction
(c) constant in magnitude
(d) constant

Key Concepts

Rolling Without Slipping

This concept refers to the condition where a rolling object has no relative motion at the point of contact with the surface. It establishes the relationship v = ?R, where v is the translational velocity of the center of mass, ? is the angular velocity, and R is the radius. This principle is fundamental in analyzing the motion of rolling bodies, ensuring that the disc does not slip as it rolls along the surface.

Rigid Body Acceleration Decomposition

For any point on a rigid body, the total acceleration can be broken down into the translational acceleration of the center of mass and the rotational acceleration relative to the center of mass. In the given scenario, because the disc moves with a constant velocity, the translational acceleration is zero. Thus, the acceleration of any point on the disc arises solely from its rotation about the center.

Centripetal Acceleration in Rotational Motion

When a body rotates with a constant angular velocity, any point on the body (apart from the center) experiences centripetal acceleration, which is directed toward the center of rotation. Its magnitude is given by ?²R, which remains constant provided that the angular velocity is constant. However, while the magnitude of this acceleration remains unchanged, its direction continuously varies as the point rotates around the center. This key idea explains why the acceleration of a point on the disc is constant in magnitude but not in direction.

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