In the given figure, a solid sphere is placed on a plank...

In the given figure, a solid sphere is placed on a plank having acceleration $a_{0}$ (shown in the figure). Then: (a). if $a_{p}=a_{0}$, pure rolling takes place (b) if $v_{p}=v_{0}$ pure rolling takes place (c) if $a_{p}=a_{0}, v_{p} \neq v_{0}$ pure rolling takes place (d) if $a_{p}=a_{0}, v_{p}=v_{0}$, pure rolling takes place

In the given figure, a solid sphere is placed on a plank having acceleration $a_{0}$ (shown in the figure). Then:
(a). if $a_{p}=a_{0}$, pure rolling takes place
(b) if $v_{p}=v_{0}$ pure rolling takes place
(c) if $a_{p}=a_{0}, v_{p} \neq v_{0}$ pure rolling takes place
(d) if $a_{p}=a_{0}, v_{p}=v_{0}$, pure rolling takes place

Key Concepts

Pure Rolling

Pure rolling, also known as rolling without slipping, is when an object rolls such that the point in contact with the surface has zero relative velocity with respect to that surface. This condition ensures that there is no sliding between the object and the surface, meaning static friction is at work to maintain the relationship between the object’s rotational and translational motions.

Rolling Without Slipping Condition

The condition for rolling without slipping is given by the equality between the translational speed of the center of mass and the product of the angular speed and the radius of the rolling body (v = ?r). For accelerations, the relation a = ?r must hold, ensuring that both the velocity and acceleration of the point of contact are matched with those of the surface, thereby preventing slip.

Relative Motion

Relative motion analysis is crucial when studying systems where an object interacts with a moving surface. In the context of rolling, pure rolling occurs when the velocity and acceleration of the point of contact relative to the surface are zero. This concept helps in understanding the dynamics of the object from different frames of reference, ensuring that the no-slip conditions are met irrespective of the motion of the surface.

Rotational and Translational Kinematics

Rotational kinematics involves the angular motion of an object, while translational kinematics deals with the linear motion of its center of mass. In rolling motion, both these aspects are interconnected by the condition that the linear velocity equals the angular velocity times the radius (and similarly for accelerations), ensuring synchrony between the translational and rotational motion mechanisms so that pure rolling is achieved.

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