Assertion: Work done by frictional force on a sphere rolling...

Assertion: Work done by frictional force on a sphere rolling without slipping on an inclined plane is negative.
Reason: Work done by the force $F, W=\int \vec{F} \cdot d \vec{S}$
(A) A
(B) $\mathrm{B}$
(C) C
(D) D

Key Concepts

Line Integral of Force

The line integral of a force is a mathematical tool used to calculate the work done by that force along a curved or straight path. By integrating the dot product of the force vector and the infinitesimal displacement vector over the trajectory, it provides a way to account for the force's contribution to work at every point along the path.

Rolling Without Slipping

Rolling without slipping is a dynamic condition in which an object's rotational and translational motions are synchronized, such that the point of contact on the rolling object has zero velocity relative to the surface. This condition is crucial in linking the object's angular and linear velocities, and it determines how friction behaves in the system.

Friction

Friction is a resistive force that arises from the interaction between two surfaces in contact. It acts to oppose relative motion and can either hinder motion (as kinetic friction) or prevent the initiation of motion (as static friction). In rolling scenarios, static friction plays a critical role in ensuring that the rolling object does not slip.

Work

Work is the energy transferred to or from an object via the application of force along a displacement. It is generally calculated using the line integral of the force along the path of the object, expressed as W = ? F · ds, which represents the cumulative effect of the force's component in the direction of motion.

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