A uniform solid cylinder rolling without slipping along a...

A uniform solid cylinder rolling without slipping along a horizontal plane suddenly encounters a plane inclined at angle $\theta$ as shown in figure. The value of $\theta$ which could bring the cylinder immediately to rest after impact, is : (a) $90^{\circ}$ (b) $60^{\circ}$ (c) $120^{\circ}$ (d) $30^{\circ}$

A uniform solid cylinder rolling without slipping along a horizontal plane suddenly encounters a plane inclined at angle $\theta$ as shown in figure. The value of $\theta$ which could bring the cylinder immediately to rest after impact, is :
(a) $90^{\circ}$
(b) $60^{\circ}$
(c) $120^{\circ}$
(d) $30^{\circ}$

Key Concepts

Impact Dynamics

Impact dynamics studies the behavior of objects under sudden forces or impulses during collisions. It involves evaluating how impulsive forces change both translational and rotational motions, often using conservation laws and constraints imposed by conditions like rolling without slipping.

Angular Momentum Conservation

The principle of conservation of angular momentum states that in the absence of external torques, the total angular momentum of a system remains constant. This concept is critical in impact dynamics, where analyzing angular momentum about a suitable pivot (such as the instantaneous center) helps determine the conditions for changes in motion.

Rotational Inertia

Rotational inertia, or moment of inertia, measures an object’s resistance to angular acceleration about a given axis. It is essential in linking torque, angular momentum, and rotational kinetic energy, and plays a key role in determining how an object’s rotational state changes during interactions such as impacts.

Rolling Without Slipping

Rolling without slipping is a condition where the point of contact between a rolling object and the surface does not move relative to the surface. This coupling between translational and rotational motion is expressed by the relation v = ?R and is fundamental in understanding the dynamics of rolling bodies.

Instantaneous Center of Rotation

The instantaneous center of rotation is the point about which all parts of a rolling object momentarily rotate. In rolling without slipping, the contact point plays this role, making it a useful reference for analyzing rotational effects and simplifying the computation of angular momentum during impacts.

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