The solid cylinder is rolling without slipping on a plane...

The solid cylinder is rolling without slipping on a plane having inclination $\theta$ and the coefficient of static friction $\mu_{s}$. The relation between $\theta$ and $\mu_{s}$ is (a) $\tan \theta>3 \mu_{s}$ (b) $\tan \theta \leq 3 \mu_{j}$ (c) $\tan \theta<3 \mu_{s}^{2}$ (d) none of these

The solid cylinder is rolling without slipping on a plane having inclination $\theta$ and the coefficient of static friction $\mu_{s}$. The relation between $\theta$ and $\mu_{s}$ is
(a) $\tan \theta>3 \mu_{s}$
(b) $\tan \theta \leq 3 \mu_{j}$
(c) $\tan \theta<3 \mu_{s}^{2}$
(d) none of these

Key Concepts

Rolling Without Slipping

This concept refers to the condition where an object rolls over a surface with no relative sliding at the point of contact. It requires that the translational velocity of the center of mass is equal to the product of the angular velocity and the radius, ensuring a fixed relationship between linear and rotational motion.

Static Friction

Static friction is the force that prevents relative motion between two surfaces in contact. Its maximum value is given by the product of the coefficient of static friction and the normal force. In the context of rolling, static friction prevents slipping and ensures that the rolling motion is maintained.

Force Analysis on an Inclined Plane

When an object is on an inclined plane, gravitational forces are decomposed into components parallel and perpendicular to the plane. The parallel component drives the motion down the slope, while the perpendicular component, combined with other forces, determines the normal force, which directly affects the maximum static friction available for rolling without slipping.

Rotational Dynamics and Moment of Inertia

Rotational dynamics deals with the motion of rotating bodies and involves the moment of inertia, which quantifies an object’s resistance to changes in its rotational motion. In rolling motion, the moment of inertia is critical as it links the frictional force to the angular acceleration, thereby governing the object's overall acceleration on the incline.

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