A solid cylinder is rolling without slipping down an inclinc...

A solid cylinder is rolling without slipping down an inclinc of inclination $\theta$. Minimum coc[ficicnt of friction so that the cylinder docs not slip on the incline is (a) $\tan \theta$ (b) $\frac{\operatorname{lan} \theta}{2}$ (c) $\frac{\tan \theta}{3}$ (d) $\tan \left(\frac{\theta}{3}\right)$

A solid cylinder is rolling without slipping down an inclinc of inclination $\theta$. Minimum coc[ficicnt of friction so that the cylinder docs not slip on the incline is
(a) $\tan \theta$
(b) $\frac{\operatorname{lan} \theta}{2}$
(c) $\frac{\tan \theta}{3}$
(d) $\tan \left(\frac{\theta}{3}\right)$

Key Concepts

Rolling Without Slipping

This concept refers to a motion where an object rotates as it translates, without any sliding between its surface and the contact surface. The rolling condition links linear acceleration to angular acceleration by the radius of the object, ensuring that the point of contact has zero velocity relative to the surface.

Rotational Dynamics and Moment of Inertia

Rotational dynamics deals with the relationship between the net torque acting on an object, its moment of inertia, and its angular acceleration. The moment of inertia quantifies how the mass is distributed with respect to the axis of rotation and plays a crucial role in determining how much torque is needed for a given angular acceleration.

Inclined Plane Forces

Objects on an inclined plane are subject to gravitational force components that act parallel and perpendicular to the surface. The parallel component (mg sin?) influences the acceleration down the slope, while the perpendicular component (mg cos?) determines the normal force, both of which are foundational in analyzing the motion on an incline.

Static Friction in Rolling Motion

Static friction prevents slipping by providing the necessary torque for rotation without energy loss through sliding. It ensures that the rolling condition is maintained and must be sufficient to counteract the slipping tendency induced by gravity on an incline. The minimum coefficient of static friction is determined by the balance between the frictional force needed for rotation and the maximum available static friction related to the normal force.

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