What should be the minimum coefficient of static friction...

What should be the minimum coefficient of static friction between the plane and the cylinder, for the cylinder not to slip on an inclined plane?
(a) $\frac{1}{3} \tan \theta$
(b) $\frac{1}{3} \sin \theta$
(c) $\frac{2}{3} \tan \theta$
(d) $\frac{2}{3} \sin \theta$

Key Concepts

Rotational Dynamics and Torque

Rotational dynamics involves the study of torques, angular accelerations, and moments of inertia. For a rolling object, the net torque produced by friction leads to rotational acceleration, which must be analyzed alongside translational acceleration to fully understand the motion of the object.

Dynamics on an Inclined Plane

Analyzing dynamics on an inclined plane involves decomposing gravitational forces into components parallel and perpendicular to the plane. This approach is essential for understanding how the net force, friction, and torque interact to govern the motion of objects, including their potential to roll without slipping.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It acts to prevent movement when a force is applied, and its maximum value is proportional to the normal force between the surfaces.

Coefficient of Static Friction

The coefficient of static friction is a dimensionless number that represents the ratio of the maximum static frictional force to the normal force. It quantifies the roughness of the contacting surfaces, determining how much force is needed to initiate motion.

Rolling Without Slipping

Rolling without slipping describes the motion of an object wherein its point of contact with the surface is instantaneously at rest relative to the surface. This condition creates a relationship between the translational speed of the object and its rotational speed, ensuring that friction provides the necessary torque to maintain this relationship.

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