An imperfectly rough sphere moves from rest down a plane...

An imperfectly rough sphere moves from rest down a plane inclined at an angle $\alpha$ to the horizontal. The coefficient of friction between the inclined plane and sphere is $\mu$. Then (a) if $\mu<\frac{2}{7} \tan \alpha$, then the sphere never rolls (b) if $\mu=\frac{2}{7} \tan \alpha$, then the maximum friction always being exerted (c) $\mu>\frac{2}{7} \tan \alpha$, then pure rolling takes place (d) all the above

An imperfectly rough sphere moves from rest down a plane inclined at an angle $\alpha$ to the horizontal. The coefficient of friction between the inclined plane and sphere is $\mu$. Then
(a) if $\mu<\frac{2}{7} \tan \alpha$, then the sphere never rolls
(b) if $\mu=\frac{2}{7} \tan \alpha$, then the maximum friction always being exerted
(c) $\mu>\frac{2}{7} \tan \alpha$, then pure rolling takes place
(d) all the above

Key Concepts

Role of Friction in Rolling Motion

Friction, particularly static friction, is essential in initiating and sustaining rolling without slipping. It acts to provide the necessary torque that sets the object into rotation while simultaneously preventing relative sliding at the contact point. The available friction must be sufficient to meet the dynamic requirements without exceeding its maximum limit, thereby linking the conditions of motion directly to the coefficient of friction.

Translational and Rotational Dynamics

In the study of rolling objects, both translational and rotational dynamics must be considered. Newton’s second law governs the linear acceleration of the center of mass while a corresponding rotational equation relates the torque due to friction with the angular acceleration. The interplay between these two aspects, linked by the rolling condition, is crucial in understanding the motion and stability of rolling bodies.

Rolling Without Slipping

Rolling without slipping is the motion in which an object rotates in such a way that its point of contact with the surface remains momentarily at rest relative to the surface. This condition establishes a specific relation between the translational velocity and the angular velocity of the rolling object (v = ?r) and is key to analyzing the combined rotational and linear motion of bodies.

Dynamics on Inclined Planes

The dynamics of bodies on inclined planes involves resolving gravitational forces into components parallel and perpendicular to the surface. These components determine the acceleration along the plane and the normal force, which in turn affects the frictional forces. This analysis is fundamental in problems involving motion down a slope, especially when evaluating whether rolling or slipping occurs.

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