A uniform cylinder, a uniform solid sphere and a uniform...

A uniform cylinder, a uniform solid sphere and a uniform hollow spherical shell, all roll down on a rough inclined plane. If only one of them slip, it is
(a) solid sphere
(b) hollow sphere
(c) cylinder
(d) depending on the mass of the body.

Key Concepts

Rolling Without Slipping

This concept refers to the condition where an object rolls on a surface without any relative motion (sliding) at the point of contact. It implies a specific relationship between the linear velocity (or acceleration) of the center of mass and the angular velocity (or acceleration), typically given by v = R? or a = R?. Maintaining this condition requires that static friction be sufficient to prevent slipping during the motion.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on how its mass is distributed relative to the axis of rotation. In problems involving rolling, a lower moment of inertia (relative to mass times the square of the radius) means that the object can achieve higher acceleration for a given gravitational force because less of the potential energy goes into rotational kinetic energy.

Static Friction in Rolling Motion

Static friction is the force that acts at the point of contact between the rolling object and the surface, preventing slipping. The amount of static friction needed to maintain rolling without slipping depends on both the linear acceleration of the object and its moment of inertia. If the required static friction exceeds what is available (determined by the surface's coefficient of friction), the object will begin to slip.

Dynamics of Rigid Bodies on Inclined Planes

This concept involves analyzing both the translational and rotational motions of bodies rolling down an incline. The gravitational force component along the slope accelerates the center of mass, while friction provides the necessary torque for rotation. The balance between these forces, influenced by the object's moment of inertia, determines how quickly an object accelerates and whether the imposed friction demands can be met without slipping.

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