Λsolid spherical ball of mass M and radius R rolls without...

$\Lambda$ solid spherical ball of mass $M$ and radius $R$ rolls without slipping down a surface inclined to horizontal at an angle $\theta$. Considering that ball is uniform solid sphere and that ball and surface are perfectly rigid. Column-I Column-II (a) Friction force involved (p) 7ero (b) Minimum value of coefficient of friction for pure rolling (q) $(2 / 7) M g \sin \theta$ (c) Work done against frictional force (r) Static friction (d) Force of kinetic friction (s) $(2 / 7) \tan \theta$

$\Lambda$ solid spherical ball of mass $M$ and radius $R$ rolls without slipping down a surface inclined to horizontal at an angle $\theta$. Considering that ball is uniform solid sphere and that ball and surface are perfectly rigid.
Column-I Column-II
(a) Friction force involved
(p) 7ero
(b) Minimum value of coefficient of friction for pure rolling
(q) $(2 / 7) M g \sin \theta$
(c) Work done against frictional force
(r) Static friction
(d) Force of kinetic friction
(s) $(2 / 7) \tan \theta$

Key Concepts

Pure Rolling Condition

In rolling motion, pure rolling occurs when an object rolls without slipping, meaning the point of contact with the surface is momentarily at rest. This links the translational velocity and angular velocity through the relation v = ?R, ensuring that the object's rotation exactly compensates for its forward movement.

Static Friction

Static friction is the force that prevents relative motion at the point of contact between the rolling object and the surface. In the case of pure rolling, static friction provides the necessary torque for rotation without doing work, as there is no displacement at the point of contact relative to the surface.

Minimum Coefficient of Friction

The minimum coefficient of friction is a threshold value that ensures the available static friction is sufficient to prevent slipping during rolling. It is determined by balancing the forces acting parallel to the inclined plane and ensuring that the frictional force required for pure rolling does not exceed what static friction can provide.

Work Done by Friction

In a scenario of pure rolling without slipping, the static friction force does not perform work because the point of contact on the rolling object is instantaneously at rest relative to the surface. Therefore, even though friction is essential for initiating and sustaining the rolling motion, it does not lead to energy dissipation.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For rolling objects, particularly those with non-uniform mass distributions such as a sphere, the moment of inertia plays a critical role in determining how the gravitational energy is partitioned between translational and rotational kinetic energies during motion down an incline.

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