A solid sphere of radius 2 m and mass 100 kg is rotating initially with an angular velocity of

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A solid sphere of radius 2 m and mass 100 kg is rotating...

A solid sphere of radius $2 \mathrm{~m}$ and mass $100 \mathrm{~kg}$ is rotating initially with an angular velocity of 5 rud/sec. It is then placod on a rough ground with friction coefficient of $0.3$. Afler somc time il starts rolling Column-I Column-II (a) Time after which it starts rolling (p) $577.17$ (b) Angular velocity of sphere (q) $2.858$ (c) Velocity of centre of mass (r) $1.429$ (d) Net mechanical energy of the system (s) $0.972$

A solid sphere of radius $2 \mathrm{~m}$ and mass $100 \mathrm{~kg}$ is rotating initially with an angular velocity of 5 rud/sec. It is then placod on a rough ground with friction coefficient of $0.3$. Afler somc time il starts rolling
Column-I Column-II
(a) Time after which it starts rolling
(p) $577.17$
(b) Angular velocity of sphere
(q) $2.858$
(c) Velocity of centre of mass
(r) $1.429$
(d) Net mechanical energy of the system
(s) $0.972$

Key Concepts

Rolling Without Slipping

This concept occurs when an object rolls on a surface such that the point of contact has zero velocity relative to the surface. It establishes a kinematic relation between the translational velocity and the angular velocity (v = ?r), which is fundamental in determining when an object begins to roll properly without sliding.

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. In the context of rolling objects, friction—particularly static friction—is what enables the transition from sliding to rolling by providing the necessary torque, and it also plays a role in dissipating energy as heat.

Rotational Dynamics

Rotational dynamics deals with the effects of torques on the angular motion of objects. It involves concepts such as angular acceleration, the moment of inertia, and the work done by torques (? = I?). These principles help analyze the evolution of the object's spin and its interaction with friction.

Moment of Inertia

The moment of inertia quantifies how mass is distributed relative to an object's axis of rotation. It defines the resistance to changes in an object's rotational state and is essential in calculating angular acceleration and the rotational kinetic energy during dynamics analyses.

Energy Dissipation

Energy dissipation in a mechanical system refers to the conversion of organized kinetic energy into thermal energy due to non-conservative forces like friction. In problems involving rolling, energy losses manifest as a reduction in mechanical energy, influencing both the rotational and translational energy components of the system.

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