A uniform disc of mass m=1 kg and radius 1 m starts rolling down from A without slipping, on t

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A uniform disc of mass m=1 kg and radius 1 m starts...

A uniform disc of mass $\mathrm{m}=1 \mathrm{~kg}$ and radius $1 \mathrm{~m}$ starts rolling down from A without slipping, on the rough inner surface of a semicircular container of radius $4 \mathrm{~m} .$ The angular velocity of the disc at $\mathrm{B}$ is: (a) $\mathrm{g} \mathrm{rad} \mathrm{s}^{-1}$ (b) $2.5 \mathrm{~g} \mathrm{rad} \mathrm{s}^{-1}$ (c) $\sqrt{2 \mathrm{~g}} \mathrm{rad} \mathrm{s}^{-1}$ (d) $\sqrt{3 \mathrm{~g}} \mathrm{rad} \mathrm{s}^{-1}$

A uniform disc of mass $\mathrm{m}=1 \mathrm{~kg}$ and radius $1 \mathrm{~m}$ starts rolling down from A without slipping, on the rough inner surface of a semicircular container of radius $4 \mathrm{~m} .$ The angular velocity of the disc at $\mathrm{B}$ is:
(a) $\mathrm{g} \mathrm{rad} \mathrm{s}^{-1}$
(b) $2.5 \mathrm{~g} \mathrm{rad} \mathrm{s}^{-1}$
(c) $\sqrt{2 \mathrm{~g}} \mathrm{rad} \mathrm{s}^{-1}$
(d) $\sqrt{3 \mathrm{~g}} \mathrm{rad} \mathrm{s}^{-1}$

Key Concepts

Rolling Without Slipping

This concept refers to the condition in which an object rolls on a surface in such a way that there is no relative sliding between the object and the surface. In problems involving rolling without slipping, the point of contact is momentarily at rest relative to the surface. This leads to the kinematic relation v = ?r, where v is the translational speed of the center of mass, ? is the angular speed, and r is the radius of the rolling object.

Conservation of Mechanical Energy

In the context of dynamics, this principle states that in the absence of non-conservative forces (or if they do no work, such as static friction in rolling conditions), the total mechanical energy (potential plus kinetic) remains constant. When an object rolls down a slope or curved surface, its gravitational potential energy is converted into both translational and rotational kinetic energy.

Translational and Rotational Kinetic Energy

A rolling object possesses two forms of kinetic energy: translational kinetic energy due to the motion of its center of mass, and rotational kinetic energy due to its spin about its own axis. The translational kinetic energy is given by ½mv², while the rotational kinetic energy is given by ½I?², where I is the moment of inertia about the axis of rotation.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a given axis. For a uniform disc, the moment of inertia about its central axis is typically expressed as I = (1/2)mr². This intrinsic property of the disc is crucial in determining its rotational dynamics when subject to external forces or torques.

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