A solid cylinder of mass M and radius R rolls down an inclined plane of height h. The angular ve

step 1 Question is a solid cylinder of mass and radius are rolled downs and inclined plane of height at

A solid cylinder of mass M and radius R rolls down an...

A solid cylinder of mass $\mathrm{M}$ and radius $\mathrm{R}$ rolls down an inclined plane of height $\mathrm{h}$. The angular velocity of the cylinder when it reaches the bottom of the plane will be. $\{\mathrm{A}\}(2 / \mathrm{R}) \sqrt{(\mathrm{gh})}$ $\{\mathrm{B}\}(2 / \mathrm{R}) \sqrt{(\mathrm{gh} / 2)}$ $\{\mathrm{C}\}(2 / \mathrm{R}) \sqrt{(\mathrm{gh} / 3)}$ $\{\mathrm{D}\}(1 / 2 \mathrm{R}) \sqrt{(\mathrm{gh})}$

A solid cylinder of mass $\mathrm{M}$ and radius $\mathrm{R}$ rolls down an inclined plane of height $\mathrm{h}$. The angular velocity of the cylinder when it reaches the bottom of the plane will be.
$\{\mathrm{A}\}(2 / \mathrm{R}) \sqrt{(\mathrm{gh})}$
$\{\mathrm{B}\}(2 / \mathrm{R}) \sqrt{(\mathrm{gh} / 2)}$
$\{\mathrm{C}\}(2 / \mathrm{R}) \sqrt{(\mathrm{gh} / 3)}$
$\{\mathrm{D}\}(1 / 2 \mathrm{R}) \sqrt{(\mathrm{gh})}$

Key Concepts

Relationship between Translational and Rotational Kinetic Energy

In rolling motion, the kinetic energy is partitioned into translational and rotational components. This separation, along with the no-slip condition, allows one to relate the angular velocity of the rolling object to its speed, which is essential in solving problems involving rolling objects on inclined planes.

Moment of Inertia

The moment of inertia quantifies how the mass of an object is distributed relative to its rotational axis and determines the resistance of the object to changes in its rotational motion. For standard shapes like a solid cylinder, its value plays a key role in computing the rotational kinetic energy during motion.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces, the total mechanical energy of a system remains constant. In the context of rolling objects, the gravitational potential energy at a height is converted into both translational and rotational kinetic energy as the object descends.

Rolling Without Slipping

This condition occurs when a rolling object moves such that its point of contact with the surface is momentarily at rest. It creates a relationship between the linear velocity and angular velocity of the object (v = ?R), which is crucial for correctly relating kinetic energies in translational and rotational forms.

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