A thin uniform rod A B of mass M and length L is hinged at one end A to the horizontal floor. In

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A thin uniform rod A B of mass M and length L is hinged at...

A thin uniform rod $A B$ of mass $M$ and length $L$ is hinged at one end $\mathrm{A}$ to the horizontal floor. Initially it stands vertically. It is allowed to fall freely on the floor in the vertical plane. The angular velocity of the rod when its end $B$ strikes the floor is $\{\mathrm{A}\} \sqrt{(\mathrm{g} / \mathrm{L})}$ $\{\mathrm{B}\} \sqrt{(2 \mathrm{~g} / \mathrm{L})}$ $\{C\} \sqrt{(3 g / L)}$ $\{\mathrm{D}\} 2 \sqrt{(\mathrm{g} / \mathrm{L})}$

A thin uniform rod $A B$ of mass $M$ and length $L$ is hinged at one end $\mathrm{A}$ to the horizontal floor. Initially it stands vertically. It is allowed to fall freely on the floor in the vertical plane. The angular velocity of the rod when its end $B$ strikes the floor is
$\{\mathrm{A}\} \sqrt{(\mathrm{g} / \mathrm{L})}$
$\{\mathrm{B}\} \sqrt{(2 \mathrm{~g} / \mathrm{L})}$
$\{C\} \sqrt{(3 g / L)}$
$\{\mathrm{D}\} 2 \sqrt{(\mathrm{g} / \mathrm{L})}$

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. In the context of a falling rod, the loss of gravitational potential energy as it moves from a vertical to a horizontal position is converted into rotational kinetic energy, facilitating the calculation of angular velocity.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on the mass distribution relative to that axis. For example, a thin rod hinged at one end has a specific moment of inertia, which is critical for calculating its angular velocity when falling.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation. It is given by the formula (1/2) × I × ?², where I is the moment of inertia and ? is the angular velocity. This relationship is essential in analyzing systems where energy is transferred from potential energy to rotational motion.

Conservation of Energy

This concept states that the total energy in an isolated system remains constant. In rotational dynamics, gravitational potential energy is converted into rotational kinetic energy as an object, such as a rod, falls or rotates. This principle is used to relate the loss in potential energy to the gain in kinetic energy, allowing one to solve for variables like angular velocity.

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