A light rod carries three equal masses A, B and C as shown...

A light rod carries three equal masses $A, B$ and $C$ as shown in figure. The velocity of $B$ in vertical position of rod if it is released from horizontal position as shown in figure is: (a) $\sqrt{2 g} l$ (b) $\sqrt{\frac{18 g l}{7}}$ (c) $\sqrt{\frac{4 g l}{3}}$ (d) $\sqrt{\frac{8 g l}{7}}$

A light rod carries three equal masses $A, B$ and $C$ as shown in figure. The velocity of $B$ in vertical position of rod if it is released from horizontal position as shown in figure is:
(a) $\sqrt{2 g} l$
(b) $\sqrt{\frac{18 g l}{7}}$
(c) $\sqrt{\frac{4 g l}{3}}$
(d) $\sqrt{\frac{8 g l}{7}}$

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Key Concepts

Conservation of Energy

This principle states that in a closed system the total energy remains constant. In dynamics problems involving rotation, the loss of gravitational potential energy is converted into rotational kinetic energy, allowing one to relate the change in height of masses to their acquired speeds.

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in its rotational motion. It is calculated as the sum of the mass of each point multiplying the square of its distance from the axis of rotation. In problems involving rods and attached masses, computing the moment of inertia is essential for determining the angular motion.

Rotational Kinetic Energy

In rotational motion, the kinetic energy is expressed as (1/2) times the moment of inertia times the square of the angular velocity. Using this formulation in energy conservation calculations allows one to determine the angular velocity generated by a given change in potential energy.

Relationship Between Angular and Linear Velocity

For a point mass moving in a circular path, the linear speed is directly related to the angular velocity by the equation v = r?, where r is the distance from the rotation axis. This relationship is fundamental when converting the calculated angular velocity into the linear speed of a point on a rotating object.

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