A weightless rod of length 2 l carries two equal masses m,...

A weightless rod of length $2 l$ carries two equal masses $m$, one tied at lower end $A$ and the other at the middle of the rod at $B$. The rod can rotate in vertical planc about a fixed horizontal axis passing through $C$. The rod is released from rest in horizontal position. The speed of the mass $B$ at the instant rod becomes vertical is (a) $\sqrt{\frac{3 g l}{5}}$ (b) $\sqrt{\frac{4 g l}{5}}$ (c) $\sqrt{\frac{6 g l}{5}}$ (d) $\sqrt{\frac{7 g l}{5}}$

A weightless rod of length $2 l$ carries two equal masses $m$, one tied at lower end $A$ and the other at the middle of the rod at $B$. The rod can rotate in vertical planc about a fixed horizontal axis passing through $C$. The rod is released from rest in horizontal position. The speed of the mass $B$ at the instant rod becomes vertical is
(a) $\sqrt{\frac{3 g l}{5}}$
(b) $\sqrt{\frac{4 g l}{5}}$
(c) $\sqrt{\frac{6 g l}{5}}$
(d) $\sqrt{\frac{7 g l}{5}}$

Key Concepts

Relation between Angular Velocity and Linear Speed

For any rotating body, the linear speed at a point is related to the angular velocity and the distance from the axis of rotation. This relation, expressed as v = ?r, is key in connecting the rotational dynamics to linear measurements and is particularly important when determining the speed of a specific mass located at some distance from the pivot.

Conservation of Energy in Rotational Systems

This concept involves equating the loss in gravitational potential energy to the gain in rotational kinetic energy of a system. When an object rotates under gravity, potential energy is converted into kinetic energy, and by accounting for all energy changes in the system, one can determine quantities like angular velocity and linear speed. This principle is fundamental in solving problems where energy transitions occur without non-conservative losses.

Rotational Kinetic Energy

Rotational kinetic energy is the energy associated with the rotation of an object about an axis and is given by one half the product of the moment of inertia and the square of the angular velocity. Understanding this energy form is crucial when dealing with rotating bodies and allows for the analysis of how energy is distributed among various parts of a system in motion.

Moment of Inertia

The moment of inertia measures a body’s resistance to changes in its rotational motion about a given axis. It depends on the mass distribution relative to the axis of rotation. Calculating the total moment of inertia for a composite system often requires summing the moments of inertia of individual masses, sometimes using the parallel axis theorem if the masses are not located at the axis of rotation.

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