A uniform rod of length l and mas m is suspended by two...

A uniform rod of length $l$ and mas $m$ is suspended by two vertica inextensible string as shown figure. Then tension in the left strin when right string snaps, is :
(a) $\frac{3 m g}{4}$
(b) $\frac{m g}{4}$
(c) $\frac{m g}{2}$
(d) $\frac{m g}{8}$

Key Concepts

Reaction Forces in Constrained Motion

When a body is constrained by supports, the reaction forces (such as the tension in a string) adjust to not only balance the weight of the body but also to provide the necessary force to accelerate its center of mass, especially once the body starts moving. Understanding reaction forces is crucial for analyzing scenarios where a sudden removal or failure of one support alters the load distribution on the remaining supports.

Center-of-Mass Acceleration in Rotational Motion

In a rigid body undergoing rotation about a fixed pivot, the acceleration of the center of mass is influenced by both the rotational and translational motion of the body. This concept explains how even though the body rotates, the overall acceleration of its center of mass is determined by the net external forces acting on the body, which in turn affects the reaction forces at supports or pivot points.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear dynamics. When a body rotates about a pivot, the distribution of its mass relative to that pivot plays a crucial role in determining its angular acceleration for a given torque. It is an essential concept for calculating the rotational response of a body when subjected to external forces or torques.

Rotational Dynamics

When a sudden change, such as the removal of one support, occurs, the body begins to rotate about the remaining pivot. Rotational dynamics involves the study of torques, angular acceleration, and the resulting rotational motion of bodies. It helps in understanding how an unbalanced torque produced by forces such as gravity causes the body to start rotating, and how this rotation affects the internal force distributions.

Static Equilibrium

Before any change occurs, a rigid body like a rod held by supports must satisfy the conditions of static equilibrium, meaning that the sum of forces and the sum of torques acting on the body are zero. This concept ensures that the body is at rest and not experiencing any translational or rotational acceleration, which is foundational in understanding the initial distribution of forces (such as tensions) in the supporting elements.

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