In the adjoining Fig. 5.36 block A is of mass m and block B...

In the adjoining Fig. $5.36$ block $A$ is of mass $m$ and block $B$ is of mass $2 m .$ The spring has a force constant $k$. All the surfaces are smooth and the system is released from rest with the spring not being stretched (A) The maximum extension of the spring is $\frac{4 m g}{k}$ (B) The speed of block $A$ when extension in spring is $\frac{2 m g}{k}$, is $2 g \sqrt{\frac{2 m}{3 k}}$ (C) Net acceleration of block $B$ when the extension in the spring is maximum, is $\frac{2}{3} g$. (D) Tension in the thread for extension of $\frac{2 m g}{k}$ in spring is $m g$.

In the adjoining Fig. $5.36$ block $A$ is of mass $m$ and block $B$ is of mass $2 m .$ The spring has a force constant $k$. All the surfaces are smooth and the system is released from rest with the spring not being stretched
(A) The maximum extension of the spring is $\frac{4 m g}{k}$
(B) The speed of block $A$ when extension in spring is $\frac{2 m g}{k}$, is $2 g \sqrt{\frac{2 m}{3 k}}$
(C) Net acceleration of block $B$ when the extension in the spring is maximum, is $\frac{2}{3} g$.
(D) Tension in the thread for extension of $\frac{2 m g}{k}$ in spring is $m g$.

Key Concepts

Relative Motion in Connected Systems

In systems where multiple bodies are interconnected, the motions of the individual components are not independent. Analyzing relative motion ensures that constraints imposed by connections (such as inextensible strings or springs) are respected. This concept is vital to correctly relate the accelerations, velocities, and displacements of the objects in the system, thereby enabling a coherent and accurate analysis of the overall motion.

Free Body Diagram Analysis

Free body diagrams are essential tools in mechanics for visually representing all the forces acting on individual objects. By isolating each body and diagramming forces like tension, gravitational force, and spring force, one can systematically apply Newton's laws to solve for accelerations and tensions, simplifying the analysis of complex systems into manageable components.

Equilibrium and Extremum Conditions

Understanding equilibrium conditions and identifying extremum points, such as maximum extension or compression, is key in dynamics. At these positions, certain conditions—like the instantaneous zero relative acceleration or zero net external force on a component—are satisfied, which allows for the determination of critical parameters like maximum displacement or transition points in the motion.

Hooke's Law and Spring Forces

Hooke's Law establishes that the force exerted by an ideal spring is directly proportional to the amount by which it is stretched or compressed, with the proportionality constant being the spring’s force constant. This relationship is crucial for analyzing spring-mass systems, predicting the behavior of the spring under load, and determining quantities such as extension, compression, and the associated restoring force.

Newton's Second Law of Motion

This fundamental principle states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In many-body systems, applying this law to individual objects helps in establishing equations of motion, enabling the calculation of accelerations, tensions, and other dynamic quantities by setting up and solving relevant force balance equations.

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