In the adjoining Fig. 4.21, block A is of mass m and block B...

In the adjoining Fig. 4.21, block $A$ is of mass $m$ and block $B$ is of mass $2 \mathrm{~m}$. The spring has a force constant $k$. All the surfaces are smooth and the system is released from rest with spring unstretched, then (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 $2 g \sqrt{\frac{m}{k}}$ (C) the net acceleration of block $B$ when the extension in the spring is maximum, is $\frac{g}{2}$. (D) tension in the thread for extension of $\frac{2 m g}{k}$ in spring is $m g$.

In the adjoining Fig. 4.21, block $A$ is of mass $m$ and block $B$ is of mass $2 \mathrm{~m}$. The spring has a force constant $k$. All the surfaces are smooth and the system is released from rest with spring unstretched, then
(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 $2 g \sqrt{\frac{m}{k}}$
(C) the net acceleration of block $B$ when the extension in the spring is maximum, is $\frac{g}{2}$.
(D) tension in the thread for extension of $\frac{2 m g}{k}$ in spring is $m g$.

Key Concepts

Newton's Second Law

This fundamental law states that the acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. It is crucial in analyzing the forces acting on each element in a system, determining how they accelerate and interact, particularly when external forces and internal forces such as tension are involved.

Hooke's Law

Hooke's Law describes the behavior of springs, stating that the force exerted by a spring is directly proportional to its extension or compression and acts in the opposite direction. This principle is central to understanding systems where a spring mediates the interaction between different masses, enabling the calculation of restoring forces and energy stored in the spring.

Relative Motion in Connected Systems

In systems with interconnected objects, such as masses connected by a spring, understanding the relative motion is essential. The analysis involves determining how the individual motions of the bodies relate to each other, especially under conditions like maximum extension, when the relative acceleration between parts of the system momentarily becomes zero.

Energy Conservation in Mechanics

Energy conservation is a key concept wherein the total energy of an isolated system remains constant. In mechanical systems, the conversion between kinetic energy, gravitational potential energy, and elastic potential energy (from springs) helps predict system behavior over time, such as speeds at various displacements or extensions.

Dynamic Equilibrium Conditions

Dynamic equilibrium in a mechanical context refers to moments when the net forces or net accelerations on parts of a system become balanced even though the system may still be in motion. This concept is particularly useful for analyzing temporary states in which motion parameters, like acceleration or extension rates, reach an extremum or change direction.

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