A block A of mass 10 kg, connected to another hollow block...

A block $A$ of mass $10 \mathrm{~kg}$, connected to another hollow block $B$ of same size and negligible mass, by a spring of spring constant $500 \mathrm{~N} / \mathrm{m}$, floats in water as shown in Fig. 8.36. The compression in the spring is $\left(\rho_{\text {water }}=1 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}, g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$ (A) $10 \mathrm{~cm}$ (B) $20 \mathrm{~cm}$ (C) $50 \mathrm{~cm}$ (D) $100 \mathrm{~cm}$

A block $A$ of mass $10 \mathrm{~kg}$, connected to another hollow block $B$ of same size and negligible mass, by a spring of spring constant $500 \mathrm{~N} / \mathrm{m}$, floats in water as shown in Fig. 8.36. The compression in the spring is $\left(\rho_{\text {water }}=1 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}, g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
(A) $10 \mathrm{~cm}$
(B) $20 \mathrm{~cm}$
(C) $50 \mathrm{~cm}$
(D) $100 \mathrm{~cm}$

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its displacement from its equilibrium position, expressed mathematically as F = kx, where k is the spring constant and x is the displacement. This concept is essential when analyzing systems that involve springs, as it allows us to connect the mechanical deformation of the spring to the forces involved in the system.

Equilibrium Analysis

Equilibrium analysis involves setting the net forces acting on a system to zero in order to determine the conditions under which the system is at rest. In the context of floating objects connected by springs, this means balancing the gravitational force, buoyant force, and the force due to spring compression or extension. This approach is key to solving for unknown displacements or forces that maintain the system in a stable state.

Buoyancy

Buoyancy refers to the upward force exerted by a fluid that opposes the weight of an immersed object. This force is determined by Archimedes’ principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. In problems involving floating bodies, this concept is crucial for determining equilibrium conditions and understanding how submerged volumes contribute to the balance of forces.

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