A uniform cylinder of length ℓand mass M having cross-sectional area A is suspended, with its le

step 1 Given arrangement in the cushion where spring has spring constant k and the cylinder has length

A uniform cylinder of length ℓand mass M having...

A uniform cylinder of length $\ell$ and mass $M$ having cross-sectional area $A$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density at equilibrium position. The extension of the spring when it is in equilibrium is: (A) $\frac{M g}{k}\left(1-\frac{\ell A \sigma}{M}\right)$ (B) $\frac{M g}{k}\left(1-\frac{\ell A \sigma}{2 M}\right)$ (C) $\frac{M g}{k}\left(1+\frac{\ell A \sigma}{M}\right)$ (D) $\frac{M g}{k}$

A uniform cylinder of length $\ell$ and mass $M$ having cross-sectional area $A$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density at equilibrium position. The extension of the spring when it is in equilibrium is:
(A) $\frac{M g}{k}\left(1-\frac{\ell A \sigma}{M}\right)$
(B) $\frac{M g}{k}\left(1-\frac{\ell A \sigma}{2 M}\right)$
(C) $\frac{M g}{k}\left(1+\frac{\ell A \sigma}{M}\right)$
(D) $\frac{M g}{k}$

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Key Concepts

Static Equilibrium

Static equilibrium occurs when all the forces acting on an object balance each other so that there is no net force movement. In combined systems, such as one involving gravitational force, buoyant force, and elastic force from a spring, setting the net force to zero allows for solving for unknown quantities like the spring’s extension.

Hooke’s Law

Hooke’s Law describes the behavior of elastic springs, stating that the force exerted by a spring is proportional to its extension or compression (F = kx), where k is the spring constant. It is essential in determining how much a spring stretches or compresses under the influence of external forces.

Archimedes’ Principle

This principle states that any object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. In problems involving immersion, the displaced volume and the fluid density are used to calculate this force, which is crucial when assessing overall force balances in systems that include buoyancy effects.

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