A spring with spring constant 35 N / m is attached to the...

A spring with spring constant $35 \mathrm{N} / \mathrm{m}$ is attached to the ceiling, and a 5.0 - cm-diameter, $1.0 \mathrm{kg}$ metal cylinder is attached to its lower end. The cylinder is held so that the spring is neither stretched nor compressed, then a tank of water is placed underneath with the surface of the water just touching the bottom of the cylinder. When released, the cylinder will oscillate a few times but, damped by the water, quickly reach an equilibrium position. When in equilibrium, what length of the cylinder is submerged?

Key Concepts

Static Equilibrium

Static equilibrium occurs when all the forces acting on an object balance each other, resulting in zero net force and no acceleration. In the context of the problem, the object reaches a point where the gravitational force, the spring's restoring force, and the buoyant force from the water exactly counteract one another, establishing a stable equilibrium state.

Damped Oscillatory Motion

Damped oscillatory motion refers to the oscillation of an object where the amplitude gradually decreases over time due to resistive forces such as friction or fluid drag. This concept is relevant when considering that the cylinder initially oscillates upon release, but the water's damping effect quickly brings it to a steady equilibrium.

Archimedes' Principle

Archimedes' Principle explains that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This concept is essential for determining the submerged volume of the object at equilibrium, as the buoyant force must balance with the weight of the object and any other forces acting on it.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its rest position, described mathematically as F = -kx, where k is the spring constant and x is the displacement. This principle is crucial for understanding how the spring behaves when a mass is attached and subsequently displaced from its equilibrium position.

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