A cyclindrical buoy weighing 20 kN is to float in sea water...

A cyclindrical buoy weighing 20 kN is to float in sea water whose density is $1020 \mathrm{~kg} / \mathrm{m}^3$. The buoy has a diameter of 2 m and is 2.5 m high. Prove that it is unstable. If the buoy is anchored with a chain attached to the centre of its base, find the tension in the chain to keep the buoy in vertical position.

A cyclindrical buoy weighing 20 kN is to float in sea water whose density is $1020 \mathrm{~kg} / \mathrm{m}^3$. The buoy has a diameter of 2 m and is 2.5 m high. Prove that it is unstable.

If the buoy is anchored with a chain attached to the centre of its base, find the tension in the chain to keep the buoy in vertical position.

Key Concepts

Restraint Forces and Tension

When a floating body is inherently unstable, external restraints like chains or moorings are used to maintain its desired orientation. The tension in the restraint is calculated by analyzing the moment equilibrium about a pivotal point, taking into account the positions of the buoy's center of gravity, buoyant force, and the point of attachment of the chain. This concept is essential for determining how much additional force is required to counteract the overturning moments and stabilize the buoy.

Static Equilibrium

Static equilibrium in the context of floating bodies requires that the sum of forces and moments acting on the buoy is zero. This involves balancing the upward buoyant force against the downward weight and any additional forces, such as the tension from an anchoring chain. Ensuring equilibrium is essential when determining the forces needed, such as the chain tension, to keep an unstable buoy in a vertical position.

Metacentric Height

Metacentric height is the distance between the center of gravity and the metacenter, which is the point where the buoyant force intersects the buoy’s vertical line when it is slightly tilted. A positive metacentric height indicates stable equilibrium, whereas a negative value implies that the body is unstable. This concept is pivotal when proving the instability of the buoy as it quantifies its initial stability upon tilting.

Center of Buoyancy

The center of buoyancy is the point at which the resultant buoyant force, due to displaced fluid, is considered to act. It is typically located at the centroid of the displaced volume. In analyzing the stability of floating bodies, the relative positions of the center of buoyancy and the center of gravity determine the tendency of the body to right itself or overturn.

Archimedes' Principle

This principle states that a fluid exerts an upward buoyant force on a submerged object equal to the weight of the fluid it displaces. It is a foundational concept in hydrostatics that governs the buoyant force acting on floating bodies and is crucial for determining whether the buoy’s weight is balanced by the displaced seawater.

Center of Gravity

The center of gravity is the point through which the entire weight of the buoy can be considered to act. Its position in relation to the center of buoyancy and the metacenter is critical in assessing the stability of a floating body. An elevated center of gravity relative to the center of buoyancy can destabilize the buoy by promoting overturning.

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