The United States possesses the eight largest warships in...

The United States possesses the eight largest warships in 0the world - aircraft carriers of the Nimitz class-and is building two more. Suppose one of the ships bobs up to float 11.0 $\mathrm{cm}$ higher in the water when 50 fighter planes take off from it in 25 minutes, at a location where the free-fall acceleration is 9.78 $\mathrm{m} / \mathrm{s}^{2}$ . Bristling with bombs and missiles, the planes have an average mass of 29000 $\mathrm{kg}$ . Find the horizontal area enclosed by the waterline of the $\$ 4$ billion ship. By comparison, its flight deck has area 18000 $\mathrm{m}^{2}$ . Below decks are passageways hundreds of meters long, so narrow that two large men cannot pass each other.

The United States possesses the eight largest warships in 0the world - aircraft carriers of the Nimitz class-and is building two more. Suppose one of the ships bobs up to float 11.0 $\mathrm{cm}$ higher in the water when 50 fighter planes take off from it in 25 minutes, at a location where the free-fall acceleration is 9.78 $\mathrm{m} / \mathrm{s}^{2}$ . Bristling with bombs and missiles, the planes have an average mass of 29000 $\mathrm{kg}$ . Find the horizontal area enclosed by the waterline of the
$\$ 4$ billion ship. By comparison, its flight deck has area 18000 $\mathrm{m}^{2}$ . Below decks are passageways hundreds of meters long, so narrow that two large men cannot pass each other.

Key Concepts

Waterline Geometry

The horizontal area enclosed by the waterline (essentially the waterplane area) of a ship is a key geometric feature that helps determine how much the ship will rise or sink for a given change in weight. Here, by recognizing that the rise in the ship corresponds to a specific volume of water no longer being displaced, one can calculate the waterline area by dividing the removed volume (determined from the lost weight) by the vertical change in draft.

Density, Weight, and Gravitational Acceleration

To relate mass to the volume of water displaced, one must use the density of water along with the acceleration due to gravity. This conversion is critical in problems involving buoyancy, as it allows one to determine how much volume corresponds to a given weight. In this problem, knowing the gravitational acceleration helps translate the aircrafts' mass into the weight that was formerly contributing to the water displacement.

Buoyant Force and Displacement

When a ship is floating, the buoyant force, which is the upward force exerted by the displaced water, is equal to the weight of the ship. The change in the ship’s draft (i.e., how high it sits in the water) can be used to determine its waterline area, because the lost weight corresponds to a reduced volume of water displaced, directly linking the vertical change in draft to the horizontal area of the hull below the waterline.

Archimedes' Principle

This principle states that a floating body displaces an amount of fluid whose weight is equal to the weight of the body. In the context of the problem, once mass (in the form of fighter planes) is removed from the ship, the ship needs to displace less water to balance its weight, which explains why the ship rises higher in the water.

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