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Chapter 14: Problem 37

The United States possesses the eight largest warships in the world-aircraft carriers of the Nimilz 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 fighters take off from it in 25 min, at a location where the free-fall acceleration is \(9.78 \mathrm{m} / \mathrm{s}^{2} .\) Bristling with bombs and missiles, the planes have average mass 29 000 kg. Find the horizontal area enclosed by the waterline of the S1-billion ship. By comparison, its flight deck has area 18 000 \(\mathrm{m}^{2}\). Below decks are passageways hundreds of meters long, so narrow that two large men cannot pass each other.

Short Answer

Expert verified
The horizontal area enclosed by the waterline of the ship is approximately 12,609 square meters.

Step by step solution

01

Calculate the mass of the aircrafts

Firstly, determine the total mass of the 50 fighters that took off from the ship. As it is given that the average mass of each fighter is 29,000 kg, multiply this by 50 to find the total mass. Thus, the total mass \(m\) is \(m = 50 * 29,000 kg = 1,450,000 kg\).
02

Determine the force exerted by the aircrafts and calculate the volume of water displaced

The force exerted by the aircraft can be found by multiplying the total mass of the aircraft by the acceleration due to gravity \(g\), which would give us \(F = m * g = 1,450,000 kg * 9.78 m/s^2 = 1.41 * 10^7 N\). The volume of the water displaced (\( V \)) when the ship bobs up can be calculated using the equation of buoyancy, \( F = 蟻 * g * V \), where \( 蟻 \) is the density of sea water (1025 kg/m^3). Solving for \( V \), we get \( V = F / (蟻 * g) = 1.41 * 10^7 N / (1025 kg/m^3 * 9.78 m/s^2) = 1,387 m^3 \).
03

Calculate the horizontal area enclosed by the waterline of the ship

We have the volume of the water displaced and the height the ship has floated up, so we can calculate the horizontal surface area enclosed by the waterline of the ship with the formula for volume, \(V = A * H\), where \(A\) is the area and \(H\) is the height of 11.0 cm = 0.11 m. Rearranging for \(A\), we get \(A = V / H = 1,387 m^3 / 0.11 m = 12,609 m^2, rounded to the nearest whole number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Displacement
Understanding volume displacement is crucial when discussing buoyancy. When an object is submerged or partially submerged in a fluid, it displaces a certain volume of that fluid. This is essentially the volume of the object that is under the surface of the water. In the given problem, when the fighters take off, the warship bobs up 11.0 cm, indicating that it no longer displaces the same volume of water.

To determine this volume displacement, we can use Archimedes' principle. This principle states that the upward buoyant force exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. The volume of displaced water in this scenario was calculated using the formula:
  • \[ V = \frac{F}{\rho \cdot g} \]
Here, \(F\) is the force exerted by the aircraft, \(\rho\) is the density of sea water, and \(g\) is the acceleration due to gravity. This provides a deeper look into how a ship's buoyancy changes with different external weights, affecting its displacement and waterline.
Force Calculation
Force calculation is a fundamental step in analyzing the problem of buoyancy. The force exerted by the weight of the aircraft must be considered to find how much water is displaced when the ship rises. This involves calculating gravitational force, which is the force with which Earth pulls objects towards its center.

To find the force exerted by the aircraft that took off, we use the formula:
  • \[ F = m \times g \]
Here, \(m\) represents the total mass of the aircraft and \(g\) is the acceleration due to gravity, provided as 9.78 m/s虏. This calculation helps to understand how weight impacts buoyancy as the force exerted downward must be equal to the upward buoyant force under static conditions. Identifying this force helps predict changes in the ship鈥檚 draft, which is the vertical distance between the waterline and the bottom of the hull.
Mass and Weight Relationship
The relationship between mass and weight is essential to solving buoyancy problems. Mass refers to the amount of matter in an object and is typically measured in kilograms. Weight, on the other hand, is the force exerted by gravity on that mass, which changes depending on the gravitational force present. This concept often confuses students, as they tend to use the terms interchangeably.

Weight can be calculated using:
  • \[ \text{Weight} = m \times g \]
In this calculation, \(m\) is the mass and \(g\) is the gravity. As the gravitational force may vary slightly from one location to another (as in the given example where \(g\) is 9.78 m/s虏 instead of the standard 9.81 m/s虏), this affects the object's weight. By understanding this relationship, it is easier to comprehend how changes in mass affect buoyancy, displacement, and the ship's rising in water.

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Most popular questions from this chapter

\- A \(10.0-\mathrm{kg}\) block of metal measuring \(12.0 \mathrm{cm} \times 10.0 \mathrm{cm} \times\) \(10.0 \mathrm{cm}\) is suspended from a scale and immersed in water as in Figure P14.25b. The 12.0-cm dimension is vertical, and the top of the block is \(5.00 \mathrm{cm}\) below the surface of the water. (a) What are the forces acting on the top and on the bottom of the block? (Take \(P_{0}=1.0130 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) ) (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.

In \(1983,\) the United States began coining the cent piece out of copper-clad rinc rather than pure copper. The mass of the old copper penny is \(3.083 \mathrm{g},\) while that of the new cent is 2.517 g. Calculate the percentage of sinc (by volume) in the new cent. The density of copper is \(8.960 \mathrm{g} / \mathrm{cm}^{3}\) and that of zinc is \(7.133 \mathrm{g} / \mathrm{cm}^{3} .\) The new and old coins have the same volume.

A hypodermic syringe contains a medicine with the density of water (Figure \(\mathbf{P} 1 \mathbf{1} . \mathbf{5 3}\) ). The barrel of the syringe has a cross sectional area \(A=2.50 \times 10^{-5} \mathrm{m}^{2},\) and the needle has a cross-sectional area \(a=1.00 \times 10^{-8} \mathrm{m}^{2} .\) In the absence of a force on the plunger, the pressure everywhere is 1 atm. \(A\) force \(\mathbf{F}\) of magnitude \(2.00 \mathrm{N}\) acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

The human brain and spinal cord are immersed in the cerebrospinal fluid. The Auid is normally continuous between the cranial and spinal cavities. It normally exerts a pressure of 100 to \(200 \mathrm{mm}\) of \(\mathrm{H}_{2} \mathrm{O}\) above the prevailing atmospheric pressure. In medical work pressures are often measured in units of millimeters of \(\mathrm{H}_{2} \mathrm{O}\) because body fluids, including the cercbrospinal fluid, typically have the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap, as illus. treated in Figure P14.21. A hollow tube is inserted into the spinal column, and the height to which the fluid rises is observed. If the fluid rises to a height of \(160 \mathrm{mm}\), we write its gauge pressure as \(160 \mathrm{mm} \mathrm{H}_{2} \mathrm{O} .\) (a) Express this pres. sure in pascals, in atmospheres, and in millimeters of mercury. (b) Sometimes it is necessary to determine if an accident victim has suffered a crushed vertebra that is blocking flow of the cerebrospinal fluid in the spinal column. In other cases a physician may suspect a tumor or other growth is blocking the spinal column and inhibiting flow of cerebrospinal fluid. Such conditions can be investigated by means of the Queckensied test. In this procedure, the veins in the patient's neck are compressed, to make the blood pressure rise in the brain. The increase in pressure in the blood vessels is transmitted to the cercbrospinal fluid. What should be the normal effect on the height of the fluid in the spinal tap? (c) Suppose that compressing the veins had no effect on the fluid level. What might account for this?

A Pilot tube can be used to determine the velocity of air flow by measuring the difference between the total pressure and the static pressure (Fig. \(\mathrm{P} 14.49) .\) If the fluid in the tube is mercury, density \(\rho_{\mathrm{Hg}}=13600 \mathrm{kg} / \mathrm{m}^{3},\) and \(\Delta h=5.00 \mathrm{cm},\) find the speed of air flow. (Assume that the air is stagnant at point \(A,\) and take \(\rho_{\mathrm{air}}=1.25 \mathrm{kg} / \mathrm{m}^{3} .\) )
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