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Chapter 2: Problem 11

A submarine submerges by admitting seawater \((S=1.03)\) into its ballast tanks. The amount of water admitted is controlled by air pressure, because seawater will cease to flow into the tank when the internal pressure (at the hull penetration) is equal to the hydrostatic pressure at the depth of the submarine. Consider a ballast tank, which can be modeled as a vertical half- cylinder \((R=8 \mathrm{ft}\) \(L=20 \mathrm{ft}\) ) for which the air pressure control valve has failed shut. The failure occurred at the beginning of a dive from 60 ft to 1000 ft. The tank was initially filled with seawater to a depth of \(2 \mathrm{ft}\) and the air was at a temperature of \(40^{\circ} \mathrm{F}\). As the weight of water in the \(\operatorname{tank}\) is important in maintaining the boat's attitude, determine the weight of water in the tank as a function of depth during the dive. You may assume that tank internal pressure is always in equilibrium with the ocean's hydrostatic pressure and that the inlet pipe to the tank is at the bottom of the tank and penetrates the hull at the "depth" of the submarine.

Short Answer

Expert verified
The weight of the water in the tank as a function of depth during the dive is given by the sum of the initial weight of the water (when the tank is filled to a depth of 2ft) and the weight of the additional water that enters the tank as the submarine descends, which increases proportionally with the depth until the tank is full. This weight can be calculated at any depth using the steps provided, assuming the tank's internal pressure is always in equilibrium with the ocean's hydrostatic pressure.

Step by step solution

01

Calculate initial volume and weight of seawater in tank

Using the formula for the volume of a cylinder \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height, calculate the initial volume of seawater in the tank when it is filled to a depth of 2ft. Then, multiply this volume by the specific gravity of seawater, \(S=1.03\), to determine the initial weight of the water.
02

Determine the increase in volume with depth

The key concept here is that as the submarine descends, more seawater will enter the tank to equalize the pressure. This increase in volume corresponds to an increased depth in the tank, and can be modeled as increases in the 'height' of the cylinder. Given that the inlet pipe is at the bottom of the tank and that the tank pressure is always in equilibrium with ocean hydrostatic pressure, the volume of water inside the tank increases proportionally with the submarine's depth. Therefore, for every foot the submarine descends, an additional foot of water will enter the tank until it is full.
03

Calculate the weight of water as a function of depth

The weight of water in the tank at any depth \(d\) is the sum of the initial weight calculated in Step 1 and the weight of the additional water at that depth. To find this additional weight, calculate the volume of the additional water using the formula in Step 1 (with \(h = d - 2\) to account for the initial 2ft filling) and multiply it by the specific gravity of seawater. Then sum this with the initial weight to determine the total weight of water as a function of depth during the dive.

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