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Chapter 15: Problem 22

A water storage tower, like the one shown in the accompanying photo, is filled with freshwater to a depth of \(6.4 \mathrm{m}\). What is the pressure at (a) \(4.5 \mathrm{m}\) and (b) \(5.5 \mathrm{m}\) below the surface of the water? \((c)\) Why are the metal bands on such towers more closely spaced near the base of the tower?

Short Answer

Expert verified
(a) The pressure at 4.5 m is \( P_0 + 44,145 \text{ Pa} \). (b) At 5.5 m, it's \( P_0 + 53,955 \text{ Pa} \). (c) Bands are closer at the base due to higher pressure.

Step by step solution

01

Understanding Pressure in Fluids

Pressure in a fluid at a specific depth is given by the formula \( P = P_0 + \rho g h \), where \( P_0 \) is the atmospheric pressure at the surface, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \) on Earth), and \( h \) is the depth within the fluid.
02

Calculate Pressure at 4.5 m Below the Surface

Using the formula from Step 1 and setting \( h = 4.5 \text{ m} \), the pressure at this depth is \( P = P_0 + \rho g (4.5) \). Assuming the density of water is \( 1000 \text{ kg/m}^3 \), we have \( P = P_0 + 1000 \times 9.81 \times 4.5 \). Calculate this value to find the pressure at 4.5 m.
03

Calculate Pressure at 5.5 m Below the Surface

Using the formula \( P = P_0 + \rho g h \) and setting \( h = 5.5 \text{ m} \), the pressure is \( P = P_0 + 1000 \times 9.81 \times 5.5 \). Calculate this value to find the pressure at 5.5 m.
04

Explanation of Metal Bands on Water Towers

The reason for the closer spacing of metal bands near the base of the tower is due to the increase in pressure with depth. As depth increases, the force exerted by the water on the structure increases, necessitating stronger reinforcement to withstand these greater forces.

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Key Concepts

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

Density of Fluid
The density of a fluid is a measure of how much mass is contained in a given volume. For water, which is the fluid in this exercise, the density is typically given as approximately \(1000 \text{ kg/m}^3\). This means that one cubic meter of water has a mass of 1000 kilograms. Understanding density is crucial in fluid mechanics because it directly affects the pressure calculation. In any fluid pressure scenario, the density will determine how much pressure a specific depth will exert. Knowing that water has a high density helps us calculate how the water's weight increases with depth, thereby increasing pressure.
Acceleration Due to Gravity
Acceleration due to gravity is denoted by \(g\) and is approximately \(9.81 \text{ m/s}^2\) on Earth. This is the acceleration that objects experience when falling freely under the influence of Earth's gravitational field. In the context of fluid pressure, gravity plays a significant role because it is the force pulling the fluid downwards, creating what we feel as pressure. The formula for pressure (\( P = P_0 + \rho g h \)) includes gravity, showing its role in increasing pressure with depth. Thus, whenever you're calculating the pressure at a certain depth in a fluid, always remember that gravity is central to this calculation.
Water Tower Design
Water towers are designed to store water at a height to create sufficient water pressure that can be distributed to homes and businesses. The height of the water in the tower creates potential energy, using gravity to maintain water pressure throughout the distribution system. When designing a water tower, engineers must account for the pressure at different depths. This involves considering the tower's shape and the spacing of reinforcements like metal bands. A well-designed water tower will ensure even, reliable water pressure for the system it supports, while also being structurally sound to handle the pressure exerted by the stored water.
Pressure Calculation
Pressure calculation in fluids involves understanding how different factors like depth, fluid density, and gravity interact. The basic formula is \( P = P_0 + \rho g h \), where you add the atmospheric pressure (\( P_0 \)) to the pressure due to the fluid column above the point you are measuring.
  • \( \rho \) is the density of the fluid.
  • \( g \) is the acceleration due to gravity.
  • \( h \) is the depth of the point within the fluid.
Calculating pressure at a given depth becomes a straightforward task using this formula. It helps determine how much potential force is exerted on any submerged surface. By breaking down this calculation step-by-step, one can predict how structural elements of containment, like the metal bands on a water tower, need to be designed to withstand increasing pressures at varying depths.

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Most popular questions from this chapter

A wooden block with a density of \(710 \mathrm{kg} / \mathrm{m}^{3}\) and a volume of \(0.012 \mathrm{m}^{3}\) is attached to the top of a vertical spring whose force constant is \(k=540 \mathrm{N} / \mathrm{m}\). Find the amount by which the spring is stretched or compressed if it and the wooden block are (a) in air or (b) completely immersed in water. [The density of air may be neglected in part (a).]

On a planet in a different solar system the acceleration of gravity is greater than it is on Earth. If you float in a pool of water on this planet, do you float higher than, lower than, or at the same level as when you float in water on Earth?

Predict/Explain A block of wood has a steel ball glued to one surface. The block can be floated with the ball "high and dry" on its top surface. (a) When the block is inverted, and the ball is immersed in water, does the volume of wood that is submerged increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. When the block is inverted the ball pulls it downward, causing more of the block to be submerged. II. The same amount of mass is supported in either case, therefore the amount of the block that is submerged is the same. III. When the block is inverted the ball experiences a buoyant force, which reduces the buoyant force that must be provided by the wood.

A cylindrical container \(1.0 \mathrm{m}\) tall contains mercury to a certain depth, \(d\). The rest of the cylinder is filled with water. If the pressure at the bottom of the cylinder is two atmospheres, what is the depth \(d ?\)

A geode is a hollow rock with a solid shell and an airfilled interior. Suppose a particular geode weighs twice as much in air as it does when completely submerged in water. If the density of the solid part of the geode is \(2500 \mathrm{kg} / \mathrm{m}^{3}\), what fraction of the geode's volume is hollow?
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