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Chapter 14: Problem 44

A legendary Dutch boy saved Holland by plugging a hole in a dike with his finger, which is \(1.20 \mathrm{cm}\) in diameter. If the hole was \(2.00 \mathrm{m}\) below the surface of the North Sea (density \(\left.1030 \mathrm{kg} / \mathrm{m}^{3}\right),(\mathrm{a})\) what was the force on his finger? (b) If he pulled his finger out of the hole, how long would it take the released water to fill 1 acre of land to a depth of \(1 \mathrm{ft}\), assuming the hole remained constant in sixe? (A typical U.S. family of four uses 1 acre-foot of water, \(1234 \mathrm{m}^{3},\) in 1 year.

Short Answer

Expert verified
a) The force on his finger can be calculated using these steps. b) The time it would take to fill 1 acre-foot of land can also be calculated using the steps outlined above.

Step by step solution

01

Calculate the pressure under water

The pressure under water can be found using the formula \(P = \rho gh\), where \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity and \( h \) is the height of fluid column above the point in question. Substituting \( \rho = 1030 \, \mathrm{kg}/\mathrm{m}^{3}\), \( g = 9.81 \, \mathrm{m}/\mathrm{s}^{2}\) (approximate value of gravity) and \( h = 2 \, \mathrm{m}\) gives \( P = 1030 \, \mathrm{kg}/\mathrm{m}^{3} \times 9.81 \, \mathrm{m}/\mathrm{s}^{2} \times 2 \, \mathrm{m}\).
02

Calculate the force on the boy's finger

Once we have the pressure, the force on the boy's finger is calculated by the formula \(F = PA\), where \( P \) is the pressure and \( A \) is the area. The area, \( A \), of the hole is calculated using the formula for the area of a circle, \( A = \pi r^{2} \), with radius \( r = 0.012 \, \mathrm{m}/2\). Now, we can substitute \( P \) and \( A \) into our force equation to find \( F \).
03

Calculate water outflow speed

The speed of the water outflow can be calculated using Torricelli's theorem, which states that the speed, \( v \), of efflux from a hole in a fluid container is given by \( v = \sqrt{2gh} \) where \( h \) is the height of the fluid above the hole. We substitute \( g = 9.81 \, \mathrm{m}/\mathrm{s}^{2}\) and \( h = 2 \, \mathrm{m}\) to find \( v \).
04

Calculate time to fill an acre of land

The volume of water needed to fill 1 acre of land to a depth of 1 foot is 1234 cubic meters. The time it would take to fill this volume can be calculated by \( t = V/Q \) where \( V \) is the volume and \( Q \) is the volume flow rate. The volume flow rate, \( Q \), is equal to the speed of flow, \( v \), times the area of the hole, \( A \). From the earlier steps, we have obtained all these values and we can substitute into our equation to find \( t \). Note that the time might be in seconds and you might need to convert it to a more suitable unit.

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Key Concepts

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

Understanding Hydrostatic Pressure
Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It's an important concept in fluid dynamics because it affects how fluids behave under different conditions. The pressure increases with depth since there is more fluid above a certain point which has its weight acting downwards due to gravity. To calculate this pressure, we use the formula:
\(P = \rho gh\),
where \( \rho \) is the fluid's density, \( g \) is the acceleration due to gravity, and \( h \) is the depth below the fluid's surface. In our textbook exercise, the calculations for hydrostatic pressure serve as the foundation for determining the force exerted on the Dutch boy's finger by the North Sea's water.
The Basics of Force Calculation
In fluid mechanics, when we need to compute the force exerted by a fluid on a surface, we need two essential pieces of information: the area of that surface and the pressure exerted on it. The formula to calculate force is
\(F = PA\),
where \(F\) is force, \(P\) is pressure, and \(A\) is area. The force exerted by the fluid can be immense, even with a small surface area; this is because fluid pressure can be quite significant at depth due to hydrostatic pressure. Contextually, when the Dutch boy put his finger over the hole, we calculate the force he felt by applying this formula using the area of the hole and the pressure under 2 meters of seawater.
Torricelli's Theorem and Fluid Flow
Torricelli's theorem is a principle that describes the velocity of a fluid flowing out of an orifice under the influence of gravity. It's found on the concept that the velocity (\(v\)) of fluid flowing from an opening is proportional to the square root of the height (\(h\)) of the fluid above the opening, assuming the fluid exits into an area of negligible pressure like the atmosphere. Mathematically, it is expressed as
\(v = \sqrt{2gh}\).
This theorem allows us to predict the speed of water exiting through the hole that the Dutch boy's finger was blocking. Torricelli's theorem helps us further to determine the volume flow rate, which is essential to solve the second part of our problem.
Volume Flow Rate Calculation
Volume flow rate, denoted as \(Q\), is the quantity of three-dimensional fluid passing a point per unit time. It is given by the product of the cross-sectional area of the flow and the velocity of the fluid. We can express this as
\(Q = A \cdot v\),
where \(A\) is the area through which the fluid is flowing and \(v\) is the velocity. In our exercise, we used Torricelli's theorem to calculate the speed at which water would pour out of the dike and then the volume flow rate to determine how long it would take to fill a defined volume of land. It is important to realize how this concept combines the previous concepts of area and velocity to reach a practical conclusion regarding the fluid's behavior over time.

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Most popular questions from this chapter

The Bernoulli effect can have important consequences for the design of buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. As originally constructed, the John Hancock building in Boston popped window panes, which fell many stories to the sidewalk below. (a) Suppose that a horizontal wind blows in streamline flow with a speed of \(11.2 \mathrm{m} / \mathrm{s}\) outside a large pane of plate glass with dimensions \(4.00 \mathrm{m} \times 1.50 \mathrm{m}\) Assume the density of the air to be uniform at \(1.30 \mathrm{kg} / \mathrm{m}^{3} .\) The air inside the building is at atmospheric pressure. What is the total force exerted by air on the window pane? (b) What If? If a second skyscraper is built nearby, the air speed can be especially high where wind passes through the narrow separation between the buildings. Solve part (a) again if the wind speed is \(22.4 \mathrm{m} / \mathrm{s}\), twice as high.

In about 1657 Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemi. spheres. Two teams of eight horses each could pull the hemispheres apart only on some trials, and then "with greatest difficulty," with the resulting sound likened to a cannon firing (Fig. P14.62). (a) Show that the force Frequired to pull the evacuated hemispheres apart is \(\pi R^{2}\left(P_{0}-P\right),\) where \(R\) is the radius of the hemispheres and \(P\) is the pressure inside the hemispheres, which is much less than \(P_{0}\). (b) Determine the force if \(P=0.100 P_{0}\) and \(R=0.300 \mathrm{m}\)

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 horizontal pipe \(10.0 \mathrm{cm}\) in diameter has a smooth reduction to a pipe \(5.00 \mathrm{cm}\) in diameter. If the pressure of the water in the larger pipe is \(8.00 \times 10^{4} \mathrm{Pa}\) and the pressure in the smaller pipe is \(6.00 \times 10^{4} \mathrm{Pa},\) at what rate does water flow through the pipes?

\- 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.
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