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Chapter 11: Problem 66

A small crack occurs at the base of a \(15.0-\mathrm{m}\) -high dam. The effective crack area through which water leaves is \(1.30 \times 10^{-3} \mathrm{m}^{2}\) (a) Ignoring viscous losses, what is the speed of water flowing through the crack? (b) How many cubic meters of water per second leave the dam?

Short Answer

Expert verified
(a) The speed of water flow is approximately 17.15 m/s. (b) The flow rate is approximately 0.0223 m鲁/s.

Step by step solution

01

Understand Torricelli's Theorem

Torricelli's Theorem states that the speed of efflux of a fluid under gravity from a hole at height \( h \) is given by the formula \( v = \sqrt{2gh} \), where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
02

Apply the Formula for Speed

Use Torricelli鈥檚 theorem to find the speed \( v \): \( v = \sqrt{2gh} \). Substitute \( g = 9.81 \, \text{m/s}^2 \) and \( h = 15.0 \, \text{m} \). Calculate:\[ v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 15.0 \, \text{m}} \approx \sqrt{294.3} \approx 17.15 \, \text{m/s}. \]
03

Calculate Volume Flow Rate

The volume flow rate \( Q \) is given by \( Q = A \times v \), where \( A \) is the area of the crack. Given \( A = 1.30 \times 10^{-3} \, \text{m}^2 \) and \( v = 17.15 \, \text{m/s} \), calculate \( Q \):\[ Q = 1.30 \times 10^{-3} \, \text{m}^2 \times 17.15 \, \text{m/s} \approx 2.23 \times 10^{-2} \, \text{m}^3/ ext{s}. \]

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Key Concepts

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

Torricelli's Theorem
Torricelli's Theorem offers a fascinating insight into fluid dynamics. Developed by Evangelista Torricelli, it allows us to calculate the speed at which a fluid will exit a hole due to gravity. The theorem is summarized by the formula \( v = \sqrt{2gh} \), where \( v \) represents the efflux speed, \( g \) is the acceleration due to gravity (about \( 9.81 \, \text{m/s}^2 \)), and \( h \) denotes the height of the fluid above the opening.
This theorem is particularly useful for determining how fast a liquid will flow from a reservoir, such as our exercise involving a dam.
  • It assumes the liquid is non-viscous, meaning it neglects internal friction and other losses.
  • The theorem applies under ideal fluid conditions, which means the only force is gravity and the liquid surface is open to the atmosphere.
On solving, Torricelli's Theorem helps find the speed at which water exits through a crack in a dam.
Efflux Speed
Efflux speed refers to the velocity at which a fluid exits an opening due to the influence of gravity. This concept is crucial in scenarios where a fluid flows freely out of a container, such as a dam or a tank.
The efflux speed can be easily determined using Torricelli鈥檚 Theorem. In the problem given, the speed of the water flowing from the base of a dam is computed using the equation \( v = \sqrt{2gh} \).
  • Substitute \( g = 9.81 \, \text{m/s}^2 \) as the value for gravity.
  • Use \( h = 15.0 \, \text{m} \), the height of the water above the crack.
Calculating gives us \( v \approx 17.15 \, \text{m/s} \), demonstrating how quickly water escapes under these conditions.
Volume Flow Rate
The volume flow rate is the quantity of fluid that flows through a cross-section per unit time, often measured in cubic meters per second \((\text{m}^3/\text{s})\). This is an essential concept when analyzing systems involving fluid transfer.
To compute the volume flow rate, you multiply the efflux speed by the area of the opening. In our scenario, this is \( Q = A \times v \).
  • Given \( A = 1.30 \times 10^{-3} \, \text{m}^2 \), which is the area of the crack.
  • The speed \( v \) is calculated as \( 17.15 \, \text{m/s} \).
The resulting volume flow rate is \( Q \approx 2.23 \times 10^{-2} \, \text{m}^3/\text{s} \), indicating the rate at which water exits the dam through the crack.
Hydrodynamics
Hydrodynamics is the branch of physics that deals with the motion of fluids, particularly liquids. It encompasses various principles and laws that explain how fluids behave when in motion, like the water exiting a crack in a dam.
Key factors influencing hydrodynamics include:
  • The fluid鈥檚 speed and direction of flow.
  • The geometrical constraints of the system, such as the area of an opening.
  • Environmental interactions like gravity, shown in Torricelli鈥檚 Theorem.
In this context, hydrodynamics helps understand how factors such as pressure differences, gravity, and flow areas impact the movement of water through a structure. Mastery of these concepts allows engineers and scientists to design better hydraulic structures and systems for efficient water management.

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Most popular questions from this chapter

In the process of changing a flat tire, a motorist uses a hydraulic jack. She begins by applying a force of 45 N to the input piston, which has a radius \(r_{1} .\) As a result, the output plunger, which has a radius \(r_{2},\) applies a force to the car. The ratio \(r_{2} / r_{1}\) has a value of \(8.3 .\) Ignore the height difference between the input piston and output plunger and determine the force that the output plunger applies to the car.

(a) The volume flow rate in an artery supplying the brain is \(3.6 \times 10^{-6} \mathrm{m}^{3} / \mathrm{s} .\) If the radius of the artery is \(5.2 \mathrm{mm},\) determine the average blood speed. (b) Find the average blood speed at a constriction in the artery if the constriction reduces the radius by a factor of 3. Assume that the volume flow rate is the same as that in part (a).

A spring is attached to the bottom of an empty swimming pool, with the axis of the spring oriented vertically. An 8.00-kg block of wood \(\left(\rho=840 \mathrm{kg} / \mathrm{m}^{3}\right)\) is fixed to the top of the spring and compresses it. Then the pool is filled with water, completely covering the block. The spring is now observed to be stretched twice as much as it had been compressed. Determine the percentage of the block's total volume that is hollow. Ignore any air in the hollow space.

(a) The mass and the radius of the sun are, respectively, \(1.99 \times 10^{30} \mathrm{kg}\) and \(6.96 \times 10^{8} \mathrm{m}\) . What is its density? (b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why? (c) Would a solid object sink or float in water if were made from a material whose density was the same as that of the planet Saturn (mass \(=5.7 \times 10^{26} \mathrm{kg},\) radius \(=6.0 \times 10^{7} \mathrm{m} ) ?\) Provide a reason for your answer.

When an object moves through a fluid, the fluid exerts a viscous force \(\overrightarrow{\mathbf{F}}\) on the object that tends to slow it down. For a small sphere of radius \(R\) moving slowly with a speed \(v\) , the magnitude of the viscous force is given by Stokes' law, \(F=6 \pi \eta R v,\) where \(\eta\) is the viscosity of the fluid. (a) What is the viscous force on a sphere of radius \(R=5.0 \times 10^{-4} \mathrm{m}\) that is falling through water \(\left(\eta=1.00 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s}\right)\) when the sphere has a speed of 3.0 \(\mathrm{m} / \mathrm{s} ? \quad\) (b) The speed of the falling sphere increases until the viscous force balances the weight of the sphere. Thereafter, no net force acts on the sphere, and it falls with a constant speed called the "terminal speed If the sphere has a mass of \(1.0 \times 10^{-5} \mathrm{kg}\) , what is its terminal speed?
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