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Chapter 13: Problem 47

A small circular hole 6.00 \(\mathrm{mm}\) in diameter is cut in the side of a large water tank, 14.0 \(\mathrm{m}\) below the water level in the tank. The top of the tank is open to the air. Find the speed at which the water shoots out of the tank.

Short Answer

Expert verified
The water exits the tank at approximately 16.57 m/s.

Step by step solution

01

Understand the Concept

The problem is about fluid flow through an orifice, and we can apply Torricelli's theorem to find the exit speed of the water. The theorem states that the speed \( v \) at which a fluid exits a hole under the influence of gravity alone is given by \( v = \sqrt{2gh} \), where \( g \) is the acceleration due to gravity, and \( h \) is the height of the fluid above the hole.
02

Identify the Known Values

From the problem, we know: \( h = 14.0 \, \text{m} \) is the water height above the hole, and the acceleration due to gravity \( g = 9.81 \, \text{m/s}^2 \).
03

Apply Torricelli's Theorem

Use the formula \( v = \sqrt{2gh} \). Substitute the known values: \( g = 9.81 \, \text{m/s}^2 \) and \( h = 14.0 \, \text{m} \). Compute the expression to find \( v \).
04

Calculate the Exit Speed

Substitute the numbers into the formula: \[ v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 14.0 \, \text{m}} \] Calculate to find \( v = \sqrt{274.68} \approx 16.57 \, \text{m/s} \).
05

Conclusion

Using Torricelli's theorem and calculations, the speed of the water as it shoots out of the tank is approximately \( 16.57 \, \text{m/s} \).

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Key Concepts

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

Fluid Flow
Fluid flow refers to the movement of liquid, in this case, water, often driven by forces such as gravity. It's a key concept in fluid mechanics and can help us understand how water behaves as it moves through different environments.

When a liquid flows from one area to another, a variety of factors like pressure differences, gravity, and channel shape affect its flow rate and velocity. In our example of the water tank, the flow results from the gravitational force acting on the water above and through the opening of the tank.
  • Gravity accelerates the water as it descends.
  • Pressure differences between the water inside the tank and the air outside cause the water to exit through the orifice.
Orifice
An orifice is simply a small opening or hole through which fluid can flow. In the context of our exercise, the orifice is the 6.00 mm diameter hole in the water tank.

Orifices are important in fluid mechanics because they control the flow of fluid and affect how fast the fluid exits a container. The speed at which the fluid exits depends not just on the size of the hole, but also on the pressure and height of the liquid above it.

In practical terms, when water flows out of an orifice:
  • The size of the orifice influences the flow rate.
  • The pressure from the fluid above pushes the liquid through the orifice.
Acceleration Due to Gravity
Acceleration due to gravity, denoted as \( g \), is a fundamental constant that affects all objects in free fall on Earth. In our calculations, it measures at approximately 9.81 m/s虏. This value is crucial in Torricelli's theorem, which is used to determine the speed of water exiting the orifice.

Without gravity, the water wouldn't exit the tank at the speed calculated. Torricelli's theorem directly ties to gravity as it dictates the kinetic energy that a fluid gains as it falls from a height.
  • Gravity provides the force needed for water to accelerate as it moves out of the tank.
  • This force and the height of the water column give rise to the speed found using Torricelli's theorem.
Fluid Mechanics
Fluid mechanics is the study of how fluids (liquids and gases) behave under various conditions. It combines the principles of physics to understand everything from the simple flow of water in a pipe to the chaotic atmospheric patterns of Earth.

In the context of the exercise, fluid mechanics helps explain why water behaves the way it does when exiting the orifice.
  • It considers forces like gravity, pressure, and friction.
  • It uses equations of motion and energy conservation to predict fluid behavior.
Understanding fluid mechanics is vital for engineers and scientists to design systems like pipelines, dams, and even weather models.

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Most popular questions from this chapter

By how many newtons do you increase the weight of your car when you fill up your 11.5 gal gas tank with gasoline? A gallon is equal to 3.788 \(\mathrm{L}\) and the density of gasoline is 737 \(\mathrm{kg} / \mathrm{m}^{3}\) .

What gauge pressure must a pump produce to pump water from the bottom of the Grand Canyon (elevation 730 \(\mathrm{m} )\) to Indian Gardens (elevation 1370 \(\mathrm{m} ) ?\) Express your result in pascals and in atmospheres.

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There is a maximum depth at which a diver can breathe through a snorkel tube (Fig. 13.42\()\) , because as the depth increases, so does the pressure difference, which the difference, if any.tends to collapse the diver's lungs. since the snorkel connects the air in the lungs to the atmosphere at the surface, the pressure inside the lungs is atmospheric pressure. What is the external-internal pressure difference when the diver's lungs are at a depth of 6.1 \(\mathrm{m}\) (about 20 \(\mathrm{ft}\) ? Assume that the diver is in fresh- water. (A scuba diver breathing from compressed air tanks can operate at greater depths than can a snorkeler, since the pressure of the air inside the scuba diver's lungs increases to match the external pressure of the water.)

Blood. (a) Mass of blood. The human body typically contains 5 L of blood of density 1060 \(\mathrm{kg} / \mathrm{m}^{3} .\) How many kilograms of blood are in the body? (b) The average blood pressure is \(13,000 \mathrm{Pa}\) at the heart. What average force does the blood exert on each square centimeter of the heart? (c) Red blood cells. Red blood cells have a specific gravity of 5.0 and a diameter of about 7.5\(\mu \mathrm{m}\) . If they are spherical in shape (which is not quite true), what is the mass of such a cell?
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