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Chapter 15: Problem 64

A water tank springs a leak. Find the speed of water emerging from the hole if the leak is \(2.7 \mathrm{m}\) below the surface of the water, which is open to the atmosphere.

Short Answer

Expert verified
The speed of water emerging is approximately 7.26 m/s.

Step by step solution

01

Identify the Relevant Principle

The principle involved in solving this problem is Torricelli's theorem, which relates the speed of fluid flowing out of an orifice to the height of the liquid column above the opening. According to Torricelli's theorem, the speed of efflux, \( v \), is given by \( v = \sqrt{2gh} \) where \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \), and \( h \) is the height of the fluid column above the hole.
02

Substitute Known Values

We substitute the given values into the equation from Torricelli's theorem. The height \( h \) is given as \( 2.7 \text{ m} \). Thus, the equation becomes: \[ v = \sqrt{2 \cdot 9.81 \text{ m/s}^2 \cdot 2.7 \text{ m}} \]
03

Calculate Speed of Efflux

First, calculate the value inside the square root:\[ 2 \cdot 9.81 \cdot 2.7 = 52.686 \]Now, take the square root:\[ v = \sqrt{52.686} \approx 7.26 \text{ m/s} \]
04

Review Assumptions

Ensure that the assumptions hold true: the hole is small compared to the tank size, and the flow is in a region where the external pressure is atmospheric. These conditions allow the use of Torricelli's theorem as derived.

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Key Concepts

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

Speed of Efflux
The concept of "speed of efflux" refers to the speed at which a fluid exits an opening. In our example, water is flowing out of a leak in a tank, and we use Torricelli's theorem to determine this speed. This theorem provides a straightforward way to calculate efflux speed, illustrating how potential energy due to height converts into kinetic energy of the fluid exiting the tank. The formula, derived from the energy conservation principle, is given by: \[ v = \sqrt{2gh} \]
where:
  • \(v\) is the speed of efflux (m/s)
  • \(g\) is the acceleration due to gravity (\(9.81 \text{ m/s}^2\))
  • \(h\) is the height of the fluid column above the hole (m)
By applying this formula, we can derive the speed at which water exits the tank, given the height from which it's falling. Knowing the height is crucial since it determines the potential energy contributing to the efflux speed.
Fluid Dynamics
Fluid dynamics is the study of fluids (liquids and gases) in motion. It's an intricate field involving several complex topics, but one of its fundamental principles is the conservation of energy. Torricelli's theorem is a classic application of this principle, specifically ideal for understanding fluid flow out of small openings. This ideal-world scenario assumes ideal flow conditions:
  • No viscosity or friction
  • Constant atmospheric pressure at the opening
  • Incompressible and steady flow
These assumptions simplify the real-world complexities, allowing us to predict fluid behavior using basic principles. Therefore, Torricelli's relevant formula helps derive practical solutions while studying systems like water tanks or dams, where the speed of fluid discharge is essential. Fluid dynamics aids engineers and scientists in designing efficient systems, ensuring safety and functionality.
Acceleration due to Gravity
The acceleration due to gravity is a familiar concept in physics, crucial for understanding many phenomena, particularly in fluid dynamics. It refers to the constant acceleration experienced by an object due to gravitational force, usually measured as \(9.81 \text{ m/s}^2\) on Earth. This constant plays a key role in Torricelli's theorem, determining how the potential energy due to fluid height converts into kinetic energy.
In our tank example, gravity aids the water to accelerate as it falls through the leak:
  • The greater the height \(h\), the higher the potential energy
  • This energy converts to kinetic energy as water exits, manifested as efflux speed
Thus, knowing the precise acceleration due to gravity allows us to accurately calculate and predict this efflux speed, making it pivotal in many physics and engineering calculations.

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Most popular questions from this chapter

BIO Vasodilation When the body requires an increased blood flow rate in a particular organ or muscle, it can accomplish this by increasing the diameter of arterioles in that area. This is referred to as vasodilation. What percentage increase in the diameter of an arteriole is required to double the volume flow rate of blood, all other factors remaining the same?

A tin can is filled with water to a depth of \(39 \mathrm{cm} .\) A hole \(11 \mathrm{cm}\) above the bottom of the can produces a stream of water that is directed at an angle of \(36^{\circ}\) above the horizontal. Find (a) the range and (b) the maximum height of this stream of water.

Water flows in a cylindrical, horizontal pipe. As the pipe narrows to half its initial diameter, the pressure in the pipe changes. (a) Is the pressure in the narrow region greater than, less than, or the same as the initial pressure? Explain. (b) \(\mathrm{Cal}-\) culate the change in pressure between the wide and narrow regions of the pipe. Give your answer symbolically in terms of the density of the water, \(\rho,\) and its initial speed \(v\).

BIO Blood Speed in an Arteriole A typical arteriole has a diameter of \(0.030 \mathrm{mm}\) and carries blood at the rate of \(5.5 \times 10^{-6} \mathrm{cm}^{3} / \mathrm{s} .\) (a) What is the speed of the blood in an arteriole? (b) Suppose an arteriole branches into 340 capillaries, each with a diameter of \(4.0 \times 10^{-6} \mathrm{m} .\) What is the blood speed in the capillaries? (The low speed in capillaries is beneficial; it promotes the diffusion of materials to and from the blood.)

IP A block of wood floats on water. A layer of oil is now poured on top of the water to a depth that more than covers the block, as shown in Figure \(15-31\). (a) Is the volume of wood submerged in water greater than, less than, or the same as before? (b) If \(90 \%\) of the wood is submerged in water before the oil is added, find the fraction submerged when oil with a density of \(875 \mathrm{kg} / \mathrm{m}^{3}\) covers the block.
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