Torricelli’s Law: Definition, Formula and Daily Life Examples 

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Torricelli’s law, also known as Torricelli’s principle or Torricelli’s theorem, states that the speed of fluid flowing out of an orifice in a tank under gravity is proportional to the square root of the vertical distance between the liquid surface and the centre of the orifice and to the square root of twice the gravity acceleration (g = 9.81 N/kg near the earth’s surface).

Torricelli’s Law in Simple Words

In simple words, Torricelli’s law is a physics law that explains the relationship between the speed of a fluid flowing out of an opening in a container and the height of the fluid in the container. This law was named after Evangelista Torricelli, an Italian physicist who discovered it in the 17th century.

The law states that the faster a fluid flows out of an opening in a container, the higher the fluid level above the opening. This means that the velocity of the fluid coming out of the opening is directly proportional to the square root of the height of the fluid in the container.

Daily Life Examples of Torricelli’s Law

Here are some examples of Torricelli’s law in daily life:

1. Filling a water bottle – the water flows out faster when the bottle is tipped, due to the increase in height and resulting increase in velocity.
2. Emptying a bathtub – the water drains out faster when the drain is open, due to the increased velocity caused by the height of the water above the drain.
3. Sprinklers – the velocity of water flowing through a sprinkler nozzle is determined by the height of the water above the nozzle.
4. Garden hose – the water flows out faster when the hose is held lower, due to the increased velocity caused by the height of the water in the hose.
5. Hydraulic systems – the velocity of fluid flow in hydraulic systems is determined by the height of the fluid column in the system.

The Formula for Torricelli’s law

The formula for Torricelli’s law is:

v = sqrt(2gh)

where “v” is the velocity of the fluid coming out of the opening, “g” is the acceleration due to gravity, and “h” is the height of the fluid above the opening. This formula assumes that the fluid is not compressed, has no friction, and that the opening is small relative to the size of the container. Also, it assumes that the fluid is only affected by the gravity of the Earth.

• “v”: Velocity of the fluid coming out of the opening
• “g”: Acceleration due to gravity
• “h”: Height of the fluid above the opening
• The formula assumes that the fluid is not compressed, has no friction, and that the opening is small relative to the size of the container.
• The formula also assumes that the fluid is only affected by the gravity of the Earth.

Torricelli’s Law is a particular case of Bernoulli’s principle.

Derivation of Torricelli’s law

1. Consider a container filled with a fluid of height “h” above an opening at the bottom.
2. Assume that the fluid is incompressible and that the opening is small relative to the size of the container.
3. Let us calculate the speed at which the fluid is flowing out of the opening. We can do this by applying the principle of conservation of energy. The potential energy of the fluid at height “h” above the opening is given by mgh, where “m” is the mass of the fluid, “g” is the acceleration due to gravity, and “h” is the height of the fluid. As the fluid flows out of the opening, it gains kinetic energy. The kinetic energy of the fluid is given by (1/2)mv^2, where “v” is the velocity of the fluid coming out of the opening, and “m” is the mass of the fluid. Therefore, the conservation of energy equation can be written as:

mgh = (1/2)mv²

4. We can simplify this equation by canceling out the mass “m” on both sides of the equation.

gh = (1/2)v²

5. Rearranging this equation to solve for “v”, we get:

v = sqrt(2gh)

This is the formula for Torricelli’s Law, which states that the velocity of fluid flowing out of a small opening in a container is proportional to the square root of the height of the fluid above the opening.

Important Points

	Torricelli’s Theorem
Definition	Torricelli’s theorem describes the relationship between the speed of a fluid flowing out of an opening in a container and the height of the fluid in the container.
Formula	The formula for Torricelli’s theorem is v = sqrt(2gh), where “v” is the velocity of the fluid coming out of the opening, “g” is the acceleration due to gravity, and “h” is the height of the fluid above the opening.
Assumptions	The fluid is incompressible. The fluid is non-viscous. The opening through which the fluid is flowing is small relative to the size of the container. The fluid is only flowing under the influence of gravity.
Applications	In fluid mechanics, Torricelli’s theorem is utilized to determine the velocity of fluids as they flow through pipes and channels. In engineering, Torricelli’s theorem can be applied to calculate the flow rate of fluids in different systems. Physicists often study the motion of fluids under the influence of gravity using Torricelli’s theorem. The principles of Torricelli’s theorem can be used to design and analyze fluid-based systems, including hydraulic and irrigation systems.

Solved Numerical Problems

Problem 1:

A water tank is filled to a height of 10 meters above an opening. Find the velocity of water coming out of the opening.

Solution:

Given:

• Height of water in the tank (h) = 10 m
• Acceleration due to gravity (g) = 9.8 m/s^2

Using Torricelli’s theorem, we can find the velocity of water coming out of the opening:

v = sqrt(2gh)

v = sqrt(2 x 9.8 x 10)

v = sqrt(196)

v = 14 m/s

Therefore, the velocity of water coming out of the opening is 14 m/s.

Problem 2:

A tank is filled with oil to a height of 6 meters above an opening. Find the time taken for the oil to flow out of the opening.

Solution:

Given:

• Height of oil in the tank (h) = 6 m
• Acceleration due to gravity (g) = 9.8 m/s^2
• Velocity of oil coming out of the opening (v) = sqrt(2gh)

Using the equation of motion, we can find the time taken for the oil to flow out of the opening:

h = (1/2)gt^2

6 = (1/2) x 9.8 x t^2

t^2 = (2 x 6) / 9.8

t^2 = 1.22

t = sqrt(1.22)

t = 1.1 s (approx.)

Therefore, the time taken for the oil to flow out of the opening is approximately 1.1 seconds.

Summary

• Torricelli’s theorem relates the velocity of a fluid flowing out of an opening in a container to the height of the fluid above the opening.
• The theorem assumes that the fluid is incompressible and non-viscous, the opening is small relative to the size of the container, and the fluid is only affected by the gravity of the Earth.
• It has various applications in fluid mechanics, engineering, and physics, and is used to calculate the velocity and flow rate of fluids in different systems.
• It is also used in the design and analysis of fluid-based systems such as hydraulic and irrigation systems.
• The theorem is related to Bernoulli’s principle and has similarities with the Venturi effect.
• The velocity of the fluid can be calculated using the formula v = sqrt(2gh), where v is the velocity of the fluid, g is the acceleration due to gravity, and h is the height of the fluid above the opening.

Multiple Choice Questions

1. What does Torricelli’s theorem relate? A) Pressure and velocity of a fluid B) Volume and temperature of a fluid C) Velocity and flow rate of a fluid D) Density and viscosity of a fluid

Answer: C) Velocity and flow rate of a fluid

2. What does the formula v = sqrt(2gh) represent in Torricelli’s theorem? A) Pressure of the fluid B) Volume of the fluid C) Velocity of the fluid D) Temperature of the fluid

Answer: C) Velocity of the fluid

3. What assumptions are made in Torricelli’s theorem? A) The fluid is compressible and viscous B) The opening is large relative to the size of the container C) The fluid is not affected by gravity D) The fluid is incompressible and non-viscous, and the opening is small relative to the size of the container

Answer: D) The fluid is incompressible and non-viscous, and the opening is small relative to the size of the container

4. In what fields is Torricelli’s theorem used? A) Mathematics and geology B) Chemistry and biology C) Fluid mechanics, engineering, and physics D) Astronomy and astrophysics

Answer: C) Fluid mechanics, engineering, and physics

5. What is the relationship between Torricelli’s theorem and Bernoulli’s principle? A) There is no relationship between them B) They are equivalent theories C) Bernoulli’s principle is a special case of Torricelli’s theorem D) Torricelli’s theorem is a special case of Bernoulli’s principle

Answer: C) Bernoulli’s principle is a special case of Torricelli’s theorem.

Practice Exam Questions

1. A tank of water is open to the atmosphere at the top and has an opening near the bottom. If the height of the water in the tank is 5 meters, what is the velocity of the water coming out of the opening? (Hint: Use the formula v = sqrt(2gh), where g = 9.8 m/s^2)
2. A water tank has a hole in the bottom of it that is 4 centimeters in diameter. The height of the water in the tank is 2 meters. What is the flow rate of the water coming out of the hole? (Hint: Use the formula Q = Av, where A is the area of the hole and v is the velocity of the water coming out of the hole)
3. A cylindrical tank with a height of 6 meters and a radius of 2 meters is filled with water to a height of 4 meters. What is the velocity of the water coming out of a hole located at the bottom of the tank? (Hint: Use the formula v = sqrt(2gh), where h is the height of the water above the hole)
4. A pipe with a diameter of 5 centimeters is connected to a water tank that has a height of 10 meters. If the water level in the tank is at a height of 8 meters, what is the flow rate of the water coming out of the pipe? (Hint: Use the formula Q = Av, where A is the area of the pipe and v is the velocity of the water coming out of the pipe, which can be calculated using v = sqrt(2gh))
5. A water tank has an opening in the bottom that is 2 meters below the surface of the water. If the height of the water in the tank is 5 meters, what is the velocity of the water coming out of the opening? (Hint: Use the formula v = sqrt(2gh), where h is the height of the water above the opening)

Conclusion

Torricelli’s law, discussed on this post, explains a formula used to calculate the velocity of fluids flowing through pipes and channels. This law assumes that the fluid is non-viscous, incompressible, and flowing under the influence of gravity only. The article highlights the various fields in which this law is used, such as fluid mechanics, engineering, and physics. Additionally, the article provides solved numerical problems and multiple-choice questions to help readers test their understanding of Torricelli’s law.

About the Author

Umair Javaid is a materials science Masters degree holder who has been contributing to whatinsight.org since 2020. He is passionate about science, particularly physics, chemistry, and engineering, and aims to provide accessible and insightful articles to readers.

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