How Does Torricelli’s Law Describe Fluid Outflow in Physics?
The law of Torricelli is also called Torricelli's theorem. This is a theorem in fluid dynamics which is simply relating the speed of fluid that is flowing from an orifice to the height of fluid above the opening.
The law states that the speed denoted as v that is of efflux of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth ‘h’ is the same as the speed that a body that is in this case a drop of water would acquire in freely falling from a height that is h which is written as:
V = \[\sqrt{2gh}\]
where the letter g is the acceleration which is due to gravity that is 9.81 m/s2 near the Earth surface.
On this page, we are going to learn more about the law.
Torricelli's Theorem
The theorem of Torricelli is also known as Torricelli’s principle or the equation of Torricelli equation. That statement is the speed v of a liquid that is flowing under the force of gravity out of an opening in a tank is proportional jointly to the square root of the vertical distance. The distance denoted by h which is between the liquid surface and the centre of the opening and to the square root of twice the acceleration caused by gravity that is 2g or simply v = 2gh1/2.
(The value which is of the acceleration caused by gravity at the surface of Earth is about 32.2 feet per second per second or 9.8 metres per second per second. The theorem is said to be named after Evangelista Torricelli who discovered it in 1643).
The speed of water which is flowing through an opening in a tank at a given distance denoted by ‘h’ is the same as the speed that would be attained by a drop of water falling freely under the force of gravity alone.' This is considered neglecting effects of air which is through the same distance as h.
The speed of efflux is said to be independent of the flow of the direction. At the point of the opening, the speed is given by this equation irrespective of whether the opening is directed upward or downward or even horizontally.
Torricelli Experiment
In the early 1600s, Sir Galileo argued that a pump of suction was able to draw water from a well because of the "force of vacuum" which is inside the pump. Then after Galileo's death, there was an Italian Mathematician and Physicist Evangelista Torricelli 1608-1647 who proposed another explanation. He suggested that the air which is in our atmosphere has weight and that the force that is of the atmosphere pushing down on the surface of the water usually drives the water into the suction pump when it is evacuated.
In the experiment conducted, some but not all of the mercury was drained out of the glass tube and it slipped into the dish. The explanation of Torricelli was that by assuming that mercury drains from the glass tube until the force of the column of mercury pushing down on the inside of the tube exactly balances the force of the atmosphere which is pushing down on the liquid surface from outside of the tube.
Torricelli predicted that the height of the column of mercury would generally change from day to day as the pressure of the atmosphere changed. Today his apparatus is called a barometer derived from the Greek “baros” that means "weight," because it literally measures the weight of the atmosphere. Thus, we can say that a standard unit of pressure called the atmosphere pressure was defined as follows.
1 atm = 760 mmHg
1 torr = 1 mmHg
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On a sunny day, at sea level, there is the weight of a 760-mm column of mercury which is inside a glass tube that generally balances the weight of the atmosphere that is generally pushing down on the pool of mercury that surrounds the tube as well. Therefore, we can say that the pressure of the atmosphere is said to be equivalent to 760 mm Hg.
Torricelli Equation
In terms of Physics, the equation of Torricelli's or the formula of Torricelli's is an equation created by Evangelista Torricelli. The equation itself is:
Vfx2 = Vix2 + 2axΔx
The term Vfx is the object's velocity that is final along the x-axis on which the acceleration is constant.
The term Vix is the object's velocity which is the initial along the x-axis.
The term ax is the object's acceleration which is along the x-axis, given as a constant.
The term Δx is the change of the object's position along the x-axis, also known as displacement.
This equation is said to be valid along any axis on which the acceleration is constant.
FAQs on Torricelli’s Law Explained: Formula, Experiment & Examples
1. What is Torricelli's Law in simple terms?
Torricelli's Law states that the speed at which a fluid flows out of a small opening (orifice) in a container is equal to the speed that an object would acquire if it were to fall freely from the surface of the fluid to the opening. This speed is known as the velocity of efflux. In essence, the greater the height of the fluid above the hole, the faster the fluid will exit.
2. What is the formula for Torricelli's Law and what do its variables represent?
The formula used to calculate the velocity of efflux according to Torricelli's Law is: v = √(2gh).
In this equation:
- v represents the velocity of the fluid exiting the orifice.
- g is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
- h is the vertical distance (height) from the centre of the orifice to the surface of the fluid in the container.
3. What is a common example that demonstrates Torricelli's Law?
A classic example is the 'spouting can' experiment. If you take a tall container, like a plastic bottle, and poke holes at different heights, you will observe that the water stream from the lowest hole travels the farthest and fastest. This is because the lowest hole has the greatest height 'h' of water above it, leading to the highest exit velocity as predicted by Torricelli's Law.
4. How is Torricelli's Law derived from Bernoulli's principle?
Torricelli's Law is a special case of Bernoulli's principle. By applying Bernoulli's equation to the fluid at the top surface of the container and at the exit orifice, we can derive the law. We assume the pressure at both points is atmospheric pressure and that the fluid velocity at the large top surface is negligible compared to the exit velocity. This simplification of the Bernoulli equation for an ideal fluid under these conditions results directly in the formula v = √(2gh).
5. What are the ideal conditions and assumptions for Torricelli's Law to be accurate?
For Torricelli's Law to hold true, several ideal conditions are assumed. In real-world scenarios, these factors can cause slight deviations. The key assumptions are:
- The fluid is incompressible and non-viscous (an ideal fluid).
- The opening or orifice is very small compared to the cross-sectional area of the container.
- There is no energy loss due to friction at the orifice.
- The flow is steady and irrotational (laminar flow).
6. Why does a liquid flow out faster from a hole at the bottom of a tank than one near the top?
This happens because of the relationship between pressure and depth. The pressure within a fluid increases with depth due to the weight of the fluid above. A hole at the bottom of a tank experiences much greater pressure than a hole near the top. This greater pressure exerts a larger force on the fluid at the opening, pushing it out at a higher velocity. This is a direct consequence of converting more potential energy (due to greater height 'h') into kinetic energy.
7. What is the difference between Torricelli's law of efflux and his famous barometer experiment?
These are two different principles discovered by Evangelista Torricelli. Torricelli's Law of Efflux, as explained here, deals with fluid dynamics and describes the speed of a fluid flowing from an orifice. In contrast, his barometer experiment involved a tube of mercury inverted in a dish. It demonstrated that atmospheric pressure could support a column of mercury, proving that a vacuum could exist and providing the first measurement of air pressure.
