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Bernoulli’s principle

Suppose a fluid is moving but its depth is constant—that is, h 1 = h 2 . Under this condition, Bernoulli’s equation becomes

p 1 + 1 2 ρ v 1 2 = p 2 + 1 2 ρ v 2 2 .

Situations in which fluid flows at a constant depth are so common that this equation is often also called Bernoulli’s principle    , which is simply Bernoulli’s equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) Bernoulli’s principle reinforces the fact that pressure drops as speed increases in a moving fluid: If v 2 is greater than v 1 in the equation, then p 2 must be less than p 1 for the equality to hold.

Calculating pressure

In [link] , we found that the speed of water in a hose increased from 1.96 m/s to 25.5 m/s going from the hose to the nozzle. Calculate the pressure in the hose, given that the absolute pressure in the nozzle is 1.01 × 10 5 N/m 2 (atmospheric, as it must be) and assuming level, frictionless flow.


Level flow means constant depth, so Bernoulli’s principle applies. We use the subscript 1 for values in the hose and 2 for those in the nozzle. We are thus asked to find p 1 .


Solving Bernoulli’s principle for p 1 yields

p 1 = p 2 + 1 2 ρ v 2 2 1 2 ρ v 1 2 = p 2 + 1 2 ρ ( v 2 2 v 1 2 ) .

Substituting known values,

p 1 = 1.01 × 10 5 N/m 2 + 1 2 ( 10 3 kg/m 3 ) [ (25.5 m/s) 2 (1.96 m/s) 2 ] = 4.24 × 10 5 N/m 2 .


This absolute pressure in the hose is greater than in the nozzle, as expected, since v is greater in the nozzle. The pressure p 2 in the nozzle must be atmospheric, because the water emerges into the atmosphere without other changes in conditions.

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Applications of bernoulli’s principle

Many devices and situations occur in which fluid flows at a constant height and thus can be analyzed with Bernoulli’s principle.


People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment . Entrainment devices have been in use since ancient times as pumps to raise water to small heights, as is necessary for draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in [link] .

Figure A is a drawing of a Bunsen burner: air and natural gas enter at the bottom and move upwards. Figure B is a drawing of an atomizer: a squeeze bulb creates a horizontally flowing jet of air that pulls moving drops of perfume upwards and out of the bottle. Figure C is a drawing of a common aspirator that has water moving from the top to the bottom and combines with the flow of air that enters from a side. Figure D is a drawing of a chimney in which a flow of hot air moves upwards combining with the cool air entering from the side.
Entrainment devices use increased fluid speed to create low pressures, which then entrain one fluid into another. (a) A Bunsen burner uses an adjustable gas nozzle, entraining air for proper combustion. (b) An atomizer uses a squeeze bulb to create a jet of air that entrains drops of perfume. Paint sprayers and carburetors use very similar techniques to move their respective liquids. (c) A common aspirator uses a high-speed stream of water to create a region of lower pressure. Aspirators may be used as suction pumps in dental and surgical situations or for draining a flooded basement or producing a reduced pressure in a vessel. (d) The chimney of a water heater is designed to entrain air into the pipe leading through the ceiling.

Velocity measurement

[link] shows two devices that apply Bernoulli’s principle to measure fluid velocity. The manometer in part (a) is connected to two tubes that are small enough not to appreciably disturb the flow. The tube facing the oncoming fluid creates a dead spot having zero velocity ( v 1 = 0 ) in front of it, while fluid passing the other tube has velocity v 2 . This means that Bernoulli’s principle as stated in

Questions & Answers

when the frame of reference in which we apply the laws is an inertial frame of reference.please check internet for details about inertial frame of references.
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No there is Something
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Verify that nothing has zero energy
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what is the formula for simple harmonic motion?
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At what time does the expression for the velocity become unphysical
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velocity is a physician vector quantity; both magnitude and direction needed to define it. the scalar absolute value ( magnitude) of velocity is called "speed being a coherent derived unite whose quantity is measured in SI ( metric system) as metres per second (m/s) or SI base unit of (m . s^-1).
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number of lines passing through area which is placed at some angle. these line are are produced by charge(+ or -).
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Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at 43° with respect to the 50-yard line (the +x-axis) and covers 7.8 m in 1 s. He then runs straight down the field at 90° with respect to the 50-yard line (that is, in the +y-direction) for 17 m, with an elapsed time of 2.9 s. (Express your answers in vector form.) (a) What is Matthews's final displacement (in m) from the start of the play?
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What is his average velocity (in m/s)?
Practice Key Terms 2

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