# 12.2 Bernoulli’s equation

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## Making connections: take-home investigation with two strips of paper

For a good illustration of Bernoulli’s principle, make two strips of paper, each about 15 cm long and 4 cm wide. Hold the small end of one strip up to your lips and let it drape over your finger. Blow across the paper. What happens? Now hold two strips of paper up to your lips, separated by your fingers. Blow between the strips. What happens?

## Velocity measurement

[link] shows two devices that measure fluid velocity based on Bernoulli’s principle. The manometer in [link] (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 ${P}_{1}+\frac{1}{2}{\mathrm{\rho v}}_{1}^{2}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}$ becomes

${P}_{1}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}\text{.}$

Thus pressure ${P}_{2}$ over the second opening is reduced by $\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}$ , and so the fluid in the manometer rises by $h$ on the side connected to the second opening, where

$h\propto \frac{1}{2}{\mathrm{\rho v}}_{2}^{2}\text{.}$

(Recall that the symbol $\text{∝}$ means “proportional to.”) Solving for ${v}_{2}$ , we see that

${v}_{2}\propto \sqrt{h}\text{.}$

[link] (b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air speed indicators in aircraft.

## Summary

• Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
${P}_{1}+\frac{1}{2}{\mathrm{\rho v}}_{1}^{2}+\rho {\mathrm{gh}}_{1}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}+\rho {\text{gh}}_{2}.$
• Bernoulli’s principle is Bernoulli’s equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding
${P}_{1}+\frac{1}{2}{\mathrm{\rho v}}_{1}^{2}={P}_{2}+\frac{1}{2}{\mathrm{\rho v}}_{2}^{2}.$
• Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

## Conceptual questions

You can squirt water a considerably greater distance by placing your thumb over the end of a garden hose and then releasing, than by leaving it completely uncovered. Explain how this works.

Water is shot nearly vertically upward in a decorative fountain and the stream is observed to broaden as it rises. Conversely, a stream of water falling straight down from a faucet narrows. Explain why, and discuss whether surface tension enhances or reduces the effect in each case.

Look back to [link] . Answer the following two questions. Why is ${P}_{\text{o}}$ less than atmospheric? Why is ${P}_{\text{o}}$ greater than ${P}_{\text{i}}$ ?

a car move 6m. what is the acceleration?
depends how long
Peter
What is the simplest explanation on the difference of principle, law and a theory
how did the value of gravitational constant came give me the explanation
how did the value of gravitational constant 6.67×10°-11Nm2kg-2
Varun
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor.
9.8m/s?
Sqrt(2*1.5m*9.81m/s^2)
Richard
0.5m* mate.
0.05 I meant.
Guess your solution is correct considering the ball fall from 1.5m height initially.
Sqrt(2*1.5m*9.81m/s^2)
Deepak
How can we compare different combinations of capacitors?
find the dimension of acceleration if it's unit is ms-2
lt^-2
b=-2 ,a =1
M^0 L^1T^-2
Sneha
what is botany
Masha
it is a branch of science which deal with the study of plants animals and environment
Varun
what is work
a boy moving with an initial velocity of 2m\s and finally canes to rest with a velocity of 3m\s square at times 10se calculate it acceleration
Sunday
.
Abdul
6.6 lol 😁😁
Abdul
show ur work
Sunday
Abdul
Abdul
If the boy is coming to rest then how the hell will his final velocity be 3 it'll be zero
Abdul
re-write the question
Nicolas
men i -10 isn't correct.
Stephen
using v=u + at
Stephen
1/10
Happy
ya..1/10 is very correct..
Stephen
hnn
Happy
how did the value 6.67×10°-11Nm2kg2 came tell me please
Varun
Work is the product of force and distance
Kym
physicist
Michael
what is longitudinal wave
A longitudinal wave is wave which moves parallel or along the direction of propagation.
sahil
longitudinal wave in liquid is square root of bulk of modulus by density of liquid
harishree
Is British mathematical units the same as the United States units?(like inches, cm, ext.)
We use SI units: kg, m etc but the US sometimes refer to inches etc as British units even though we no longer use them.
Richard
Thanks, just what I needed to know.
Nina
What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?
yes.
Abdul
Yes
Albert
sure
Ajali
yeap
Sani
yesssss
bilal
hello guys
Ibitayo
when you will ask the question
Ana
bichu
is free energy possible with magnets?
joel
no
Mr.
you could construct an aparatus that might have a slightly higher 'energy profit' than energy used, but you would havw to maintain the machine, and most likely keep it in a vacuum, for no air resistance, and cool it, so chances are quite slim.
Mr.
calculate the force, p, required to just make a 6kg object move along the horizontal surface where the coefficient of friction is 0.25
Gbolahan
Albert
if a man travel 7km 30degree east of North then 10km east find the resultant displacement
11km
Dohn
disagree. Displacement is the hypotenuse length of the final position to the starting position. Find x,y components of each leg of journey to determine final position, then use final components to calculate the displacement.
Daniel
1.The giant star Betelgeuse emits radiant energy at a rate of 10exponent4 times greater than our sun, where as it surface temperature is only half (2900k) that of our sun. Estimate the radius of Betelgeuse assuming e=1, the sun's radius is s=7*10exponent8metres
2. A ceramic teapot (e=0.20) and a shiny one (e=0.10), each hold 0.25 l of at 95degrees. A. Estimate the temperature rate of heat loss from each B. Estimate the temperature drop after 30mins for each. Consider only radiation and assume the surrounding at 20degrees
James
Is our blood not red
If yes than why when a beam of light is passing through our skin our skin is glowing in red colour
because in our blood veins more red colour is scattered due to low wavelength also because of that scattered portion comes on skin and our skin act as a thinscreen.
so you saying blood is not red?
Donny
blood is red that's why it is scattering red colour!
like if u pass light frm red colour solution then it will scatter red colour only.. so same it is with our skin..red colour blood is moving inside the veins bcz of thinkness of our fingers.. it appears to be red.
No I am not saying that blood is not red
then ur qtn is wrong buddy.. 😊
Blood is red. The reason our veins look blue under our skin, is because thats the only wavelength on light that can penetrate our skin.
Mr.
Red light is reflected from our blood but because of its wavelength it is not seen. While in the other hand blue light has a longer wavelength allowing it to pass the our skin and to our eyes.
Nina
Thus, our veins appear blue while they are really red... THE MORE YOU KNOW...(;
Nina
So in conclusion our blood is red but we can only see blue spectrum because of our skin. The more longer a wavelength is the more durable it is to reflection, so blue light cant pass thew skin completely causing a reflection which causes veins to appear blue. While the red light is scatter around.
Nina
the reason why when we shine a light at our skin it appears red is because the red light is increased and more goes to your eyes. So in other words it increases the amount of red light vs it being scatterd around everywhere.
Nina
I think the blood is only a mixture of colors but red is predominant due to high level of haemoglobin.
stanley
As a side note, the heme part of hemoglobin is why blood is red
Sedlex
a car starts from rest acceleration and moves with a uniform acceleration a, in time t. the distance covered during the motion is expressed as?.
distance=a×(t^2)
Emmanuel
1/2at.t
David