# 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 ${P}_{\text{i}}$ ?

please guys help, what is the difference between concave lens and convex lens
convex lens brings rays of light to a focus while concave diverges rays of light
Christian
for mmHg to kPa yes
Matthew
it depends on the size
Vincent
a lens which diverge the ray of light
rinzuala
concave diverges light
Matthew
thank you guys
Vincent
A diverging lens
Yusuf
What is isotope
Yusuf
each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element. "some elements have only one stable isotope
Karthi
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I would like to know this too
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Calculate . ..
Olufunsho
What is light wave
What is wave
Sakeenah
What is light
Sakeenah
okay
True
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Jeff
Great science students
Omo
A wave is a disturbance which travels through the medium transferring energy from one form to another without causing any permanent displacement of d medium itself
OGOR
Light is a form o wave
OGOR
Neurons communicate by sending message through nerves in coordination
OGOR
What are petrochemicals, give two examples
OGOR
light has dual nature, particle as well as wave. when we want to explain phenomena like Interference of light, then we consider light as wave.
Lalita
what is it as in the form of it or how to visualize it or what it contains
Matthew
particles of light are like small packets of energy called photons, and flow or motion of photons is wave like
Lalita
light is just the energy of which photons emit
Matthew
the wave is how they travel
Matthew
photons do not emitt energy, they are energy. They are massless particles.
Lalita
a wave is a disturbance through the medium. Have you ever thrown a stone in still water? the disturbance produced travels in form of wave, the wave produced by throwing stone in still water are circular in nature.
Lalita
a photon does contain mass when in motion. it doesnt contain mass when at rest
Matthew
when would it ever be at rest
Bob
a wave is a disturbance of which energy travels
Matthew
that's darkness. darkness has no mass because the photons within in aren't moving or producing energy
Matthew
Hi guys. Please I've been trying to understand the concept of SHM, but it's not been really easy, could someone please explain it to me or suggest a site I could visit? Thank you.
Odo
Matthew
effective mass of photons only comes into picture when we consider it accelerating in gravitational field, mass of photon has no meaning as it is always travelling with speed of light and is never at rest. with that high speed, Energy and momentum are equivalent. and darkness is absense of photons.
Lalita
darkness is absense of light. not the presence of 'resting photons'. photons are never at rest.
Lalita
photons are present in darkness but don't give off any light because they are stationary with no mass or energy. once a force makes them move again they will gain mass and give off light
Matthew
this theory is presented in Einsteins theory of special relativity
Matthew
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density and the viscosity iter of the liquid .use the dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square ×p all over 4×iter ×l
True
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density (rho)and the viscosity (iter)of the liquid. Use the method of dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square x p all over 4 x iter x l
True
Matthew, photons ARE light. there is no such thing as a photon that isn't moving. in fact the speed they move at is called C (for constant) in physics. through a vacuum they always travel at this speed no matter what. they can not slow down; except in another medium.
The reason why a photon can go at this speed is BECAUSE it had no mass. nothing can go this speed or faster because it needs to have no mass or negative mass. that's why it's called the constant.
when a photon hits something that is opaque, this is the only way to "stop"it. it isn't merely stopped but absorbed and turned into heat energy, then the remaining energy is reflected in different wavelengths. that reflection is what we call color. the darker something is, the less photons are ther
e. complete blackness is the absolute absence of photons altogether. I believe what you're referring to is not speed, but wavelength, which is indirectly proportional to the amount of energy a particular photon is made up of.
in order for a photon to have zero wavelength, it must (at least theoretically) have infinite energy.
about mass: you may have photons confused with electrons. elections have a mass so small that people say they are without mass, but they do. it is called electron mass or Me-.
you may also be getting electrons and photons confused because of the cherenkov effect. that is what happens when a particle travels faster than light IN THAT PARTICULAR MEDIUM. I emphasize that because no other particle besides photons can go the speed of c.
when a particle goes faster than light in a particular medium, a blue light is emitted, called cherenkov radiation. this is why nuclear reactors glow blue.
nuclear reactors release so much energy that when they emit electrons, those electrons are given enough energy to go faster than light in that medium (in this case water), releasing blue light. if you put the reactor in air or a vacuum, this effect wouldn't happen because the speed of light in air
is very close to c, which is the universal speed limit. I'd you did go faster than c, time would go backwards and you would have infinite theoretical mass and probably spagghettify, like with a black hole.
*if
*electrons
light waves can travel through a vacuum, and do not require a medium. In empty space, the wave does not dissipate (grow smaller) no matter how far it travels, because the wave is not interacting with anything else.
Salim
Please is there any instructional material for sounds Waves, Echo, light waves
Salami
how far there is hot topic that is boarding me now
Abraham
linear motion
Ahmed
kinematic
Abraham
Akinsanya
kinematic
Emma
kinematics disscuss the motion without cuases ...
ghulam
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Godslight
How does the Geiger tube works
pls he do we find for tension
tension is equal to the weight of the object. so for example if something weighs 45 Newtons then the tension in the Rope holding it is 45 Newtons. and because it is in equilibrium if the object is 45N and there are three ropes holding it there would be 15 N of tension in each to equal the weight
Shii
does that work for you?
Shii
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Belinda
very correct
Kudzy
A prankster applies 450 V to an 80.0 µF capacitor and then tosses it to an unsuspecting victim. The victim’s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change?
what is mass
the quantity of matter that a body contains, as measured by its acceleration under a given force or by the force exerted on it by a gravitational field.
Aliyu
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Nikita
correct
Jalil
or in easier terms the amount of stuff in an object (stuff meaning whatever element or material that makes the object heavy) the object composition
Shii
an object's resistance to change in motion?
Kudzy
your mass never changes but your weight changes based on the gravitational pull of a system or planet. your mass is just the amount of matter with a certain object
Matthew
please can someone help, why a bats can fly in the night without heating anything, how does he managed an see in darkness?
Vincent
...hit an obstacle and has bounced back.
Odo
Hi Vincent. From what I can remember and what I've learnt, bats do not have a good eye sight and so they make use of waves, when they send out waves and they do not return back to them, they realise that the site is free of obstacles but if it sends a wave and it returns back, it means it hit an
Odo
obstacle and bounced back
Odo
So bats make use of waves in place of eye sight. I think but confirm from a few sites .
Odo
they us echolocation to make a sort of mental map. many sea animals use the same method. this can also be compared to sonar which works in the same way
Matthew
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yes
Lajpat
its a question Lajpat Rai
EDWIN
Google Quantum Physics or refer to the text
Shii
atoms are considered the smallest unit of matter. it is further divided into three subatomic particles: protons, neutrons, and electrons.
Arshiya
what's the dimensions of moment of inertia?
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what is important property of thermistor
Naveedkhan
a horizontal force of 10n is applied to a 4.0kg block that is at rest on a perfectly smooth, level surface.find the speed of the block and how far has it gone after 6.0s.
10n/4.0kg=2.5m/s^2 v=6.0sx2.5m/s^2= 15m/s (15m/s x 15m/s x 2)÷ (2.5m/s^2 )= 180m
could yoy guve me formula
Denz
v=u+at, a=F/m=10/4, t=6, u=0, so v=15m/s, v2-u2=2as, so s=45m
praveen
a=F/m. 10n/4kg =2.5m/s^2 a=v/t but v=at = 2.5m/ s^2×6s=15m/s v^2=2as. but s=v^2/2a =15^2÷(2×2.5^2)=45m
Thank yoy guys. i'm having a bad time on our Physics class We're now studying work energy theorem can someone interested teaching me?
Denz
what is resistance of copper wire length
Naveedkhan
what is acceleration
is the rate of change of displacement
caramen
velocity
Naveedkhan
rate of change of velocity called acceleration
Nikita
is the rate of change in velocity with time taken
Anyebem
an unbalanced force
Matthew
EDWIN
M/S2
Achmet
Achmet
Aliyu
how quickly velocity changes over a time interval (period)
Shii