# 14.6 Bernoulli’s equation  (Page 4/8)

 Page 4 / 8
${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}$

becomes

${p}_{1}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}.$

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

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

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

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

Part (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. Measurement of fluid speed based on Bernoulli’s principle. (a) A manometer is connected to two tubes that are close together and small enough not to disturb the flow. Tube 1 is open at the end facing the flow. A dead spot having zero speed is created there. Tube 2 has an opening on the side, so the fluid has a speed v across the opening; thus, pressure there drops. The difference in pressure at the manometer is 1 2 ρ v 2 2 , so h is proportional to 1 2 ρ v 2 2 . (b) This type of velocity measuring device is a Prandtl tube , also known as a pitot tube.

## A fire hose

All preceding applications of Bernoulli’s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli’s equation in which pressure, velocity, and height all change.

## Calculating pressure: a fire hose nozzle

Fire hoses used in major structural fires have an inside diameter of 6.40 cm ( [link] ). Suppose such a hose carries a flow of 40.0 L/s, starting at a gauge pressure of $1.62\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}{\text{N/m}}^{2}$ . The hose rises up 10.0 m along a ladder to a nozzle having an inside diameter of 3.00 cm. What is the pressure in the nozzle? Pressure in the nozzle of this fire hose is less than at ground level for two reasons: The water has to go uphill to get to the nozzle, and speed increases in the nozzle. In spite of its lowered pressure, the water can exert a large force on anything it strikes by virtue of its kinetic energy. Pressure in the water stream becomes equal to atmospheric pressure once it emerges into the air.

## Strategy

We must use Bernoulli’s equation to solve for the pressure, since depth is not constant.

## Solution

Bernoulli’s equation is

${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}+\rho g{h}_{1}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}+\rho g{h}_{2}$

where subscripts 1 and 2 refer to the initial conditions at ground level and the final conditions inside the nozzle, respectively. We must first find the speeds ${v}_{1}$ and ${v}_{2}$ . Since $Q={A}_{1}{v}_{1}$ , we get

${v}_{1}=\frac{Q}{{A}_{1}}=\frac{40.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}{\text{m}}^{3}\text{/}\text{s}}{\pi {\left(3.20\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\text{m)}}^{2}}=12.4\text{m/s}.$

Similarly, we find

${v}_{2}=56.6\phantom{\rule{0.2em}{0ex}}\text{m/s}.$

This rather large speed is helpful in reaching the fire. Now, taking ${h}_{1}$ to be zero, we solve Bernoulli’s equation for ${p}_{2}$ :

${p}_{2}={p}_{1}+\frac{1}{2}\rho \left({v}_{1}^{2}-{v}_{2}^{2}\right)-\rho g{h}_{2}.$

Substituting known values yields

$\begin{array}{cc}\hfill {p}_{2}& =1.62\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}+\frac{1}{2}\left(1000\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}\right)\left[{\text{(12.4 m/s)}}^{2}-{\text{(56.6 m/s)}}^{2}\right]\hfill \\ & \phantom{\rule{0.1em}{0ex}}-\phantom{\rule{0.1em}{0ex}}\left(1000\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}\right)\left(9.80\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}\right)\left(10.0\phantom{\rule{0.2em}{0ex}}\text{m}\right)\hfill \\ & =0.\hfill \end{array}$

## Significance

This value is a gauge pressure, since the initial pressure was given as a gauge pressure. Thus, the nozzle pressure equals atmospheric pressure as it must, because the water exits into the atmosphere without changes in its conditions.

## 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:
${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}+\rho g{h}_{1}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}+\rho g{h}_{2}.$
• Bernoulli’s principle is Bernoulli’s equation applied to situations in which the height of the fluid is constant. The terms involving depth (or height h ) subtract out, yielding
${p}_{1}+\frac{1}{2}\rho {v}_{1}^{2}={p}_{2}+\frac{1}{2}\rho {v}_{2}^{2}.$
• Bernoulli’s principle has many applications, including entrainment and velocity measurement.

what is friction
a force act by surface between two bodies whose are always oppose the relative motion .....
Raghav
when two rough bodies are placed in contact and try to slip each other ... than a force act them and it's ippse the relative motion between them
Raghav
thats friction force and roughnes of both bodies is define friction of surface
Raghav
what is a progressive wave
What is the wake for therapist
can u like explain your question with clear detail
Chikamso
who would teach me vectors?
what's chemistry
branch of science dt deals with the study of physical properties of matter and it's particulate nature
Josiah
Good
Daniel
actually
Nathz
Y acctually do u hav ur way of defining it? just bring ur iwn idear
Daniel
well, it deals with the weight of substances and reaction behind them as well as the behavior
Josiah
buh hope Esther, we've answered ur question
Josiah
what's ohms law
CHIJIOKE
ohms law states that, the current flowing through an electric circuit is directly proportional to the potential difference, provided temperature and pressure are kept constant
Josiah
what is sound
James
ohms law states that the resistance of a material is directly proportional to the potential difference between two points on that material, if temperature and other physical conditions become constant
Chikamso
How do I access the MCQ
As I think the best is, first select the easiest questions for you .and then you can answer the remaining questions.
lasitha
I mean I'm unable to view it
Abraham
when I click on it, it doesn't respond
Abraham
ohhh,try again and again ,It will be showed
lasitha
okay
Abraham
what is centripetal force
هي قوة ناتجة من الحركة الدائرية ويكون اتجاهها إلى المركز دائماً
meaning of vector quantity
vector quantity is any quantity that has both magnitude in terms of number (units) and direction in terms of viewing the quantity from an origin using angles (degree) or (NEWS) method
LEWIS
vector quantity is physical quantity has magnitude and direction
vector is a quantity that is use in measuring size of physical properties and their direction
Bitrus
what difference and similarities between work,force,energy and power?
Anes
power
mehreen
power
saba
enery is the ability to do work. work is job done, force is a pull or push. power has to do with potential. they belong to different categories which include heat energy, electricity.
Andrew
force refers to a push or pull... energy refers to work done while power is work done per unit time
Shane
mathematically express angular velocity and angular acceleration
it depends on the direction. an angular velocity will be linear and angular acceleration will be an angle of elevation.
Andrew
The sonic range finder discussed in the preceding question often needs to be calibrated. During the calibration, the software asks for the room temperature. Why do you suppose the room temperature is required?
Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See [link] .) (a) Calculate the echo times for temperatures of 5.00°C5.00°C and 35.0°C.35.0°C. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty an
Shaina
give a reason why musicians commonly bring their wind instruments to room temperature before playing them.
Shaina
The ear canal resonates like a tube closed at one end. (See [link]Figure 17_03_HumEar[/link].) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be 37.0°C,37.0°C, which is the same as body tempe
Shaina
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0°C10.0°C to 30.0°C30.0°C ? That is, find the ratio of the frequencies at those temperatures.
Shaina
what are vector quantity
Quantities that has both magnitude and direction
NNAEMEKA
what is lenses
Rhoda
vector quantities are those physical quantites which have magnitude as well as direction and obey the laws of vector algebra.
Huzaif
electric current has both magnitude and direction but it doesn't obey the laws of vector algebra, hence it is not a vector quantity.
Huzaif
what is momentum
Momentum=mv
Nana
what is dimension
what is vector
A quantity having both magnitude and direction
Zubair
quality having both magnitude and direction
ZINA
they have magnitude and direction.
Andrew
What is biology
Jimmy
Jimmy nkhoma
Jimmy
What is biology
Jimmy
what is math
Tamzid
Biology is d study of living things
Nana By   By   By   By