# 12.2 Bernoulli’s equation  (Page 2/7)

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

Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with $m$ replaced by $\rho$ . In fact, each term in the equation has units of energy per unit volume. We can prove this for the second term by substituting $\rho =m/V$ into it and gathering terms:

$\frac{1}{2}{\mathrm{\rho v}}^{2}=\frac{\frac{1}{2}{\text{mv}}^{2}}{V}=\frac{\text{KE}}{V}\text{.}$

So $\frac{1}{2}{\mathrm{\rho v}}^{2}$ is the kinetic energy per unit volume. Making the same substitution into the third term in the equation, we find

$\rho \mathrm{gh}=\frac{\mathrm{mgh}}{V}=\frac{{\text{PE}}_{\text{g}}}{V},$

so $\rho \text{gh}$ is the gravitational potential energy per unit volume. Note that pressure $P$ has units of energy per unit volume, too. Since $P=F/A$ , its units are ${\text{N/m}}^{2}$ . If we multiply these by m/m, we obtain $\text{N}\cdot {\text{m/m}}^{3}={\text{J/m}}^{3}$ , or energy per unit volume. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

## Making connections: conservation of energy

Conservation of energy applied to fluid flow produces Bernoulli’s equation. The net work done by the fluid’s pressure results in changes in the fluid’s $\text{KE}$ and ${\text{PE}}_{\text{g}}$ per unit volume. If other forms of energy are involved in fluid flow, Bernoulli’s equation can be modified to take these forms into account. Such forms of energy include thermal energy dissipated because of fluid viscosity.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, we will look at a number of specific situations that simplify and illustrate its use and meaning.

## Bernoulli’s equation for static fluids

Let us first consider the very simple situation where the fluid is static—that is, ${v}_{1}={v}_{2}=0$ . Bernoulli’s equation in that case is

${P}_{1}+\rho {\mathrm{gh}}_{1}={P}_{2}+\rho {\mathrm{gh}}_{2}\text{.}$

We can further simplify the equation by taking ${h}_{2}=0$ (we can always choose some height to be zero, just as we often have done for other situations involving the gravitational force, and take all other heights to be relative to this). In that case, we get

${P}_{2}={P}_{1}+\rho {\mathrm{gh}}_{1}\text{.}$

This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by ${h}_{1}$ , and consequently, ${P}_{2}$ is greater than ${P}_{1}$ by an amount $\rho {\mathrm{gh}}_{1}$ . In the very simplest case, ${P}_{1}$ is zero at the top of the fluid, and we get the familiar relationship $P=\rho \mathrm{gh}$ . (Recall that $P=\mathrm{\rho gh}$ and $\text{Δ}{\text{PE}}_{\text{g}}=\text{mgh}.$ ) Bernoulli’s equation includes the fact that the pressure due to the weight of a fluid is $\rho \text{gh}$ . Although we introduce Bernoulli’s equation for fluid flow, it includes much of what we studied for static fluids in the preceding chapter.

## Bernoulli’s principle—bernoulli’s equation at constant depth

Another important situation is one in which the fluid moves but its depth is constant—that is, ${h}_{1}={h}_{2}$ . Under that condition, Bernoulli’s equation becomes

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

Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli’s principle    . It is 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.) As we have just discussed, pressure drops as speed increases in a moving fluid. We can see this from Bernoulli’s principle. For example, 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.

#### Questions & Answers

what is temperature
temperature is the measure of degree of hotness or coldness of a body. measured in kelvin
a characteristic which tells hotness or coldness of a body
babar
Average kinetic energy of an object
Kym
average kinetic energy of the particles in an object
Kym
Mass of air bubble in material medium is negative. why?
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
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Abdul
6.6 lol 😁😁
Abdul
show ur work
Sunday
sorry..the answer is -10
Abdul
your question is wrong
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?
can I ask questions?
yes.
Abdul
Yes
Albert
sure
Ajali
yeap
Sani
yesssss
bilal
hello guys
Ibitayo
when you will ask the question
Ana
anybody can ask here
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