St. Norbert School
Session:-2023-24
Topic:- Bernoulli’s Theorem
Submitted To:Ms.Sujata Godhke
Submitted By:Anant Soni
Class:-XI – ‘B’
Roll no:- ‘4’
CERTIFICATE
This is to certify that ‘Anant Soni’ of class 11th
‘B’ has completed the project work in physics in
the year 2023-2024 on “Bernoulli’s Theorem”
under the guidance of ‘Ms. Sujata Godhke’ as
prescribed by CBSE course.
Internal Examiner
Principal’s Signature
School Stamp
Acknowledgement
I would like to express my special thanks for
attitude to my subject teacher ‘Ms.Sujata
Godhke Mam’ to give her guidance to me
in the successful completion of this project.
I also want to give special thanks to our
principal ‘Rev.Sr.Gracy’ who gave his
golden opportunity to do this wonderful
project on the topic ‘Bernoulli, s
theorem’so that I will get to know about it
information for the same.
INDEX
 Pressure
 Bernoulli’s Equation
 Derivation of Bernoulli’s
Equation
 Torricelli and his Orifice
 Derivation of Torricelli’s
Equation
 Streamlines
 Applications Of Bernoulli’s
Theorem
1. Venturi tube
2. Aerodynamic Lift
3. Bernoulli Theorem in Medical Field
4. Blown Roofs
Conclusion
 Bibliography
PRESSURE
1. Pressure is defined as force per unit
area.
2. Standard unit is Pascal, which is N/m2
3. For liquid pressure, the medium is
considered as a continuous
distribution of matter.
4. For gas pressure, it is calculated as
the average pressure of molecular
collisions on the container.
5. Pressure acts perpendicular on the
surface.
6. Pressure is a scalar quantity –
pressure has no particular direction
(i.e.
acts in every direction).
Bernoulli’s
Equation
Where p is the pressure, ρ is the density, v
is the velocity, h is elevation, and g is
gravitational acceleration.
Derivation of
Bernoulli’s Equation

●
Restrictions
Incompressible
● Non-viscous fluid (i.e. no
friction)
● Following a streamline motion (no
turbulence)
●
Constant density
*There exists an extended form of
equation that takes friction and
compressibility into account, but that is too
complicated for our level of study.
Etotal = 1/ 2m v 2
W = F/ A* A* d =
 Consider the change in total energy
of the fluid as it moves from the
inlet to the outlet.
Δ Etotal = Wdone on fluid - Wdone by fluid
Δ Etotal = (1/2mv22 + mgh1) – (1/2mv12 + mgh2)
Wdone on fluid - Wdone by fluid = (1/2mv22 + mgh1)
– (1/2mv12 + mgh2)
P2V2 - P1V1 = (1/2mv22 + mgh1) – (1/2mv12 +
mgh2)
P2 – P1 = (1/2ρ v12 + ρ gh1) – (1/2ρ v12 + ρ gh1)
Torricelli and his
Orifice
 In 1843, Evangelista Torricelli proved
that the flow of liquid through an
opening is proportional to the square
root of the height of the opening.
 Q = A*√(2g(h1-h2)) where Q is flowrate,
A is area, h is height.
 Depending on the contour and shape of
the opening, different discharge
coefficients can be applied to the
equation (of course we assume simpler
situation here -
Derivation of Torricelli’s
Equation
1. We use the Bernoulli Equation:
2. In the original diagram A1 [top] is
much larger than A2 [the opening].
Since A1V1 = A2V2 and A1 >> A2, V1 ≈
0
3. Since both the top and the opening
are open to atmospheric pressure,
P1 = P2 = 0 (in gauge pressure).
The equation simplifies down to:
ρgh1 = 1/2 ρv22 + ρgh2
1
/2 ρv22 = pg(h1-h2)
V22 = 2g(h1-h2)
∴ V2 = √(2g(h1-h2))
Q = Av2 = A √ (2g(h1-h2))
Streamlines
1. A streamline is a path traced out by a mass less
particle as it moves withthe flow.
2. Velocity is zero at the surface.
3. As you move away from the surface, the
velocity uniformly approaches the free stream
value (fluid molecules nearby the surface are
dragged due toviscosity).
4. The layer at which the velocity reaches the free
stream value is called boundary layer. It does not
necessarily match the shape of the object –
boundary layer can be detached, creating
turbulence (wing stall in aerodynamic terms).
Applications Of Bernoulli ‘s
Principle
Venturi Tube
Venturi metre is a device used to
measure the flow speed of
incompressible fluid it consist of a tube
with a broad diameter and a small
constriction at the middle as shown in
figure above.
1.
A2 < A1 ; V2 > V1
2. According to Bernoulli’s Law,
pressure at A2 is lower.
3. . This is useful in controlling fluid
velocity.
P2 + 1/ 2ρ v1^2 = P1 + 1/ 2ρ v1^2 ;
Δ P = ρ/ 2* ( v2^2 – v1^2)
Aerodynamic Lift
1. Lift is a Fort that keeps an aircraft in the air.
2. In Bernoulli's view lift is produced by
difference of pressure (faster velocity on top and
slower velocity in the bottom of the wing)
3. In Newtonian view left to the reaction force
that result from the downward deflection of the
Air.
4. The fast flowing air decreases the surrounding
air pressure. Because the air pressure is greater
below the airfoil than above, a resulting lift force
is created.
Bernoulli’s in medical
Field
Bernoulli's principle helps in explaining
blood flow in arteries. The artery may get
constricted due to the accumulation of
plaque on its inner walls. In order to drive
the blood through this constriction, a greater
demand is placed on the activity of the heart.
Blown roofs during a storm
You will notice how tinted roofs and roofs of
huts get blown away during severe storms
without causing significant damage to the rest
of the property. The principle behind minimal
damage to the property is an example
of Bernoulli’s principle:



The low pressure is created at the top of the
house when the wind blows.
The pressure created below the roof is greater
than the pressure on top of the roof.
Thus, the roof gets blown with the wind.
Conclusion
1. Bernoulli's law states that if a non-viscous fluid is
flowing along a pipe of varying cross section, then
the pressure is lower at constrictions where the
velocity is higher, and the pressure is higher where
the pipe opens out and the fluid stagnate.
2. Many people find this situation paradoxical when
they first encounter it (higher velocity, lower
pressure).
3. Venturi-meter, Bernoulli’s principle is used in
venturi-meter to find the rate offlow of a liquid.
4. It is used in a carburetor to mix air and petrol
vapor in an internal combustion engine.
5. Wings of Aero plane Wings of an aero plane are
made tapering. The upper surface is made
convex and the lower surface is made concave.
Due to this shape of the wing, the air
currents at thetop have a large velocity than
at the bottom.
6. Consequently the pressure above the surface of thewing is less as
compared to the lower surface of thewing. This difference of
pressure is helpful in giving a vertical lift to the plane.
BIBLIOGRAPHY
1. Help from Internet
 www.sceincefare.com
 www.mycbsegide.com
2. Help from books
 Referenced from H.C. Verma
 Referenced from physics
NCERT