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Applications of Bernoulli's Theorem(I) is considered one of the most asked concept.
51 Questions around this concept.
In the diagram shown, the difference in the two tubes of the manometer is 5 cm, the cross section of the tube at A and B is 6 mm2 and 10 mm2 respectively. The rate at which water flows through the tube is (g=10 ms-2)
The figure shows a liquid of a given density flowing steadily in a horizontal tube of varying cross–section. Cross-sectional areas at A are 1.5 cm2, and B is 25 mm2, if the speed of liquid at B is 60 cm/s then (PA – PB) is: (Given PA and PB are liquid pressures at A and B points)
Density
A and B are on the axis of the tube
Bernoulli's equation is written in form as -
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A hole is made at the bottom of the tank filled with water (density=100 kg/). If the total pressure at the bottom of the tank is three atmospheres(1 atm= N/), then the velocity of efflux-
If the velocity head of a stream of water is equal to 10cm, then its speed of flow is equal to-
A tank is filled to a height of H. The range of water coming out of a hole which is a depth H/4 from the surface of the water level is-
A water tank standing on the floor has two small holes punched in the vertical wall one above the other. The holes are 2.4cm and 7.6cm above the floor. If the jest of water from the holes hits the floor at the same point, then the height of water in the tank is-
Water flows along a horizontal pipe of non-uniform cross-section. The pressure is 1cm of Hg, where the velocity is 35 cm/s. At a point, where the velocity is 65 cm/s, the pressure will be-
The pressure of water in a pipe when the tap is closed is . When the tap gets open, the pressure reduces to . The velocity with water comes out on opening the tap is-
Airstream flows horizontally past an aeroplane wing of surface area 4 . The speed of air over the top surface is 60m/s and under the bottom surface is 40m/s. the force of lift on the wing is (density of air = 1kg/).
For a point in a fluid flow, Bernoulli's Theorem relates between its pressure, its velocity and its height from a reference point.
Bernoulli's Theorem states that the total energy (Pressure energy, Potential energy, and Kinetic energy ) per unit volume or mass of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow. (Provided that there is no source or sink of the fluid along the length of the pipe).
Mathematically for a liquid flowing through a pipe.
We can write Bernoulli's equation as
Bernoulli's Theorem can be proved with the help of work-energy theorem.
Bernoulli's equation also represents the conservation of mechanical energy in case of moving fluids.
Bernoulli's theorem for the unit mass of liquid flowing through the pipe is given by
If we divide the above equation by g we get
Where
h=gravitational head
For the below figure
With the help of Bernoulli's equation
We can write
The velocity of Efflux or Torricelli's Theorem-
If a liquid is filled in a vessel up to height H and a hole is made at a depth h below the free surface of the liquid as shown in fig.
Now take the level of the hole as reference level (i.e., zero point of potential energy)
And by applying Bernoulli's equation we get
This v is called the Velocity of Efflux.
This formula is only valid when (Area of Hole) <<< (Area of the vessel)
Thus Torricelli's Theorem relates the speed of fluid flowing out of an orifice.
(Note- The speed that an object would acquire in falling
from rest through a distance h is equal to
And this is same as that of Velocity of Efflux.)
The velocity of efflux is independent of the nature of liquid (), the quantity of liquid in the vessel and the area of the orifice/hole.
The velocity of efflux depends on h (i.e depth below the free surface)
I.e Means Greater is the distance of the hole from the free surface of the liquid, greater will be the velocity of efflux
As the distance of hole from the ground is (H-h) and its initial vertical velocity at hole is zero.
So Time taken by liquid to reach the Ground =t is given by
Where
H - the height of the vessel
And h= depth below the free surface
Range (x)-
During time t liquid is moving horizontally with constant velocity v,
And it will hit the base level at a horizontal distance x as shown in the above figure.
This horizontal distance x is also called a Horizontal range.
Using x=vt
We get Range as
This range will be maximum when
And Maximum value of the range is H
Means
REACTION FORCE DUE TO EJECTION OF LIQUID
A cylindrical vessel has an opening of cross-sectional area ' a ' near the bottom. A disc is held against the opening to prevent liquid of density $\rho$ from coming out. If the height of the liquid above the opening is h. Let us analyse the force on the disc in this situation.
The disc experiences hydrostatic pressure from the liquid inside the vessel. Pressure at the level of the disc is
The air pressure on the outside of disc is .
The net outward force =.
Now the disc is moved a short distance away in the horizontal direction. The liquid comes out, strikes the disc inelastically and drops vertically downward. The water in this case will impart impulsive (impact) force on the disc.
When the disc is moved away, the liquid moves out with speed .
The (mass per second), i.e, the rate of mass coming out of the opening is given by .
Momentum per second imparted by water rightward is given by
The change in momentum per second after striking the disc is
Taking the rightward direction as positive, the force on the liquid is towards the left and its reaction with the disc is towards the right. The force obtained is twice the hydrostatic force. The force due to atmospheric pressure is cancelled out on both sides.
Note:
If the velocity of a liquid (or gas) of density $\rho$ coming out through an opening of the area of cross-section a is v, then there will be a thrust force due to liquid coming out the opening. The direction of the thrust force will be just opposite to the velocity direction.
Venturimeter-
It is a device is used for measuring the rate of flow of liquid through pipes.
This device based on application Bernoulli's theorem.
The image of the Venturi Meter device is given below
For the above figure
a1 and a2 are an area of cross-section of tube A and B respectively
And v1 and v2 are the Velocities of the flow of liquid through A and B respectively
And are the Liquid pressure at A and B respectively
Then ….(1)
Where = density of flowing liquid
And h=difference of fluid level between the vertical tube D and E
Now applying Bernoulli's equation for the horizontal flow of liquid we get
If V=rate of the flow of liquid through a pipe
Then from the continuity equation, we can write
From equation (1) and (2) and (3)
We get
V=rate of the flow of liquid through pipe
As
Aspirator pumps-
This works on the principle of Bernoulli's Theorem.
Example of Aspirator pumps is paint-gun, scent-spray or insect-sprayer, etc.
In such devices, high-speed air is passed over a tube T with the help of motion of a piston P in a cylinder C and this helps to spray the liquid L as shown in the above figure.
The high-speed air creates low pressure in the tube and because of the low-pressure liquid rise in it. And thus liquid gets sprayed with expelled air.
Change of plane of motion of spinning ball-
This can be with the help of the principle of Bernoulli's Theorem
Magnus effect- When a spinning ball is thrown it deviates from its usual path in flight. This effect is called the Magnus effect.
This effect plays a very important role in sports like cricket, tennis, and football, etc.
Working of an aeroplane-
This is also based on Bernoulli's principle.
During a tornado or hurricane, blowing off roofs by wind storms can be explained
with the help of the principle of Bernoulli's Theorem
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