Fluid Mechanics - Pressure Variation with Height , Pascal, Archimedes, Bernouilli's and Torricelli's Principles

Basic Mechanics Tutorials- At a glance

Get Excited about Fluid Mechanics! Bernoulli Theorem Demo


Fluid Dynamics - with Interesting Examples and Solved Problems - related to the Continuity Equation, Barometers, Pressure Variation with Height , Pascal, Archimedes, Bernouilli's and Torricelli's Principles

Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Physics B, AP Physics C, A Level, Singapore/GCE A-Level; 

Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

This might also be helpful in studying topics required by Common Core Physics. 


A Quick Summary of what we'll study in this chapter and the kind of problems we'll solve

(after this intro, there is a comprehensive document with study material as well as solutions to problems.)


Introduction

A fluid, in contrast to a solid, is a substance that can flow. Fluids conform to the boundaries of any container in which we put them. They do so because a fluid cannot sustain a force that is tangential to its surface i.e. a fluid is a substance that flows because it cannot withstand a shearing stress. It can, however, exert a force in the direction perpendicular to its surface. Both liquids and gases are fluids i.e. which can flow.


Here are the topics covered in this tutorial

Fluid Pressure :

A fluid cannot withstand shearing stress. It can, however, exert a force perpendicular to its surface. That force is described in terms of pressure

P = lim∆S→0F∆S where F is the force acting on a surface element of area ∆S. The SI unit of pressure is Nm-2 called pascal and abbreviated as Pa.

Pressure Variation with Height : 

P2 – P1 = -ρgh

Pascal’s Principle

A change in the pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. As an example, suppose a glass fitted with a piston is filled with a liquid. Let an external force F be applied on the piston. If the cross-sectional area of the piston is A, the pressure just below the piston is increased by F/A. Now, consider a point B at a distance z below A. The pressure at B also increases by the same amount F/A for the fluid to remain in vertical equilibrium. If the pressure at B does not change by the same amount, there would be a resultant pressure difference at the two points (which will be different from ρgz) which will cause a resultant acceleration of the fluid in the vertical direction but that cannot happen as there is no empty space to go to and the fluid is incompressible.

Archimedes’ Principle: 

Archimedes’ Principle states that ‘when a body is partially or fully dipped into a fluid at rest, the fluid exerts an upward force of buoyancy equal to the weight of the displaced fluid. This can be understood more clearly by considering the following situation. Suppose the body dipped in the fluid is replaced by the same fluid of equal volume. As the entire fluid now becomes homogeneous, all parts will remain in equilibrium. The part of the fluid substituting the body also remains in equilibrium. Forces acting on this substituting fluid are:
(a) the weight mg of this part of the fluid
(b) the resultant B of the contact forces by the remaining fluid
As the substituting fluid is in equilibrium, these two should be equal and opposite. Thus, B = mg and it acts in the vertically upward direction. Now, the situation does not change much when the substituting fluid is replaced by the body. The forces acting on the remaining fluid remain exactly the same as before and from Newton’s third law the forces acting on the body are equal and opposite to the forces acting on the surrounding fluid. As those forces do no change whether there is a dipped body or its all the same fluid, thus the forces acting on the dipped body are the same as the forces which would act on the substituting fluid.

Floating

When a solid body is dipped into a fluid, the fluid exerts an upward force of buoyancy on the solid. If the force of buoyancy equals the weight of the solid, the solid will remain in equilibrium. This is called floatation. This can happen only when the overall density of the solid is less than or equal to that of the fluid.

Flow of Ideal Fluids

An ideal fluid is incompressible and nonviscous. The first condition means that the density of the liquid is independent of the variations in pressure and thus remains constant. The second condition means that parts of the liquid in contact do not exert any tangential force on each other. Thus, there is no friction between the adjacent layers of the liquid.

The flow of an ideal fluid is steady and irrotational.
Consider a liquid passing through a glass tube. Concentrate on a particular point A in the tube and look at the particles arriving at A. If the velocity of all the particles arriving at A is same at all time, such a flow of fluid is called steady flow or streamline flow. As a particle goes from A to another point B its velocity may change, but all the particles reaching A will have the same velocity and all these particles will have the same velocity at B.
On the other hand, in turbulent flow, the velocities of different particles passing through the same point may be different and change erratically with time. For example, the motion of water in a high fall.

The path followed by an individual fluid particle in a flowing fluid is called its line of flow or streamline. A tube of flow is a bundle of streamlines. As the streamlines do not cross each other fluid flowing through different tubes of flow cannot intermix, although there is no physical partition between the tubes. When a liquid is passed slowly through a pipe, the pipe itself is one tube of flow. The flow within any tube of flow obeys the equation of continuity:

Av = a constant

where Av is the volume flow rate, A is the cross-sectional area of the tube of flow at any point, and v is the speed of the fluid at that point.
This equation expresses the law of conservation of mass in fluid dynamics, i.e. the total mass of fluid going into a tube of flow through any cross section must be equal to the total mass coming out of the same tube through any other cross section in the same time.

Bernoulli’s Equation : 

Applying the principle of conservation of mechanical energy to the flow of an ideal fluid leads to Bernoulli’s equation: P + ρgh + ½ ρv2 = a constant

Torricelli’s Theorem : 

The speed of liquid coming out through a hole at a depth h below the free surface is the same as that of a particle fallen freely through the height h under gravity. The speed of the liquid coming out is called the speed of efflux.


 

Here are some of the problems which will be covered and solved in this tutorial :



Q: A barometer kept in an elevator reads 76 cm when it is at rest. If the elevator goes up with increasing speed, will the reading be greater than or less than 76 cm?
Q: If water is used to construct a barometer, what would be the height of water column at standard atmospheric pressure (76 cm of mercury) ?
Q: A metal piece of mass 200 g lies in equilibrium inside a glass of water. The piece is in the touch with the bottom of the glass through a small number of points. If the density of the metal is 5000 kgm-3, find the normal force exerted by the bottom of the glass on the metal piece. Take g = 10 ms-2.
Q: A cube of ice floats partly in water and partly in kerosene oil. Find the ratio of the volume of ice immersed in water to that in kerosene oil. Specific gravity of kerosene oil is 0.8 and that of ice is 0.9.
Q: A cylindrical object of diameter 12 cm, height 24 cm and density 7500 kgm-3 is supported by a  vertical spring and is half dipped in water as shown in figure.  (a) Find the elongation of the spring in equilibrium condition. (b) If the object is slightly depressed and released, find the time period of resulting oscillations of the object. The spring constant is 450 N/m. Take g = 10 ms-2.
Q: A U-tube containing a liquid is accelerated horizontally with a constant acceleration a0. If the separation between the vertical limbs is ‘l ’, find the difference in the heights of the liquid in the two arms.
Q: A garden hose with an internal diameter of 2 cm is connected to a (stationary) lawn sprinkler that consists merely of a container with 25 holes, each 0.12 cm in diameter. If the water in the hose has a speed of 0.9 m/s, at what speed does it leave the sprinkler holes?
Solution: We use the equation of continuity. Let v1 be the speed of water in the hose and v2 be its speed as it leaves one of the holes.
v1A1 = v2(NA2)
0.9 x п(0.01)2 = v2(25 X п(0.0006)2)
or v2 = 10 m/s


Q: What is the acceleration of a rising hot-air balloon if the ratio of the air density outside the balloon to that inside is 1.42? Neglect the mass of the balloon fabric and the basket. Take g = 10ms-2.
Q: An object hangs from a spring balance. The balance registers 40 N in air, 25 N when this object is immersed in water, and 30 N when the object is immersed in another liquid of unknown density. What is the density of that other liquid?
Q: An iron casting containing a number of cavities weighs 6500 N in air and 4500 N in  water. What is the total volume of all the cavities in the casting? The density of iron (that is, a sample with no cavities) is 7.85 g/cm3.
Q: Suppose that you release a small ball from rest at a depth of 0.5 m below the surface in a pool of water. If the density of the ball is 0.25 that of water and if the drag force on the ball from the water is negligible, how high above the water surface will the ball shoot as it emerges from the water? (Neglect any transfer of energy to the splashing and waves produced by the emerging ball.) Take g = 10 ms-2.
Q: A cubical block of wood of edge 2.5 cm floats in water. The lower surface of the cube just touches the free end of a vertical spring fixed at the bottom of the pot. Find the maximum weight that can be put on the block without wetting it. Density of wood = 800 kgm-3 and spring constant of the spring = 60 Nm-1. Take g = 10 ms-2.
Q: A wooden plank of length 1 m and uniform cross section is hinged at one end to the bottom of a tank as shown in figure. The tank is filled with water up to a height of 0.6 m. The specific gravity of the plank is 0.5. Find the angle θ that the plank makes with the vertical in the equilibrium position. (Exclude the case θ = 0.)
Q: A cylindrical block of wood of mass ‘M’ is floating in water with its axis vertical. It is depressed a little and then released. Show that the motion of the block is simple harmonic and find its frequency.
Q: The area of cross section of a large tank is 0.6 m2. It has an opening near the bottom having area of cross section 1 cm2. A load of 25 kg is applied on the water at the top. Find the velocity of the water coming out of the opening at the time when the height of water level is 40 cm above the bottom. Take g = 10 ms-2.
Q: Water level is maintained in a cylindrical vessel upto a fixed height ‘H’.  The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel so that the water stream coming out of the hole strikes the horizontal plane at the greatest distance from the vessel?
Q: A venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipe; the cross-sectional area A of the entrance and exit of the meter matches the pipe’s cross-sectional area. Between the entrance and exit, the fluid flows from the pipe with speed V and then through a narrow “throat” of cross-sectional area a with speed v. A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid’s speed is accompanied by a change ∆P in the fluid’s pressure, which causes a height difference ‘h’ of the liquid in the two arms of the manometer. (Here ∆P means pressure in the throat minus pressure in the pipe.) (a) By applying Bernoulli’s equation and the equation of continuity to points 1 and 2 in figure, show that, V=2a2∆Pρ(a2-A2)
where ρ is the density of the fluid. (b) Suppose that the fluid is fresh water, that the cross-sectional areas are 64 cm2 in the pipe and 32 cm2 in the throat, and that the pressure is 55 kPa in the pipe and 41 kPa in the throat. What is the rate of water flow in cubic meters per second?


Complete Tutorial Document with Examples, Solved Problems and Figures: