Physics Ii: Technological University of The Philippines-Taguig Electrical Engineering Department
Physics Ii: Technological University of The Philippines-Taguig Electrical Engineering Department
Physics Ii: Technological University of The Philippines-Taguig Electrical Engineering Department
PHYSICS II
Instructor: Engr. Esmeralda Tapiz
I. FLUIDS AT REST
1.1 Properties a. Density b. Specific Gravity 1.2 Hydrostatic Pressure a. Pressure in liquids at rest b. Pressure in gases c. Pascals Law d. Boyles Law e. Archimedes Principle
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Introduction
The Hydrostatics is the science of fluids with no motion. A fluid is defined as a substance that continually flows under an applied shear stress regardless of how small the applied stress. All liquids and all gases are fluids. The term fluid is usually mistaken as liquid, but it actually covers a lot of the phases of matter (liquids, gases, plasmas and others).
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Density
is defined as the ratio of mass per unit volume. It is generally represented by the Greek letter rho, , and measured in terms of kilograms/cubic meter, or kg/m3 , slugs/ft.
Since the volume of a fluid expands and contracts, the density of fluids vary with temperature. The most common fluid, water, has maximum density of 1000 kg/m3 at 4C. Air, a mixture composed principally of the gases nitrogen (78%) and oxygen (21%), has a density of 1.29 kg/m3 at 0C and 1.20 kg/m3 at 20C.
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What is a Slug?
The slug is the unit of mass in the US common system of units, where the pound is the unit of force. The pound is therefore the unit of weight since weight is defined as the force of gravity on an object.
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What is a Slug?
The comparison of the slug and the pound makes it clear why the size of the pound is more practical for commerce. But at the precision obtainable in current scientific work, it is undesirable to have the weight of an object as a standard because the value of g does change measurably at different points on the Earth. It is much better to have a standard in terms of mass. The standard kilogram is the mass reference for scientific work.
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Specific Gravity
- is the ratio of the density of a substance to the
density of a reference substance. - it is the ratio of densities, is a dimensionless quantity How a liquid's density compares to that of water at 4C is called its specific gravity. If a liquid has a specific gravity of 0.9, then its density is 0.9 times that of water, or 0.9 x 1000 = 900 kg/m3.
Specific Volume
- the specific volume of a substance is the ratio of the
substance's volume to its mass. It is the reciprocal of density and is an intrinsic property of matter:
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1 problems 4
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sample
and solutions
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Sample 1
The specific gravity of gold is 19. (a) What is the mass of 1 cm 3 of gold? (b) What is the weight of 1 in. 3 of gold?
) Since the density of water is 1 g/cm 3, the density of gold is 19 g/cm 3 and 1 cm 3 has a mass of 19 g.
(a
(b) Since the weight density of water is 62 lb/ft 3, the weight density of gold is dg = (19)(62) lb/ft 3 = 1200 lb/ft 3. Because 1 ft 3 = (12 in.)(12 in.)(12 in.) = 1728 in. 3, a cubic inch of gold weighs
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Sample 2
The density of mammals is roughly the same as that of
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Sample 3
How much does the air in a room 12 ft square and 10 ft
high weigh? The weight density of air is 0.08 lb/ft 3 at sea level.
The volume of the room is V = (12 ft)(12 ft)(10 ft) = 1440 ft 3. Hence the weight of the air is
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Sample 4
Two liquids of different densities (1=1500 kg/m, 2=500
kg/m) are poured together into a 100 L tank, filling it. If the resulting density of the mixture is 800 kg/m.Find the respective quantities of liquids used.
Solution: V1+V2=Vmix =m/v M= x V m1/1 + m2/2=100 L ( 1m/ 1000L) m1/1500 Kg/m + m1/ 500 Kg/m=0.1 m
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Pressure
- is defined as the ratio of force per unit area
where the force is perpendicular to the cross sectional area. Pressure is a scalar quantity measured in Pascals,where 1 Pa = 1 N/m2
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Pressure
Pressures are properly expressed in pascals (1 Pa = 1 N/m 2) or in pounds per square foot, but other units are often used:
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GAUGE PRESSURE
Pressure gauges measure the difference between an
unknown pressure and atmospheric pressure. What they measure is known as gauge pressure, and the true pressure is known as absolute pressure:
A tire whose gauge pressure is 2 bar contains air at an absolute pressure of about 3 bar, since sea-level atmospheric pressure is about 1 bar.
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Samples
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Sample 1
which has a circular base 1 cm in radius. How much pressure does she exert on the ground?
The area of the heel is A = r 2 = 3.14 cm 2 = 3.14 10 4 m 2. Hence the pressure is
Since atmospheric pressure is 1.013 bar, this pressure is 20 times atmospheric pressure.
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Sample 2
The weight of a car is equally supported by its four tires. The gauge
pressure of the air in the tires is 2.0 bar and each tire has an area of 140 cm 2 in contact with the ground. What is the mass of the car?
The load on each tire consists of one-quarter of the cars weight plus the weight of the column of air directly above the area of the tire in contact with the ground, since this part of the tire has no air under it to provide an equal upward force. Therefore, only the gauge pressure of the air in the tires, which is the excess over atmospheric pressure, acts to support the cars weight. Since p gauge = 2.0 bar = 2.0 10 5 Pa, each tire supports a weight of
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Sample 3
The flat roof of a house is 30 ft long and 25 ft wide and
weighs 15,000 lb. Before a severe storm the doors and windows of the house are closed so tightly that the air pressure inside remains at a normal 14.7 lb/in. 2even when the outside pressure falls to 14.3 lb/in. 2. Compare the upward force on the roof with its weight.
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Solution (3)
The area of the roof is A = (30 ft)(25 ft) = 750 ft 2. The
difference between the pressure on the inside and outside of the roof is p = (14.7 14.3) ft/in. 2 = 0.4 lb/in. 2. Because the pressure on the inside of the roof is greater, the net force on it is upward with the magnitude
This is nearly three times the roofs weight. If the roof is not securely attached to the walls of the house and if the windows do not break first, the roof will be lifted off during the storm. Evidently a building should not be sealed when a large drop in pressure due to a storm is forecast.
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between two adjacent level surfaces in the liquid is given by hpg, where h represents the vertical distance between the two surfaces.
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Pressure is a useful quantity where fluids (gases and liquids) are concerned because of the following properties of fluids: 1. The forces that a fluid exerts on the walls of its container, and those that the walls exert on the fluid, always act perpendicular to the walls. 2. The force exerted by the pressure in a fluid is the same in all directions at a given depth. 3. An external pressure exerted on a fluid is transmitted uniformly throughout the fluid. This does not mean that pressures in a fluid are the same everywhere, because the weight of the fluid itself exerts pressures that increase with increasing depth. The pressure at a depth h in a fluid of density d due to the weight of fluid above is Hence the total pressure at that depth is When a body of fluid is in an open container, the atmosphere exerts an external pressure on it.
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Sample 1
The interior of a submarine located at a depth of 50 m in
seawater is maintained at sea-level atmospheric pressure. Find the force acting on a window 20 cm square. The density of seawater is 1.03 10 3 kg/m 3. The pressure outside the submarine is p = p atm + dgh, and the pressure inside is p atm. Hence the net pressure p acting on the window is
Since the area of the window is A = (0.2m)(0.2m) = 0.04 m 2, the force acting on it is
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Pascals principle states that in a confined fluid, an externally applied pressure is transmitted uniformly in all direction.
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In a hydraulic system, Pascals principle is applied as a force multiplier. The force multiplier of a hydraulic system can be represented by the equation:
Output force Input force = Output piston area Input piston area
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APPLICATION
A hydraulic lift for automobiles is an
example of a force multiplied by hydraulic press, based on Pascal's principle. The fluid in the small cylinder must be moved much further than the distance the car is lifted.
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Problem Solving
Example 1: In hydraulic brake, a force of 80 N is applied to a piston with area of 4 cm2. 1. What is the pressure transmitted throughout the liquid? 2. If the piston at the wheel cylinder has an area of 8 cm2, what is the force exerted on it?
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Solution
(a) P = F/A = 80 N/4 cm2 = 20 N cm-2
F=PxA = 20 N cm-2 x 8 cm2 = 160 N
(b)
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Problem Solving
Example 2: The figure shows a 10 N weight balancing a X N weight placed on a bigger syringe. What is the value of X ?
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Solution:
F1/ A1 = F2/ A2 10 N / 1.5 cm2 = X N / 4.5 cm2 Therefore X = 10 / 1.5 x 4.5 = 30 N
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Problem Solving
Example 3:
The mass of X is 2 kg. It is placed at a piston A. The cross section areas of A and B are 5 cm2 and 80 cm2 respectively.
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Problem Solving
(a) Calculate the force which acts on piston A
(b) Find the pressure which is exerted on piston B. (c) Find the mass of Y which can be lifted by piston B. (d) If piston A moves down by 20 cm, then piston B will
go up by
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Solutions
(a) Calculate the force which acts on piston A F = mg = 2 x 10 = 20 N (b)Find the pressure which is exerted on piston B. P = F/A = 20 N / 5 x 10-4 m2 = 40 000 N m-2
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Solutions
(c)Find the mass of Y which can be lifted by piston B. F1/A1 = F2/A2 2 x 10/ 5 x 10-4 = m x 10 / 80 x 10-4 m = 32 kg (d)If piston A moves down by 20 cm, then piston B will go up by 5 x 20 cm = 80 x l cm3 l = 1.25 cm
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A little history In the first century BC, the Roman architect Vitruvius related a story of how Archimedes, the Greek scientist, discovered that a goldsmith had tried to cheat King Hiero II The king had given the goldsmith a particular amount of gold to melt down and make into a crown. When the crown was made and returned to the king, the king was suspicious that the goldsmith had stolen some of the gold and replaced it with an equal weight of silver. The king turned to Archimedes for help
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So, what, exactly, had he found? He realized that if the bath were completely filled in the beginning, the volume of water that would overflow (that was displaced) would have to equal the volume of the person or object placed into the water!! He now had a way to measure the volume of the irregularly-shaped crown He discovered that the crown displaced more water than a chunk of gold of equal weight did. Its volume was greater because it contained some silver, a metal less dense than gold!!
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Imagine a very light, thin-walled sack filled with water that is in equilibrium in a pool. upward force Clearly, there must be an _______ exerted on the sack to balance its weight (the pull of gravity down) This upward force is actually the vector sum of all the forces acting on the object due to the surrounding water, and is Pressure is greater buoyant force. called the _________ on bottom of sack!
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The magnitude of the buoyant force is given by Archimedes Principle. It states that A body fully or partially submerged in a fluid is buoyed weight of up by a force that is equal to the _______ the ___________________. displaced fluid
Archimedes Principle
more detail, consider the simple case of a cubic shaped object (height h, cross sectional area A) floating submerged in a fluid of density fluid as in the figure.
Use the definition of pressure in terms of area
cube causes a downward force Fdown = PtopA on the upper face. The pressure Pbot at the bottom of the cube causes an upward force Fup = PbotA on the lower face.
Total force the fluid exerts on the cube: B = (Pbot Ptop)A (1) Use the dependence of pressure on depth: (Pbot Ptop) = fluidgh (2) Combining (1) & (2) gives B = fluidghA = fluidgV = Mg. Mg is the weight of the FLUID displaced by the object!
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An Example:
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Wobject __ > Fb
So
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Weight of canoe plus occupants = weight of displaced fluid ___________________ (maximum possible shaded in red) Electrical Engineering Department 71
Imagine that the canal is filled with water, and then the ship is slowly lowered into the canal. If the shape of the canal exactly matches the ship, and if the canal is slightly larger than the ship, then all but a thin layer of water all the way around will be displaced. So. this thin layer between the ship and the canal is really all that is necessary!!
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To dive, a sub must become heavier, so the tanks water in while venting ____ air out. allow _______
To surface, a sub must become lighter. A supply of water compressed air on the sub is used to force ______ back out of the ballast tanks.
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Six years to build it, 500 million euros later, and 918 meters long.....
This is a channel-bridge over the River Elbe and joins the former East and West Germany, as part of the unification project. It is located in the city of Magdeburg, near Berlin. The photo was taken on the day of inauguration. Did this bridge have to be designed to withstand the additional weight of ship and barge traffic, or just the weight of the water?
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Answer:
It only needs to be designed to withstand the weight of the water!
Why?
A ship always displaces an amount of water that weighs the same as the ship, regardless of how heavily a ship may be loaded.
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Sample 1
when it is immersed in seawater? The density of iron is 7.8 10 3 kg/m 3 and that of seawater is 1.03 10 3 kg/m 3.
The volume of the anchor is
Thus the buoyant force on the anchor is 129 N, and the net force needed to support it in seawater is
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Fb Wiceberg
water gVsubmerged ice gVtotal
Vsubmerged Vtotal
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Gases are similar to liquids in that they flow; hence both are called fluids. The primary difference between gases and liquids is the distance between molecules. In a liquid , the molecules are close together, where they continually experience forces from the surrounding molecules. These forces strongly affect the motion of the molecules. In a gas, the molecules are far apart, allowing them to move freely between collisions. When two molecules in a gas collide, if one gains speed in the collisions, the other loses speed, such that their kinetic energy is unchanged.
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Atmospheric Pressure The atmosphere is a layer of air surrounding the earth; its thickness has been estimated as about 500 to 600 mi. The density of the air decreases with increasing altitude. Since air has weight, this layer of air produces a pressure, called the atmospheric pressure, at the surface of the earth. The atmospheric pressure varies from day to day by about 5 per cent, the variations often accompanying changes in the weather. The pressure of the air is measured by a barometer, which often consists of an evacuated tube inverted in a dish of mercury, as shown in Figure.
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Mercury Barometer
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Barometers
The atmosphere exerts a pressure P on the open surface of the mercury in the dish, and this is transmitted to the liquid in the tube. This pressure is balanced by the pressure due to the mercury in the tube at a height h above the open surface of the dish. Recall that the pressure is always the same at any level surface in a liquid. Outside the barometer tube the pressure at this surface is entirely due to the atmosphere, so that the pressure of the mercury here is atmospheric pressure. Hence the pressure of the mercury within the tube at the level of the surface of the mercury in the dish is also atmospheric pressure. Knowing the density of mercury and the height to which the column of mercury rises within the evacuated barometer tube.
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Boyles Law
Pressure depends on density of the gas
Pressure is just the force per unit area exerted by the
molecules as they collide with the walls of the container Double the density, double the number of collisions with the wall and this doubles the pressure
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Boyles Law
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Boyles Law
Density is mass divided by volume. Halve the volume and you double the density and thus the pressure.
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Boyles Law
At a given temperature for a given quantity of gas, the
P V P V 1 1 2 2
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Boyles Law
Illustrative sample
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Fluid Flow
Fluid flow can be described in terms of two main types: streamline flow and turbulent flow.
Streamline flow, also known as Laminar flow, is illustrated here -- flow through a pipe and flow around an airplane wing.
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Fluid Flow
In streamline flow, the motion of a particle after it passes a particular point is the same as the motion of the particle that preceded it at that point. The path that a particle takes is called a stream line. Every particle that passes any particular point will follow the stream line that goes through that point. A bundle of stream lines, like the ones here, is known as a stream tube. Fluid never crosses the surface of a stream tube.
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Fluid Flow
Turbulent flow is illustrated here-flow through a small constriction in a pipe and flow around an airplane wing which is inclined at a steep angle. In turbulent flow, the motion of a particle after it passes a particular point may be quite different from the motion of the particle that preceded it at that point. Turbulent flow is characterized by randomness or irreproducibility of the motion of individual particles. It usually occurs in fluids moving at high speeds. As you might expect, friction is far greater in turbulent flow. We will concentrate most of our attention on streamline flow.
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VISCOSITY
The resistance to flow of a fluid and the resistance to the movement of an object through a fluid are usually stated in terms of the viscosity of the fluid. Experimentally, under conditions of laminar flow, the force required to move a plate at constant speed against the resistance of a fluid is proportional to the area of the plate and to the velocity gradient perpendicular to the plate. The constant of proportionality is called the viscosity
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where the script F is the volume flowrate through the tube. This volume flowrate can also be expressed by
where L is the length of the pipe, r is its radius, p is the pressure difference between the ends of the pipe, and is the viscosity of the liquid. Evidently the rate of flow depends most strongly on the radius of the pipe.
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Sample Problem
The grease nipple on a bearing has a hole 1 mm in diameter and 6 mm long. The grease being used has a viscosity of 80 P. How much pressure is needed to force 0.2 cm 3 of grease into the nipple in 5 s?
4 3
Here r = 0.5 mm = 5 10
m, L = 6 mm = 6 10
m, and
Water
0.01
Viscosity has the SI units Pascal seconds (Pa s) which is called the Poiseuille. More commonly used is the dyne sec/cm2 which is called Poise. One Pa s is 10 Poise. The Poise is used in the table because of its more common usage. Data from Gustafson. These viscosities are at 20C except for the blood and blood plasma which are at body temperature, 37C, and for steam which is at 100C.
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Surface Tension
The surface tension of water provides the necessary wall tension for the formation of bubbles with water. The tendency to minimize that wall and of liquid droplets. tension pulls the bubbles into spherical shapes (LaPlace's law).
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bubble depends upon the surface tension and the radius of the bubble. The relationship can be obtained by visualizing the bubble as two hemispheres and noting that the internal pressure which tends to push the hemispheres apart is counteracted by the surface tension acting around the circumference of the circle. For a bubble with two surfaces providing tension tension, the pressure relationship is:
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Capillary Action
Capillary action is the result of adhesion and surface tension. Adhesion of water to the walls of a vessel will cause an upward force on the liquid at the edges and result in a meniscus which turns upward. The surface tension acts to hold the surface intact, so instead of just the edges moving upward, the whole liquid surface is dragged upward.
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Capillary action occurs when the adhesion to the walls is stronger than the cohesive forces between the liquid molecules. The height to which capillary action will take water in a uniform circular tube is limited by surface tension. Acting around the circumference, the upward force is
The height h to which capillary action will lift water depends upon the weight of water which the surface tension will lift:
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Cohesion is the tendency of certain similar particles to cling together due to attractive forces. While adhesion is the tendency of certain dissimilar particles to cling together due to attractive forces.
The water surface is curving upward and it wets the tube wall because the attractive force among the waters particles is smaller than the attractive force between the waters particles and the tube wall (cohesion is smaller than adhesion) On the other hand, the mercurys surface is curving downward and does not wets the tube wall because the attractive force among the mercurys particles is greater than the attractive force between the mercurys particles and the tube wall (cohesion is greater than adhesion).
Water
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Equation of Continuity
It is common for R to be expressed in such units as gallons per minute and liters per second instead of the proper units of cubic feet per second and cubic meters per second (1 U.S. gal = 0.134 ft 3 and 1 L = 10 3 m 3 = 10 3cm 3).
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Equation of Continuity
We are not creating nor destroying mass. The mass m1 that flows into a region must equal the mass m2 that flows out of the region. That is,
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SOLVED PROBLEM
1. Water flows through a pipe at the rate of 0.35 m3/min. What is its speed where the pipe has a diameter of 1.5 cm? Where the diameter is 2.5 cm?
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SOLVED PROBLEM
1. A garden hose has an inside diameter of 12 mm, and water flows through it at 2.5 m/s. (a) What nozzle diameter is needed for the water to emerge at 10 m/s? (b) At what rate does water leave the nozzle?
(a) The cross-sectional areas of hose and nozzle are in the same ratio as the squares of their diameters,
since A =r 2 = d 2/4. From v 1A 1 = v 2A 2 we obtain (b) The rate of flow is Because 1 m 3 = 10 3 liters (L), R = 0.283 L/s. The same result, of course, would be obtained from R = v 2A 2.
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Bernoulli's Equation
From the Equation of Continuity,
we know that the fluid must be moving slower at position 1 where the cross section A1 is larger and it must be moving faster at position 2 where cross section A2 is smaller. That is, the fluid must accelerate as is moves from position 1 to position 2. That means the pressure on the fluid at position 1 must be greater than the pressure at position 2 in order to provide a net force to cause this acceleration.
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Bernoulli's Equation
This is an example of Bernoulli's Principle that the pressure exerted by a moving fluid is greater where the speed of the fluid is smaller and the pressure is smaller where the speed of the fluid is greater. Now consider fluid that flows -- along a stream tube -- with a change in cross sectional area and a change in height. Work must be done on the fluid to change its kinetic energy and its potential energy.
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Bernoulli Equation
The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy.
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Bernoulli's Equation
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Bernoullis Equation
According to this equation, which is derived from the law of conservation of energy, the quantity
has the same value at all points in the motion of such a fluid,
where p is the absolute pressure, h is the height above an arbitrary reference level, and v is the fluid velocity. Thus at the two locations 1 and 2
The quantity dgh is the potential energy of the fluid per unit volume, and is its kinetic energy per unit volume. Each term of this equation has the units of pressure.
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Another form of Bernoullis equation is obtained by dividing each term of the above equation by dg, which gives
Each term of this equation has the dimensions of a length and is called a head, where
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Solved Problems
Water flows through the pipe shown at the rate of 80
L/s. If the pressure at point 1 is 180 kPa, find (a) the velocity at point 1, (b) the velocity at point 2, and (c) the pressure at point 2.
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(b) From v 1A 1 = v 2A 2 we have (c) We now substitute the known quantities p 1, v 1, and v 2 with h 1 = 0 and h 2 = 2 m into Bernoullis equation This yields
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Solved Problems
A horizontal pipe 1 in. in radius is joined to a pipe 4 in. in radius, as in Fig. 17-3. (a) If the velocity of seawater (d = 2.00 slugs/ft 3) in the small pipe is 20 ft/s and the pressure there is 30 lb/in. 2, find the velocity and pressure in the large pipe. (b) What is the rate of flow through the pipe in pounds per minute?
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(a) The cross-sectional areas of the pipes are in the same ratio as the squares of their radii, since A = r 2. Fromv 1A 1 = v 2A 2 we obtain pipes are horizontal, h 1 = h 2, and Bernoullis equation becomes Because both
which is (b) min = 60 s, the rate of flow in the required units is Since dg = 64 lb/ft 3 and 1
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Torricellis Theorem
Torricelli's Theorem is Bernoulli's equation with certain assumption made. This deals with the set up where you have a large tank with a narrow opening allowing for the the liquid to flow out. Both the Tank and the the narrow opening (nozzle) are open to the atmosphere. The fluid velocity of the tank (water level) is very much slower than the fluid velocity of the nozzle.
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Torricellis Theorem
The pressure will be the same because they are open to the atmosphere. The fluid velocity at region 2 is much slower then the fluid velocity at region 1. Therefore the fluid velocity at region 2 is negligible. The value of y1=0. The equation simplifies to
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Sample Problems
A barrel 80 cm high is filled with kerosene. When a
tap at the bottom of the barrel is opened, with what velocity does the kerosene emerge?
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Sample Problems
A boat strikes an underwater rock that punctures a
hole 20 cm 2 in area in its hull 1.5 m below the waterline. At what rate does water enter the hull?
From Torricellis theorem, the velocity with which water enters the hull is through an orifice of area A is R = vA when the fluid velocity is v, 10 3m 2, so water enters the hull at the rate . Since the rate of flow . Here A = 20 cm 2 = 2
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