# Unit 7. Pressure in fluids

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1 -- Unit 7. Pressure in fluids Index 1.- Pressure Fluids Pressure in fluids Pascal's principle Archimedes principle Atmospheric pressure Torricelli and the barometric pressure...8 Practice exam...9 Page 1 of 10

2 1.- Pressure Pressure (symbol: P) is the ratio of force to the area over which that force is distributed. Unit SI: The SI unit of pressure is the newton per square metre, which is called the pascal (Pa) A pressure of 1 Pa is small; it approximately equals the pressure exerted by a dollar bill resting flat on a table, so we often use other units such as: 1 atm = 760 mm Hg = 1013 mbar = Pa P = F/A Activities: 1.- Who exerts a bigger pressure on the ground? a) a two-tonne elephant that is standing on just one of its legs, which has a surface of 500cm2. b) a fifty-kilogram ballerina standing on the toes of one of her feet, with a surface of 3cm2. (Answer: elephant: Pa Ballerina: 1, Pa) 2.- Calculate the pressure produced by a force of 800 N acting on an area of 2.0 m 2 Sol: 400 Pa 3.-. A boy of weight 500 N has feet with a total area of 200 cm 2. What is the pressure between him and the ground if he stands on both feet? Sol: Pa 4. The pressure between a car tyre and the road is Pa. If the car has a weight of N what is the area of contact between the tyre and the road? Sol: 0.1 m 2 5. The pressure in a boiler is Pa. If the area of one end is 0.6 m 2. What is the force on that end? Sol: N 2.- Fluids - What is it? Liquids and gases - Characteristics: mass volume shape Can they flow? Can be mixed? Can be compressed? solid liquid gas constant or variable? constant or variable? constant or variable? Yes or no? Yes or no? Yes or no? Page 2 of 10

3 - Density It is the relationship between the mass of a body and the volume it occupies d = m/v DENSITY is a physical property of matter, as each element and compound has a unique density associated with it. Density Comparison to Water: In chemistry, the density of many substances is compared to the density of water. Does an object float on water or sink in the water? If an object such as a piece of wood floats on water it is less dense than water vs. if a rock sinks, it is more dense than water. Ice: Everyone knows that ice floats on water, but did you know that this is an abnormal physical property of solid/liquid state of water? The more normal physical property is for the solid of a compound to sink in its own liquid. 3.- Pressure in fluids Pressure is defined as force divided by the area on which the force is pushing. You can write this as an equation, if you wanted to make some calculations: where P = F/A P = pressure F = force A = area Pressure due to gravity Since the weight of an object or material is equal to the force it exerts due to gravity, An object can exert downward pressure due to its weight and the force of gravity. The pressure you exert on the floor is your weight divided by the area of the soles of your shoes. If the force is due to the weight (W) of the object, the equation is then: P = W / A W = m g = d V g then P = d g h because h = V/A Static Fluid Pressure The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the fluid, and the acceleration of gravity. The pressure in a static fluid arises from the weight of the fluid and is given by the expression P static fluid = dgh where ρ = m/v = fluid density g = acceleration of gravity h = depth of fluid Page 3 of 10

4 Sea water density = kg/m 3 Sol Pa Physics and chemistry The pressure from the weight of a column of liquid of area A and height h is The most remarkable thing about this expression is what it does not include. The fluid pressure at a given depth does not depend upon the total mass or total volume of the liquid. The above pressure expression is easy to see for the straight, unobstructed column, but not obvious for the cases of different geometry which are shown. Activities: 6.- Find the difference in pressure that can be found between two places situated 5m and 9 m deep in sea water (d= 1, kg/m3). This pressure should be expressed in Pascals, atmospheres and mm of mercury. 7.- Which pressure must sea animals living at a depth of 5 000m bear? Answer: 40,376 Pa, atm, mm Hg 8.- The maximum pressure a human being can bear is 8 atm. How deep can a person go down into the sea without risk? (1 atm = Pa) Sol: 80.3 m 9.-A bath tap is round and has a 4 cm radius. If we fill the bathtub with water to a height of 50 cm, calculate the force that must be applied because of the pressure inside the liquid to lift the plug. Area of a circle =. R 2 Sol: 24.6 N 10. Calculate the difference between the pressures that must be born by two fish in a reservoir if one is 5 metres above the other. Sol: Pa 11. The side window of a submarine has an 80 cm diameter. Calculate the force that is borne by it when the submarine is at a depth of 8 km (sea water density kg/m 3.) Area of a circle =. R 2. Sol: Pa 12. You might have read that if a car falls into a river or into the sea, we must wait until it has filled up with water before opening its door. Why do you think this is so? Page 4 of 10

5 4.- Pascal's principle Pascal s principle says that given a fluid in a totally enclosed system, a change in pressure at one point in the fluid is transmitted to all points in the fluid, as well as to the enclosing walls. In other words, if you have a fluid enclosed in a pipe (with no air bubbles) and change the pressure in the fluid at one end of the pipe, the pressure changes all throughout the pipe to match. The fact that pressure inside an enclosed system is the same (neglecting gravitational differences) has an interesting consequence. Because P = F/A, you get the following equation for force: F = P A So if the pressure is the same everywhere in an enclosed system but the areas you consider are different, can you get different forces? A hydraulic system magnifies force. To make this question clearer, look at the figure, which shows a system of enclosed fluid with two hydraulic pistons, one with a piston head of area A 1 and one with a piston head of area A 2. You apply a force of F 1 on the smaller piston. What is the force on the other piston, F 2? Pressure at each point is F/A. According to Pascal s principle, the pressure is the same everywhere inside the fluid, so: Solving for F 2 gives you the force at Point 2: Page 5 of 10

6 That means that you can develop a huge force from a small force if the ratio of the piston sizes is big. For example, say the area of Piston 2 is bigger than Piston 1 by a factor of 100. Does that mean that any force you apply to piston 1 will be multiplied by 100 times on piston 2? Yes, indeed that s how hydraulic equipment works. By using a small piston at one end and a large piston at the other, you can create huge forces. Backhoes and other hydraulic machines, such as garbage trucks and hydraulic lifts, use Pascal s principle to function. What s the catch here? If you push on Piston 1 and get 100 times the force on Piston 2, you seem to be getting something for nothing. The catch is that you have to push the smaller piston 100 times as far as the second piston will move. Activities: 13.- Calculate the force that bust be applied on the small piston of a hydraulic press, if we need to lift a mass of 100 kg with the big one. The area of the small piston is 15 cm 2 and the area of the big one is 1200 cm 2. Which mass could we place on the small piston to get such a force? sol: 12,24 N; 1,25 kg 14.- The surfaces of the pistons in a hydraulic press are : 20cm 2 the small one and 500 cm 2 the big one. If we want to lift a mass of 2000 kg with it, a) Which force must we apply on the small piston? b) If we apply a maximum force of 900 N, which is the biggest amount of mass we would be able to lift? Draw it. 5.- Archimedes principle Sol: a) 784 N b) 2296 Kg Physical law of buoyancy, discovered by the ancient Greek mathematician and inventor Archimedes, stating that any body completely or partially submerged in a fluid (gas or liquid) at rest is acted upon by an upward, or buoyant, force the magnitude of which is equal to the weight of the fluid displaced by the body. The volume of displaced fluid is equivalent to the volume of an object fully immersed in a fluid or to that fraction of the volume below the surface for an object partially submerged in a liquid. B = W B = m L g B = d L V g The apparent weight is the weight of a body as affected by the buoyancy of a fluid in which it is immersed, being the true weight minus the weight of the displaced fluid B = W- W' Page 6 of 10

7 Activities: 15.- A solid body has a volume of m 3 and is placed in a liquid whose density is 1030kg/m 3. Calculate: a) The buoyancy it experiments. b) The apparent weight of the solid body, if you know that its mass is 0.15 kg. c) The density of the solid body. Answer: a) 0.27 N b) 1.2 N c) 5555 kg/m An objects weights 150 N in the air, 100 N in the water and 125 N in another liquid. a) Calculate the density of the object. b) What is the density of the other liquid? Answer: a) d = 3000 kg/m 3 ; b) d= 500 kg/m The body of a 70 Kg person takes up a volume of 60 L. Calculate if a life jacket of 30 L of volume of negligible weight can surely float. Prove it A wooden boat of 800 Kg/m 3 of density displaces a maximum volume of water of 200 l. What s the maximum weight we could put on it so that it doesn t sink? Sol: m = 40 Kg 19.- A piece of aluminium weights 26,5 N in the air and 18,6 in alcohol. The density of the alcohol is 800 Kg/m 3. Calculate the thrust it undergoes, its volume and density. Sol: B = 7.9 N V = 10-3 m 3 d= 2700 kg/m A stone 10-3 m 3 of volume and 2600 Kg/m 3 of density sinks into the water. Calculate the weight of the stone in the air, the thrust it undergoes in the water and the apparent weight in that liquid. Sol: W = N B = 9.8 N W = N 21.- When you hang a small object from a dynamometer it measures 1,4N and when you sink it completely in water it measures 1,0 N. What s the volume of the object and its density? V = m 3 d = 3571 kg/m An object of 5 Kg of mass and 2 L of volume weights 12 N when being submerged in a liquid of an unknown density. Calculate the density of the liquid. Sol: 1887 kg/m A stone of 0,5 Kg has an apparent weight of 3N when being submerged into the water. Calculate the volume and density of this object. 6.- Atmospheric pressure V = m 3 d = 2579 kg/m 3 Air pressure is the force exerted on you by the weight of tiny particles that make up the air. Although this particles are invisible, they still have weight and take up space. Since there's a lot of "empty" space between molecules, air can be compressed to fit in a smaller volume. Weather forecasters measure air pressure with a barometer. Barometers are used to measure the current air pressure at a particular location in "mm of mercury" or in "millibars" (mb). Page 7 of 10

8 How much pressure are you under? Earth's atmosphere is pressing against each square inch of you with a force of 1 kilogram per square centimeter (14.7 pounds per square inch). The force on 1,000 square centimeters (a little larger than a square foot) is about a ton! Why doesn't all that pressure squash me? Remember that you have air inside your body too, that air balances out the pressure outside so you stay nice and firm and not squishy. Air pressure can tell us about what kind of weather to expect as well. If a high pressure system is on its way, often you can expect clear skies. If a low pressure system is coming, then look for storms and rain. Pressure caused by the weight of the atmosphere. At sea level it has a mean value of one atmosphere but reduces with increasing altitude Torricelli and the barometric pressure P = d g h Galileo's secretary and scientific successor, Evangelista Torricelli, helped prove the existence of air pressure through a famous experiment that he did with a mercury-filled glass tube. Some of Galileo's contemporaries had built simple barometers, using water in long tubes, to try to prove the existence of a vacuum. But in 1643, Torricelli took the barometer idea a step further. He realized that air--contrary to contemporary thought--was not weightless and that the rise and fall of liquid in a barometric tube was related to air pressure. To show this, he developed an experiment using liquid mercury or "quicksilver." Since mercury is about 14 times as heavy as water, Torricelli was able to use a much shorter tube than the earlier water barometers required. He filled the tube with mercury and then inverted it into a dish of mercury. Some of the mercury drained from the tube, but then it stopped at a certain level--14 times less than the level water stopped at, to be exact. Torricelli postulated that air exerted pressure on the mercury in the dish, pushing hard enough on the surface area to keep most of the mercury inside the tube. Thus, it was not the attraction of the vacuum at the top of the tube which held the mercury inside (as other scientists thought), but rather, the phenomenon was the result of the pressure exerted by air. Thus, the barometer could be used to measure air pressure. Later generations of scientists, most notably Blaise Pascal, developed the barometer further. You can demonstrate air pressure's effects through an experiment similar to Torricelli's. Stick a plastic straw in a glass of juice or other colored drink, and suck enough liquid into the straw to fill it about halfway up. When you suck on the straw, a partial vacuum is created in the top of the straw. Air pressure on the liquid in the glass forces the juice up the straw and into your mouth. Now, hold your finger over the top of the straw and slowly pull the straw out of the glass. While your finger is pressed over the top, a partial vacuum is maintained in the top of the straw. The air pressure is greatest underneath the straw and will keep the liquid from dripping out. Activities: 24.- What is the value of the atmospheric pressure that Torricelli got, having into account that the above pressure is the same as the one exerted by a column of mercury 76 cm high (density= Kg/m 3 )? Calculate it in Pascals. Sol: Pa Page 8 of 10

9 25.- How high would a liquid go in Torricelli s column if we used: a. water? b. carbon tetrachloride (density 1600 Kg/m 3 )? Sol: a m b m 26.- Calculate the variation in the pressure experienced for every 10 metres we raised into the atmosphere, supposing that it s homogeneous. (0,001 Kg/m 3 ) and the variation of pressure in the water when getting 10 m deeper (density 1000 Kg/m 3 ). Sol: air Pa water Pa 27.- Change the following units: 750 mm Hg atm 1,2 atm mm Hg 1,2 atm pa 76 cm Hg mm Hg 1, Pa atm. Sol: atm b. 912 mm Hg c Pa d. 760 mm Hg e atm. Page 9 of 10

10 Practice exam 1. In a hydraulic press the large piston has a 1m 2 area, and the small piston 0,1m 3. We want to raise a 100 kg mass. a) Where should the mass be placed? b) Which force should be exerted? c) What will the pressure be on the minor piston?; and on the large one? Sol: b) 98 N c) 980 Pa 2. A suspended body weighs 100 N and 75 N when it is sunk in water. Calculate: its thrust, its volume and density. Sol: 25 N, V= m 3 d = 4000 kg/m 3 3.A submarine goes down to a 100 m depth in the sea, where density is 1030kg/m 3. a) Calculate the pressure exerted on the submarine and state it in atmospheres and in mm Hg b) Which force will have a crew member to use to open a 0,5m 2 surface hatch? c) How many Kg will equal the the force used to raise it? Sol: a) Pa 9.96 atm 7573 mm Hg b) N c) kg 4. A hot air balloon dragging a 300 kg crew basket has a Helium volume of 2000m 3. a) Which is the total weight of the balloon? b) Calculate the balloon thrust c) Can the balloon hold itself in the air? d air =1,3 kg/m 3 ; d Helium =0,2 kg/m 3 sol: 700 kg b) N c) yes because 6860 N < N 5. What pressure will a sewing machine needle top exert if it is pushed by a 50 N force and it has a 0,1mm 2 surface? Calculate also the weight of an iron block, standing on a one square meter surface to exert that same pressure. Sol Pa Kg 6. A body having a 4000 kg/m 3 density weights 100N in the air and 80 N when it is sunk in a unknown liquid. Calculate the density of that liquid. Sol: 800 Kg Page 10 of 10

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