Chapter 11: Fluid Statics PDF

Chapter 11: Fluid Statics Mrs. Pooja Brahmaiahchari What Is a Fluid ? Matter most commonly exists as a solid, liquid, gas. These states are known as the common phases of matter. Solids have a definite shape and a specific volume; liquids have a definite volume but their shape changes de...

Chapter 11: Fluid Statics Mrs. Pooja Brahmaiahchari What Is a Fluid ? Matter most commonly exists as a solid, liquid, gas. These states are known as the common phases of matter. Solids have a definite shape and a specific volume; liquids have a definite volume but their shape changes depending on the container in which they are held. Gases have neither a definite shape nor a specific volume as their molecules move to fill the container in which they are held.  Liquids, gases are considered to be fluids because they yield to shearing forces, whereas solids resist them. Note that the extent to which fluids yield to shearing forces (and hence flow easily and quickly) depends on a quantity called the viscosity.  Atoms in solids are in close contact, with forces between them that allow the atoms to vibrate but not to change positions with neighboring atoms. Thus, a solid resists all types of stress. A solid cannot be easily deformed because the atoms that make up the solid are not able to move about freely. Solids also resist compression, because their atoms form part of a lattice structure in which the atoms are a relatively fixed distance apart. A rock is an example of a solid. This rock retains its shape because of the forces holding its atoms together.  In contrast, liquids deform easily when stressed and do not spring back to their original shape once the force is removed because the atoms are free to slide about and change neighbors—that is, they flow (so they are a type of fluid), with the molecules held together by their mutual attraction. When a liquid is placed in a container with no lid on, it remains in the container. The atoms are closely packed, liquids, like solids, resist compression.  Atoms in gases are separated by distances that are large compared with the size of the particles. The forces between the particles are therefore very weak, except when they collide with one another. Gases thus not only flow (and are therefore considered to be fluids) but they are relatively easy to compress because there is much space and little force between the particles. When placed in an open container gases, unlike liquids, will escape. The major distinction is that gases are easily compressed, whereas liquids are not. Density Which weighs more, a ton of feathers or a ton of bricks? A ton is a ton, of course; but bricks have much greater density than feathers, and so we are tempted to think of them as heavier. Density is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid. Density is the mass per unit volume of a substance or object. In equation form, density is defined as 𝑚 𝜌= 𝑣 where the Greek letter 𝜌 (rho) is the symbol for density, m is the mass, and v is the volume occupied by the substance.  In the riddle regarding the feathers and bricks, the masses are the same, but the volume occupied by the feathers is much greater, since their density is much lower. The SI unit of density is kg/m3. The metric system was originally devised so that water would have a density of 1 g/cm3, equivalent to 103 kg/m3 1. Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?(ρ = 19.32 g/cm3 for gold) 2. What is the density of a 240-g rock that displaces 89.0 cm3 of water? 3. A rectangular gasoline tank can hold 80.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? Pressure Pressure is defined as the force divided by the area perpendicular to the force over which the force is applied, or 𝐹 𝑃= 𝐴 Where F is a force applied to an area A that is perpendicular to the force. The SI unit for pressure is the pascal, where: 1 pa = 1N/m2 (a) While the person being poked with the finger might be irritated, the force has little lasting effect. (b) In contrast, the same force applied to an area the size of the sharp end of a needle is great enough to break the skin.  We know that fluids cannot withstand shearing (sideways) forces; they cannot exert shearing forces, either. Fluid pressure has no direction, being a scalar quantity. The forces due to pressure have well-defined directions: they are always exerted perpendicular to any Pressure inside this tire exerts forces perpendicular to all surfaces it contacts. The arrows give representative directions and magnitudes of the forces exerted at various points. Note that static fluids do not exert shearing forces.  Pressure is exerted on all sides of this swimmer, since the water would flow into the space he occupies if he were not there. The arrows represent the directions and magnitudes of the forces exerted at various points on the swimmer. Note that the forces are larger underneath, due to greater depth, giving a net upward or buoyant force that is balanced by the weight of the swimmer. 4. The pressure exerted by a phonograph needle on a record is surprisingly large. If the equivalent of 1.00 g is supported by a needle, the tip of which is a circle 0.200 mm in radius, what pressure is exerted on the record in N/m2 ? Variation of Pressure with Depth in a Fluid If your ears have ever popped on a plane or ached during a deep dive in a swimming pool, you have experienced the effect of depth on pressure in a fluid. At the Earth’s surface, the air pressure exerted on you is a result of the weight of air above you. This pressure is reduced as you climb up in altitude and the weight of air above you decreases. Under water, the pressure exerted on you increases with increasing depth. In this case, the pressure being exerted upon you is a result of both the weight of water above you and that of the atmosphere above you.  You may notice an air pressure change on an elevator ride that transports you many stories, but you need only dive a meter or so below the surface of a pool to feel a pressure increase. The difference is that water is much denser than air, about 775 times as dense. The bottom of this container supports the entire weight of the fluid in it. The vertical sides cannot exert an upward force on the fluid (since it cannot withstand a shearing force), and so the bottom must support it all.  That pressure is the weight of the fluid divided by the area supporting it  The equation in the previous slide represents the pressure due to the weight of any fluid of average density ρ at any depth h below its surface. For liquids, which are nearly incompressible, this equation holds to great depths. For gases, which are quite compressible, one can apply this equation as long as the density changes are small over the depth considered. Atmospheric pressure is another example of pressure due to the weight of a fluid, in this case due to the weight of air above a given height.  The atmospheric pressure at the Earth’s surface varies a little due to the large-scale flow of the atmosphere induced by the Earth’s rotation (this creates weather “highs” and “lows”). However, the average pressure at sea level is given by the standard atmospheric pressure 𝑃 , measured to be 1 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑒 𝑎𝑡𝑚 = 𝑃 = 1.01 𝑥 10 𝑁⁄𝑚 = 101 𝑘𝑃𝑎 This relationship means that, on average, at sea level, a column of air above 1.00 m2 of the Earth’s surface has a weight of 1.01 x 105 N, equivalent to 1 atm. 5. What depth of mercury(𝜌 = 13.6 x 103 kg/m3) creates a pressure of 1.00 atm? 6. Calculate the pressure due to the ocean at the bottom of this trench, given its depth is 11.0 km and assuming the density of seawater is constant all the way down (1.025 x 103 kg/m3) Pascal’s Principle Can pressure be increased in a fluid by pushing directly on the fluid? Yes, but it is much easier if the fluid is enclosed. The heart, for example, increases blood pressure by pushing directly on the blood in an enclosed system. If you try to push on a fluid in an open system, such as a river, the fluid flows away. An enclosed fluid cannot flow away, and so pressure is more easily increased by an applied force.  What happens to a pressure in an enclosed fluid? Since atoms in a fluid are free to move about, they transmit the pressure to all parts of the fluid and to the walls of the container. Remarkably, the pressure is transmitted undiminished. This phenomenon is called Pascal’s principle, because it was first clearly stated by the French philosopher and scientist Blaise Pascal (1623– 1662): A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. Moreover, Pascal’s principle implies that the total pressure in a fluid is the sum of the pressures from different sources. Application of Pascal’s Principle One of the most important technological applications of Pascal’s principle is found in a hydraulic system, which is an enclosed fluid system used to exert forces. The most common hydraulic systems are those that operate car brakes. Let us first consider the simple hydraulic system shown in figure.  A typical hydraulic system with two fluid-filled cylinders, capped with pistons and connected by a tube called a hydraulic line. A downward force F1 on the left piston creates a pressure that is transmitted undiminished to all parts of the enclosed fluid. This results in an upward force F2 on the right piston that is larger than F1 because the right piston has a larger area. Relationship Between Forces in a Hydraulic System Note first that the two pistons in the system are at the same height, and so there will be no difference in pressure due to a difference in depth. Now the pressure due to F1 acting on area A1 is simply 𝑃 = as defined by 𝑃 = According to Pascal’s principle, this pressure is transmitted undiminished throughout the fluid and to all walls of the container. Thus, a pressure P2 is felt at the other piston that is equal to P1. That is P1= P2. Since 𝑃 = we see that, =  This equation relates the ratios of force to area in any hydraulic system, providing the pistons are at the same vertical height and that friction in the system is negligible. Hydraulic systems can increase or decrease the force applied to them. To make the force larger, the pressure is applied to a larger area. Gauge Pressure, Absolute Pressure, and Pressure Measurement If you limp into a gas station with a nearly flat tire, you will notice the tire gauge on the airline reads nearly zero when you begin to fill it. In fact, if there were a gaping hole in your tire, the gauge would read zero, even though atmospheric pressure exists in the tire. Why does the gauge read zero? Tire gauges are simply designed to read zero at atmospheric pressure and positive when pressure is greater than atmospheric. atmospheric pressure adds to blood pressure in every part of the circulatory system.  But atmospheric pressure has no net effect on blood flow since it adds to the pressure coming out of the heart and going back into it, too. What is important is how much greater blood pressure is than atmospheric pressure. Blood pressure measurements, like tire pressures, are thus made relative to atmospheric pressure. Gauge pressure is the pressure relative to atmospheric pressure. Gauge pressure is positive for pressures above atmospheric pressure, and negative for pressures below it.  In fact, atmospheric pressure does add to the pressure in any fluid not enclosed in a rigid container. This happens because of Pascal’s principle. The total pressure, or absolute pressure, is thus the sum of gauge pressure and atmospheric pressure: 𝑃 = 𝑃 + 𝑃 In most cases the absolute pressure in fluids cannot be negative. Fluids push rather than pull, so the smallest absolute pressure is zero. (A negative absolute pressure is a pull.) Thus, the smallest possible gauge pressure is 𝑃 = − 𝑃 (this makes zero). There is no theoretical limit to how large a gauge pressure can be.  An entire class of gauges uses the property that pressure due to the weight of a fluid is given by 𝑃 = ℎ𝜌𝑔 Consider the U-shaped tube shown in Figure, for example. This simple tube is called a manometer.  An open-tube manometer has one side open to the atmosphere. (a) Fluid depth must be the same on both sides, or the pressure each side exerts at the bottom will be unequal and there will be flow from the deeper side. (b) A positive gauge pressure 𝑃 = ℎ𝜌𝑔 transmitted to one side of the manometer can support a column of fluid of height. (c) Similarly, atmospheric pressure is greater than a negative gauge pressure 𝑃 by an amount ℎ𝜌𝑔. The jar’s rigidity prevents atmospheric pressure from being transmitted to the peanuts.  Mercury manometers are often used to measure arterial blood pressure. An inflatable cuff is placed on the upper arm as shown in Figure. By squeezing the bulb, the person making the measurement exerts pressure, which is transmitted undiminished to both the main artery in the arm and the manometer. When this applied pressure exceeds blood pressure, blood flow below the cuff is cut off. The person making the measurement then slowly lowers the applied pressure and listens for blood flow to resume. Blood pressure pulsates because of the pumping action of the heart, reaching a maximum, called systolic pressure, and a minimum, called diastolic pressure, with each heartbeat.  A barometer is a device that measures atmospheric pressure. This device measures atmospheric pressure, rather than gauge pressure, because there is a nearly pure vacuum above the mercury in the tube. The height of the mercury is such that ℎ𝜌𝑔 = 𝑃. When atmospheric pressure varies, the mercury rises or falls, giving important clues to weather forecasters. Archimedes’ Principle When you rise from lounging in a warm bath, your arms feel strangely heavy. This is because you no longer have the buoyant support of the water. Where does this buoyant force come from? Why is it that some things float and others do not? Do objects that sink get any support at all from the fluid? Is your body buoyed by the atmosphere, or are only helium balloons affected?  Answers to all these questions, and many others, are based on the fact that pressure increases with depth in a fluid. This means that the upward force on the bottom of an object in a fluid is greater than the downward force on the top of the object. There is a net upward, or buoyant force on any object in any fluid.  If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object will remain suspended at that depth. The buoyant force is always present whether the object floats, sinks, or is suspended in a fluid.  Just how great is this buoyant force? To answer this question, think about what happens when a submerged object is removed from a fluid (a) An object submerged in a fluid experiences a buoyant force FB. If FB is greater than the weight of the object, the object will rise. If FB is less than the weight of the object, the object will sink. (b) If the object is removed, it is replaced by fluid having weight Wfl. Since this weight is supported by surrounding fluid, the buoyant force must equal the weight of the fluid displaced. That is, FB=Wfl, a statement of Archimedes’ principle. The buoyant force on an object equals the weight of the fluid it displaces. Archimedes principle is valid in general, for any object in any fluid, whether partially or totally submerged. Density and Archimedes’ Principle Density plays a crucial role in Archimedes’ principle. The average density of an object is what ultimately determines whether it floats. If its average density is less than that of the surrounding fluid, it will float. This is because the fluid, having a higher density, contains more mass and hence more weight in the same volume. The buoyant force, which equals the weight of the fluid displaced, is thus greater than the weight of the object. Likewise, an object denser than the fluid will sink.  The extent to which a floating object is submerged depends on how the object’s density is related to that of the fluid. In Figure for example, the unloaded ship has a lower density and less of it is submerged compared with the same ship loaded. The fraction submerged is the ratio of the volume submerged to the volume of the object, or 𝑉 𝑉 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 = = 𝑉 𝑉  Now we can obtain the relationship between the densities by substituting into the expression 𝜌 =. This gives 𝑉 𝑚 ⁄𝜌 = 𝑉 𝑚 ⁄𝜌 Where , is average density of the object, and 𝜌 is density of fluid. Since the object floats, its mass and that of the displaced fluid are equal, and so they cancel from the equation, leaving 𝜌 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 = 𝜌  It is useful to define the ratio of the density of an object to a fluid (usually water) as specific gravity: 𝜌̅ 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 = 𝜌 Where 𝜌̅ is average density, 𝜌 density of water at 4oC If an object floats, its specific gravity is less than one. If it sinks, its specific gravity is greater than one. Moreover, the fraction of a floating object that is submerged equals its specific gravity. If an object’s specific gravity is exactly 1, then it will remain suspended in the fluid, neither sinking nor floating.  Archimedes’ principle states that the buoyant force on the object equals the weight of the fluid displaced. This, in turn, means that the object appears to weigh less when submerged; we call this measurement the object’s apparent weight. The object suffers an apparent weight loss equal to the weight of the fluid displaced. Alternatively, on balances that measure mass, the object suffers an apparent mass loss equal to the mass of fluid displaced. That is 𝐴𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑤𝑒𝑖𝑔ℎ𝑡𝑙𝑜𝑠𝑠 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑. Or 𝐴𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑚𝑎𝑠𝑠 𝑙𝑜𝑠𝑠 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑. Cohesion and Adhesion in Liquids Children blow soap bubbles and play in the spray of a sprinkler on a hot summer day. An underwater spider keeps his air supply in a shiny bubble he carries wrapped around him. A technician draws blood into a small-diameter tube just by touching it to a drop on a pricked finger. What is the common thread? All these activities are dominated by the attractive forces between atoms and molecules in liquids—both within a liquid and between the liquid and its surroundings.  Attractive forces between molecules of the same type are called cohesive forces. Liquids can, for example, be held in open containers because cohesive forces hold the molecules together. Attractive forces between molecules of different types are called adhesive forces. Such forces cause liquid drops to cling to windowpanes. Surface Tension Cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. This general effect is called surface tension. Molecules on the surface are pulled inward by cohesive forces, reducing the surface area. Molecules inside the liquid experience zero net force, since they have neighbors on all sides. Surface tension is proportional to the strength of the cohesive force, which varies with the type of liquid. Surface tension is defined to be the force F per unit length exerted by a stretched liquid membrane: 𝐹 𝛾= 𝐿 Pressures in the Body Blood Pressure Common arterial blood pressure measurements typically produce values of 120 mm Hg and 80 mm Hg, respectively, for systolic and diastolic pressures. When systolic pressure is chronically high, the risk of stroke and heart attack is increased. If, however, it is too low, fainting is a problem. Systolic pressure increases dramatically during exercise to increase blood flow and returns to normal afterward. Diastolic pressure can be an indicator of fluid balance. When low, it may indicate that a person is hemorrhaging internally and needs a transfusion. Pressure in the Eye The shape of the eye is maintained by fluid pressure, called intraocular pressure, which is normally in the range of 12.0 to 24.0 mm Hg. When the circulation of fluid in the eye is blocked, it can lead to a buildup in pressure, a condition called glaucoma. The net pressure can become as great as 85.0 mm Hg, an abnormally large pressure that can permanently damage the optic nerve. Pressure Associated with the Lungs The pressure inside the lungs increases and decreases with each breath. The pressure drops to below atmospheric pressure (negative gauge pressure) when you inhale, causing air to flow into the lungs. It increases above atmospheric pressure (positive gauge pressure) when you exhale, forcing air out. The lungs, in fact, would collapse due to the surface tension in the alveoli, if they were not attached to the inside of the chest wall by liquid adhesion  Some of the other pressures in the body are spinal and skull pressures, bladder pressure, pressures in the skeletal system. One hazard of high bladder pressure is that such pressure can force urine back into the kidneys, causing potentially severe damage. 7. Archimedes’ principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390.0 g in air is found to have an apparent mass of 350.5 g when completely submerged in an unknown liquid. (a) What mass of fluid does the iron displace? (b) What is the volume of iron, given density =7.8g/cm3 (c) Calculate the fluid’s density. THANK YOU