8.01SC Chapter 27: Static Fluids

Chapter 27 Static Fluids 27.1 Introduction........................................................................................................... 1 27.2 Density.................................................................................................................... 1 27.3 Pressure in a Fluid ................................................................................................ 3 27.4 Pascal’s Law: Pressure as a Function of Depth in a Fluid of Uniform Density in a Uniform Gravitational Field................................................................................. 4 Example 27.1 Pressure in the Earth’s Ocean ......................................................... 6 Example 27.2 Pressure in a Sample in a Centrifuge.............................................. 7 27.5 Compressibility of a Fluid.................................................................................... 8 Example 27.3 Compressibility of Water ............................................................... 10 27.6 Archimedes’ Principle: Buoyant Force............................................................. 10 Example 27.4 Archimedes’ Principle: Floating Wood ........................................ 13 Example 27.5 Rock Inside a Floating Salad Bowl................................................ 14 Example 27.6 Block Floating Between Oil and Water ........................................ 15 Chapter 27 Static Fluids 27.1 Introduction Water is everywhere around us, covering 71% of the Earth’s surface. The water content of a human being can vary between 45% and 70 % of body weight. Water can exist in three states of matter: solid (ice), liquid, or gas. Water flows through many objects: through rivers, streams, aquifers, irrigation channels, and pipes to mention a few. Humans have tried to control and harness this flow through many different technologies such as aqueducts, Archimedes’ screw, pumps, and water turbines. Water in the gaseous state also flows. Water vapor, lighter than air, can cause convection currents that form clouds. In the liquid state, the density of water molecules is greater than the gaseous state but in both states water can flow. Liquid water forms a surface while water vapor does not. Water in both the liquid and gaseous state is classified as a fluid to distinguish it from the solid state. At the macroscopic scale, matter can be roughly grouped into two classes, solids and fluids. There is some ambiguity in the use of the term fluid. In ordinary language, the term fluid is used to describe the liquid state of matter. More technically, a fluid is a state of matter that, when at rest, cannot sustain a shear stress and hence will flow. A solid, when at rest, can sustain a shear stress and although it may deform it will remain at rest. However there is some ambiguity in this description. Glacier ice will flow but very slowly. So for a time interval that is small compared to the time interval involved in the flow, glacial ice can be thought of as a solid. This description of a fluid applies to both liquids and gases. A gas will expand to fill whatever volume it is confined in, while a liquid placed in a container will have a well-defined volume with a surface layer separated the liquid and vapor phases of the substance. We shall The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. 27.2 Density The density of a small amount of matter is defined to be the amount of mass ΔM divided by the volume ΔV of that element of matter, ρ = ΔM / ΔV . (27.2.1) The SI unit for density is the kilogram per cubic meter, kg ⋅ m −3 . If the density of a material is the same at all points, then the density is given by ρ = M /V , (27.2.2) 27-1 where M is the mass of the material and V is the volume of material. A material with constant density is called homogeneous. For a homogeneous material, density is an intrinsic property. If we divide the material in two parts, the density is the same in both parts, (27.2.3) ρ = ρ1 = ρ2 . However mass and volume are extrinsic properties of the material. If we divide the material into two parts, the mass is the sum of the individual masses M = M1 + M 2 , (27.2.4) V = V1 + V2 . (27.2.5) as is the volume The density is tabulated for various materials in Table 27.1. Table 27.1: Density for Various Materials (Unless otherwise noted, all densities given are at standard conditions for temperature and pressure, that is, 273.15 K (0.00 °C) and 100 kPa (0.987 atm). Material Helium Air (at sea level) Styrofoam Wood Seasoned, typical Ethanol Ice Water Seawater Blood Aluminum Iron Copper Lead Mercury Gold Plutonium Density, ρ kg ⋅ m −3 0.179 1.20 75 0.7 × 103 0.81× 103 0.92 × 103 1.00 × 103 1.03× 103 1.06 × 103 2.70 × 103 7.87 × 103 8.94 × 103 11.34 × 103 13.55 × 103 19.32 × 103 19.84 × 103 27-2 Osmium 22.57 × 103 If we examine a small volume element of a fluid, it consists of molecules interacting via intermolecular forces. If we are studying the motion of bodies placed in fluids or the flow of the fluid at scales that are large compared to the intermolecular forces then we can consider the fluid to be continuous and quantities like density will vary smoothly from point to point in the fluid. 27.3 Pressure in a Fluid When a shear force is applied to the surface of fluid, the fluid will undergo flow. When a fluid is static, the force on any surface within fluid must be perpendicular (normal) to each side of that surface. This force is due to the collisions between the molecules of the fluid on one side of the surface with molecules on the other side. For a static fluid, these forces must sum to zero. Consider a small portion of a static fluid shown in Figure 27.1. That portion of the fluid is divided into two parts, which we shall designate 1 and 2, by a small mathematical shared surface element S of area AS . The force F1,2 (S) on the surface of region 2 due to the collisions between the molecules of 1 and 2 is perpendicular to the surface. Figure 27.1: Forces on a surface within a fluid  The force F2,1 (S) on the surface of region 1 due to the collisions between the molecules of 1 and 2 by Newton’s Third Law satisfies   F1,2 (S) = −F2,1 (S) . (27.3.1) Denote the magnitude of these forces that form this interaction pair by   F⊥ (S) = F1,2 (S) = F2,1 (S) . (27.3.2) 27-3 Define the hydrostatic pressure at those points within the fluid that lie on the surface S by P≡ F⊥ (S) . AS (27.3.3) The pressure at a point on the surface S is the limit F⊥ (S) . AS →0 AS P = lim (27.3.4) The SI units for pressure are N ⋅ m −2 and is called the pascal (Pa), where 1 Pa = 1 N ⋅ m −2 = 10−5 bar . (27.3.5) Atmospheric pressure at a point is the force per unit area exerted on a small surface containing that point by the weight of air above that surface. In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point. On a given surface area, low-pressure areas have less atmospheric mass above their location, whereas high-pressure areas have more atmospheric mass above their location. Likewise, as elevation increases, there is less overlying atmospheric mass, so that atmospheric pressure decreases with increasing elevation. On average, a column of air one square centimeter in cross-section, measured from sea level to the top of the atmosphere, has a mass of about 1.03 kg and weight of about 10.1 N. (A column one square inch in cross-section would have a weight of about 14.7 lbs, or about 65.4 N). The standard atmosphere [atm] is a unit of pressure such that (27.3.6) 1 atm = 1.01325 × 105 Pa = 1.01325 bar . 27.4 Pascal’s Law: Pressure as a Function of Depth in a Fluid of Uniform Density in a Uniform Gravitational Field Consider a static fluid of uniform density ρ . Choose a coordinate system such that the z axis points vertical downward and the plane z = 0 is at the surface of the fluid. Choose an infinitesimal cylindrical volume element of the fluid at a depth z , cross-sectional area A and thickness dz as shown in Figure 27.3. The volume of the element is dV = A dz and the mass of the fluid contained within the element is dm = ρ A dz . 27-4 Figure 27.2: Coordinate system for fluid The surface of the infinitesimal fluid cylindrical element has three faces, two caps and the cylindrical body. Because the fluid is static the force due to the fluid pressure points inward on each of these three faces. The forces on the cylindrical surface add to zero. On the end-cap at z , the force due to pressure of the fluid above the end-cap is downward,  F(z) = F(z)k̂ , where F(z) is the magnitude of the force. On the end-cap at z + dz , the force due to the pressure of the fluid below the end-cap is upward,  F(z + dz) = − F(z + dz)k̂ , where F(z + dz) is the magnitude of the force. The gravitational force acting on the element is given by g F = (dm)gk̂ = ( ρ dV )gk̂ = ρ A dz gk̂ . There are also radial inward forces on the cylindrical body which sum to zero. The free body force diagram on the element is shown in Figure 27.3. Figure 27.3: Free-body force diagram on cylindrical fluid element The vector sum of the forces is zero because the fluid is static (Newton’s Second Law). Therefore in the +k̂ -direction F(z) − F(z + dz) + ρ Adzg = 0 . (27.4.1) We divide through by the area A of the end-cap and use Eq. (27.3.4) to rewrite 27-5 Eq. (27.4.1) in terms of the pressure P(z) − P(z + dz) + ρ dzg = 0 . (27.4.2) P(z + dz) − P(z) = ρg . dz (27.4.3) Rearrange Eq. (27.4.2) as Now take the limit of Eq. (27.4.3) as the thickness of the element dz → 0 , lim dz→0 P(z + dz) − P(z) = ρg . dz (27.4.4) resulting in the differential equation dP = ρg . dz (27.4.5) Integrate Eq. (27.4.5), P( z ) z′=z P( z=0) z′=0 ∫ dP = ∫ ρ g dz′ . (27.4.6) Performing the integrals on both sides of Eq. (27.4.6) describes the change in pressure between a depth z and the surface of a fluid P(z) − P(z = 0) = ρ gz (Pascal's Law) , (27.4.7) a result known as Pascal’s Law. Example 27.1 Pressure in the Earth’s Ocean What is the change in pressure between a depth of 4 km and the surface in Earth’s ocean? Solution: We begin by assuming the density of water is uniform in the ocean, and so we can use Pascal’s Law, Eq. (27.4.7) to determine the pressure, where we use ρ = 1.03× 103 kg ⋅ m −3 for the density of seawater (Table 27.1). Then P(z) − P(z = 0) = ρ gz = (1.03× 103 kg ⋅ m −3 )(9.8 m ⋅s −2 )(4 × 103 m) (27.4.8) = 40 × 10 Pa. 6 27-6 Example 27.2 Pressure in a Rotating Sample in a Centrifuge In an ultra centrifuge, a liquid filled chamber is spun with a high angular speed ω about a fixed axis. The density ρ of the fluid is uniform. The open-ended side of the chamber is a distance r0 from the fixed axis. The chamber has cross sectional area A and of length L , (Figure 27.4). Figure 27.4: Schematic representation of centrifuge The chamber is spinning fast enough to ignore the effect of gravity. Determine the pressure in the fluid as a function of distance r from the fixed axis. Solution: Choose polar coordinates in the plane of circular motion. Consider a small volume element of the fluid of cross-sectional area A , thickness dr , and mass dM = ρ Adr that is located a distance r from the fixed axis. Denote the pressure at one end of the volume element by P(r) = F(r) / A and the pressure at the other end by P(r + dr) = F(r + dr) / A . The free-body force diagram on the volume fluid element is shown in Figure 27.5. Figure 27.5: Free-body force diagram showing only radial forces on fluid element in centrifuge The element is accelerating inward with radial component of the acceleration, ar = −rω 2 . Newton’s Second Law applied to the fluid element is then 27-7 (P(r) − P(r + dr)) A = −( ρ Adr)rω 2 , (27.4.9) We can rewrite Eq. (27.4.9) as P(r + dr) − P(r) = ρ rω 2 , dr (27.4.10) and take the limit dr → 0 resulting in dP = ρ rω 2 . dr (27.4.11) We can integrate Eq. (27.4.11) between an arbitrary distance r from the rotation axis and the open-end located at r0 , where the pressure P(r0 ) = 1 atm , P(r0 ) ∫ dP = ρω P(r ) r′=r0 2 ∫ r′ dr′ . (27.4.12) r′=r Integration yields P(r0 ) − P(r) = 1 ρω 2 (r0 2 − r 2 ) . 2 (27.4.13) The pressure at a distance r from the rotation axis is then 1 P(r) = P(r0 ) + ρω 2 (r 2 − r0 2 ) . 2 (27.4.14) 27.5 Compressibility of a Fluid When the pressure is uniform on all sides of an object in a fluid, the pressure will squeeze the object resulting in a smaller volume. When we increase the pressure by ΔP on a material of volume V0 , then the volume of the material will change by ΔV < 0 and consequently the density of the material will also change. Define the bulk stress by the increase in pressure change (27.5.1) σ B ≡ ΔP . Define the bulk strain by the ratio εB ≡ ΔV . V0 (27.5.2) 27-8 For many materials, for small pressure changes, the bulk stress is linearly proportional to the bulk strain, ΔV , (27.5.3) ΔP = − B V0 where the constant of proportionality B is called the bulk modulus. The SI unit for bulk modulus is the pascal. If the bulk modulus of a material is very large, a large pressure change will result in only a small volume change. In that case the material is called incompressible. In Table 27.2, the bulk modulus is tabulated for various materials. 27-9 Table 27.2 Bulk Modulus for Various Materials Material Diamond Iron Nickel Steel Copper Brass Aluminum Crown Glass Lead Water (value increases at higher pressure) Air (adiabatic bulk modulus) Air (isothermal bulk modulus) Bulk Modulus, Y, (Pa) 4.4 × 1011 1.6 × 1011 1.7 × 1011 1.6 × 1011 1.4 × 1011 6.0 × 1010 7.5 × 1010 5.0 × 1010 4.1× 1010 2.2 × 109 1.42 × 105 1.01× 105 Example 27.3 Compressibility of Water Determine the percentage decrease in a fixed volume of water at a depth of 4 km where the pressure difference is 40 MPa , with respect to sea level. Solution: The bulk modulus of water is 2.2 × 109 Pa . From Eq. (27.5.3), ΔV ΔP 40 × 106 Pa =− =− = −0.018 ; V0 B 2.2 × 109 Pa (27.5.4) there is only a 1.8% decrease in volume. Water is essentially incompressible even at great depths in ocean, justifying our assumption that the density of water is uniform in the ocean in Example 27.1. 27.6 Archimedes’ Principle: Buoyant Force When we place a piece of solid wood in water, the wood floats on the surface. The density of most woods is less than the density of water, and so the fact that wood floats does not seem so surprising. However, objects like ships constructed from materials like 27-10 steel that are much denser than water also float. In both cases, when the floating object is at rest, there must be some other force that exactly balances the gravitational force. This balancing of forces also holds true for the fluid itself. Consider a static fluid with uniform density ρ f . Consider an arbitrary volume element of the fluid with volume V and mass m f = ρ f V . The gravitational force acts on the volume  element, pointing downwards, and is given by F g = − ρ f V g k̂ , where k̂ is a unit vector pointing in the upward direction. The pressure on the surface is perpendicular to the surface (Figure 27.6). Therefore on each area element of the surface there is a perpendicular force on the surface. Figure 27.6: Forces due to pressure on surface of arbitrary volume fluid element Figure 27.7: Free-body force diagram on volume element showing gravitational force and buoyant force  Let F B denote the resultant force, called the buoyant force, on the surface of the volume element due to the pressure of the fluid. The buoyant force must exactly balance the gravitational force because the fluid is in static equilibrium (Figure 27.7),     0 = F B + F g = F B − ρ f V g k̂ . (27.6.1) Therefore the buoyant force is therefore  F B = ρ f V g k̂ . (27.6.2) The buoyant force depends on the density of the fluid, the gravitational constant, and the volume of the fluid element. This macroscopic description of the buoyant force that 27-11 results from a very large number of collisions of the fluid molecules is called Archimedes’ Principle. We can now understand why when we place a stone in water it sinks. The density of the stone is greater than the density of the water, and so the buoyant force on the stone is less than the gravitational force on the stone and so it accelerates downward. Place a uniform object of volume V and mass M with density ρo = M / V within a fluid. If the density of the object is less than the density of the fluid, ρo < ρ f , the object will float on the surface of the fluid. A portion of the object that is a beneath the surface, displaces a volume V1 of the fluid. The portion of the object that is above the surface displaces a volume V2 = V − V1 of air (Figure 27.8). Figure 27.8: Floating object on surface of fluid Because the density of the air is much less that the density of the fluid, we can neglect the buoyant force of the air on the object. Figure 27.9: Free-body force diagram on floating object  The buoyant force of the fluid on the object, F fB,o = ρ f V1gk̂ , must exactly balance the  gravitational force on the object due to the earth, Feg,o (Figure 27.9),   g 0 = F fB,o + Fe,o = ρ f V1 g k̂ − ρoV g k̂ = ρ f V1 g k̂ − ρo (V1 + V2 ) g k̂ . (27.6.3) Therefore the ratio of the volume of the exposed and submerged portions of the object must satisfy 27-12 ρ f V1 = ρo (V1 + V2 ) . (27.6.4) We can solve Eq. (27.6.4) and determine the ratio of the volume of the exposed and submerged portions of the object V2 ( ρ f − ρo ) . (27.6.5) = V1 ρo We now also can understand why a ship of mass M floats. The more dense steel displaces a volume of water Vs but a much larger volume of water Vw is displaced by air. The buoyant force on the ship is then  Fs B = ρ f (Vs + Vw ) g k̂ . (27.6.6) If this force is equal in magnitude to Mg , the ship will float. Example 27.4 Archimedes’ Principle: Floating Wood Consider a beaker of uniform cross-sectional area A , filled with water of density ρ w . When a rectangular block of wood of cross sectional area A2 , height, and mass M b is placed in the beaker, the bottom of the block is at an unknown depth z below the surface of the water. (a) How far below the surface z is the bottom of the block? (b) How much did the height of the water in the beaker rise when the block was placed in the beaker? Solution: We neglect the buoyant force due to the displaced air because it is negligibly small compared to the buoyant force due to the water. The beaker, with the floating block of wood, is shown in Figure 27.10. Figure 27.10 Block of wood floating in a beaker of water (a) The density of the block of wood is ρb = M b / Vb = M b / Ab h . The volume of the submerged portion of the wood is V1 = Ab z . The volume of the block above the surface is given by V2 = Ab (h − z) . We can apply Eq. (27.6.5), and determine that 27-13 V2 Ab (h − z) (h − z) ( ρ w − ρb ) . = = = V1 Ab z z ρb (27.6.7) We can now solve Eq. (27.6.7) for the depth z of the bottom of the block z= ρb ( M b / Ab h) Mb . h= h= ρw ρw ρ w Ab (27.6.8) (b) Before the block was placed in the beaker, the volume of water in the beaker is Vw = Asi , where si is the initial height of water in the beaker. When the wood is floating in the beaker, the volume of water in the beaker is equal to Vw = As f − Ab z , where s f is the final height of the water, in the beaker and Ab z is the volume of the submerged portion of block. Because the volume of water has not changed Asi = As f − Ab z . (27.6.9) We can solve Eq. (27.6.9) for the change in height of the water Δs = s f − si , in terms of the depth z of the bottom of the block, Δs = s f − si = Ab z. A (27.6.10) We now substitute Eq. (27.6.8) into Eq. (27.6.10) and determine the change in height of the water M (27.6.11) Δs = s f − si = b . ρw A Example 27.5 Rock Inside a Floating Salad Bowl A rock of mass mr and density ρ r is placed in a salad bowl of mass mb . The salad bowl and rock float in a beaker of water of density ρ w . The beaker has cross sectional area A . The rock is then removed from the bowl and allowed to sink to the bottom of the beaker. Does the water level rise or fall when the rock is dropped into the water? 27-14 Figure 27.11: Rock in a floating salad bowl Solution: When the rock is placed in the floating salad bowl, a volume V of water is  displaced. The buoyant force F B = ρ wVgk̂ balances the gravitational force on the rock and salad bowl, (27.6.12) (mr + mb )g = ρ wVg = ρ w (V1 + V2 )g . where V1 is the portion of the volume of displaced water that is necessary to balance just the gravitational force on the rock, mr g = ρ wV1g , and V2 is the portion of the volume of displaced water that is necessary to balance just the gravitational force on the bowl, mb g = ρ wV2 g , Therefore V1 must satisfy the condition that V1 = mr g / ρ w . The volume of the rock is given by Vr = mr / ρr . In particular V1 = ρr V . ρw r (27.6.13) Because the density of the rock is greater than the density of the water, ρr > ρ w , the rock displaces more water when it is floating than when it is immersed in the water, V1 > Vr . Therefore the water level drops when the rock is dropped into the water from the salad bowl. Example 27.6 Block Floating Between Oil and Water A cubical block of wood, each side of length l = 10 cm , floats at the interface between air and water. The air is then replaced with d = 10 cm of oil that floats on top of the water. a) Will the block rise or fall? Briefly explain your answer. After the oil has been added and equilibrium established, the cubical block of wood floats at the interface between oil and water with its lower surface h = 2.0× 10−2 m below the interface. The density of the oil is ρo = 6.5 × 102 kg ⋅ m −3 . The density of water is ρ w = 1.0 × 103 kg ⋅ m −3 . 27-15 b) What is the density of the block of wood? Solution: (a) The buoyant force is equal to the gravitational force on the block. Therefore ρb gV = ρ w gV1 + ρ a g(V − V1 ) , (27.6.14) where V1 is the volume of water displaced by the block, V2 = V − V1 is the volume of air displaced by the block V is the volume of the block, ρb is the density of the block of wood, and ρ a is the density of air (Figure 27.12(a)). Figure 27.12: (a) Block floating on water, (b) Block floating on oil-water interface We now solve Eq. (27.6.14) for the volume of water displaced by the block V1 = ( ρb − ρ a ) V. ( ρw − ρa ) (27.6.15) When the oil is added, we can repeat the argument leading up to Eq. (27.6.15) replacing ρ a by ρo , (Figure 27.12(b)), yielding ρb gV = ρ w gV1′+ ρo gV2′ , (27.6.16) where V1′ is the volume of water displaced by the block, V2′ is the volume of oil displaced by the block, V is the volume of the block, and ρb is the density of the block of wood. Because V2′ = V − V1′ , we rewrite Eq. (27.6.16) as ρb gV = ρ w gV1′+ ρo g(V − V1′) , (27.6.17) We now solve Eq. (27.6.17) for the volume of water displaced by the block, 27-16 V1′= ( ρb − ρo )V . ( ρ w − ρo ) (27.6.18) Because ρo >> ρ a , comparing Eqs. (27.6.18) and (27.6.15), we conclude that V1′> V1 . The block rises when the oil is added because more water is displaced. (b) We use the fact that V1′= l 2 h , V2′ = l 2 (l − h) , and V = l 3 , in Eq. (27.6.16) and solve for the density of the block ρb = ρ wV1′+ ρoV2′ ρ wl 2 h + ρo l 2 (l − h) h = = ( ρ w − ρo ) + ρo . 3 V l l (27.6.19) We now substitute the given values from the problem statement and find that the density of the block is ρb = ((1.0 × 103 kg ⋅ m −3 ) − (6.5 × 102 kg ⋅ m −3 )) (2.0× 10−2 m) + (6.5 × 102 kg ⋅ m −3 ) −1 (1.0× 10 m) (27.6.20) ρb = 7.2 × 102 kg ⋅ m −3 . Because ρb > ρo , the above analysis is valid. 27-17