# Chapter 9. Solids and Fluids 9.3 DENSITY AND PRESSURE

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1 9.3 DENSITY AND PRESSURE Chapter 9 Solids and Fluids The density of an object having uniform composition is defined as its mass M divided by its volume V: M V [9.6] SI unit: kilogram per meter cubed (kg/m 3 ) Densities of Some Common Substances Substance (kg/m 3 ) a Substance (kg/m 3 ) a Ice Water Aluminum Glycerin Iron Ethyl alcohol Copper Benzene Silver Mercury Lead Air.29 Gold Oxygen.43 Platinum Hydrogen Uranium Helium.79 0 a

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4 A Vacuum F (a) (b) If F is the magnitude of a force exerted perpendicular to a given surface of area A, then the pressure P is the force divided by the area: P F A [9.7] SI unit: pascal (Pa)

5 9.4 VARIATION OF PRESSURE WITH DEPTH When a fluid is at rest in a container, all portions of the fluid must be in static equilibrium at rest with respect to the observer. Furthermore, all points at the same depth must be at the same pressure. F 2 (a) F h Mg P (b) A P 2 A y 0 P 2 A P A Mg 0 y ity, we have y ty, 2 we have M V A(y y 2 ) nto Equation 9.8, canceling P 2 P g(y y 2 ) or rugby. Atmospheric pressure is also caused by a piling up of fluid in this case, the fluid is the gas of the atmosphere. The weight of all the air from sea level to the edge of space results in an atmospheric pressure of P Pa (equivalent to 4.7 lb/in. 2 ) at sea level. This result can be adapted to find the pressure P at any depth h (y y 2 ) (0 y 2 ) below the surface of the water: P P 0 gh

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7 P P 0 gh The pressure P at a depth h below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by the amount ρgh Pascal s principle A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. P = P 2 F A = F 2 A 2 x F A A 2 F 2 x 2 (a)

8 9.5 PRESSURE MEASUREMENTS 0 The pressure P is called the absolute pressure, and P P 0 is called the gauge pressure. If P in the system is greater than atmospheric pressure, h is positive. If P is less than atmospheric pressure (a partial vacuum), h is negative, meaning that the right-hand column in Figure 9.6a is lower than the left-hand column. Another instrument used to measure pressure is the barometer (Fig. 9.6b), in- P h P 0 A B P = 0 (a) h (b) P 0 Another instrument used to measure pressure is the barometer (Fig. 9.6b), invented by Evangelista Torricelli ( ). A long tube closed at one end is filled with mercury and then inverted into a dish of mercury. The closed end of the tube is nearly a vacuum, so its pressure can be taken to be zero. It follows that P 0 gh, where is the density of the mercury and h is the height of the mercury column. Note that the barometer measures the pressure of the atmosphere, whereas the manometer measures pressure in an enclosed fluid. One atmosphere of pressure is defined to be the pressure equivalent of a column of mercury that is exactly 0.76 m in height at 0 C with g m/s 2. At this temperature, mercury has a density of kg/m 3 ; therefore, P 0 gh ( kg/m 3 )( m/s 2 )( m) Pa atm

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10 9.6 BUOYANT FORCES AND ARCHIMEDES S PRINCIPLE Any object completely or partially submerged in a fluid is buoyed up by a force with magnitude equal to the weight of the fluid displaced by the object. B P 2 A P A Mg (a) (b) as been identified as a diffe B fluid V fluid g he fluid and V

11 Case I: A Totally Submerged Object. B = ρ fluid V obj g a B mg a B mg w = mg = ρ obj V obj g (a) (b) net force = B w = (ρ fluid ρ obj )V obj g

12 Case II: A Floating Object. B B = w B = ρ fluid V fluid g F g w = mg = ρ obj V obj g ρ fluid V fluid g = ρ obj V obj g ρ obj ρ fluid = V fluid V obj

13 M 2 2 A 2 v 2 t. However, because 9.7 FLUIDS IN MOTION dy, the mass that flows into the bot. The fluid is nonviscous, which means there is no internal friction force between adjacent layers. 2. The fluid is incompressible, which means its density is constant. 3. The fluid motion is steady, meaning that the velocity, density, and pressure at each point in the fluid don t change with time. 4. The fluid moves without turbulence. This implies that each element of the fluid has zero angular velocity about its center, so there can t be any eddy currents present in the moving fluid. A small wheel placed in the fluid would translate but not rotate. ust equal the mass that flows out th M 2, or Equation of Continuity Point Point 2 an incompressible fluid, A 2 A v 2 A 2 v 2 2 and x 2 v 2 A v 2 A 2 v 2 ρ = ble ρ 2 fluid, ΔM = ΔM 2 and A x v A v A 2 v 2 (a)

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15 The condition Av=constant is equivalent to the fact that the volume of fluid that enters one end of the tube in a given time interval equals the volume of fluid leaving the tube in the same interval, assuming that the fluid is incompressible and there are no leaks Av= Flow Rate TIP 9.3 Continuity Equations The rate of flow of fluid into a system equals the rate of flow out of the system. The incoming fluid occupies a certain volume and can enter the system only if the fluid already inside goes out, thereby making room.

16 e lower blue region in the figure. In 44337_09_p /28/04 :6 AM Page 292 n the upper portion in the Bernoulli s Equation Point time t is 292 Chapter 9 Solids and Fluids mass of the fluid passing through the pipe in the time interval t, then the change in kinetic energy of the volume of fluid is K E 2 mv mv 2 W P The change in the gravitational Apotential energy x is P V PE mgy 2 mgy Because the net work done by the fluid on the segment of fluid shown in Figure 9.29 changes the kinetic energy and the potential energy of the nonisolated system, we have x 2 P 2 A 2 W fluid KE PE The three terms in this equation are those we have just evaluated. Substituting expressions for each of the terms gives Point ause DANIEL BERNOULLI, by Swiss the P V P 2 V equation of continuity, v th 2 mv mv 2 mg y 2 mg y Physicist and Mathematician ( ) If we divide each term by V and recall that m/v, this expression becomes In his most famous work, Hydrodynamica, Bernoulli showed that, as the velocity of P P 2 2 v 2 x v 2 gy 2 gy fluid flow increases, its pressure decreases. In this same publication, Bernoulli also Rearrange the terms as follows: attempted the first explanation of the y behavior of gases with changing pressure P [9.6] P 2 This is Bernoulli s equation, often expressed as A 2 v 2 gy P 2 2 v2 2 gy 2 and temperature; this was the beginning of the kinetic theory of gases. Corbis-Bettmann e time t equals the volume that pas [9.7] Bernoulli s equation P 2 v 2 gy constant W 2 P 2 A 2 x 2 P 2 V Bernoulli s equation states that the sum of the pressure P, the kinetic energy W y 2 is negative per unit volume, 2 v 2, and the potential energy per because unit volume, gy, has the v the force on th same value all points along a streamline. An important consequence of Bernoulli s equation can demonstrated by considering Figure 9.30, which shows water flowing through a horizontal constricted TIP 9.4 Bernoulli s Principle Wfor Gases F x pipe from Pa region A of x large cross-sectional P area Vinto a region of smaller crosssectional area. This device, called a Venturi tube, can be used to measure the gases because they aren t Equation 9.6 isn t strictly true for of the lower net incompressible. The qualitative behavior is the speed of blue region work fluid flow. Because the pipe is horizontal, in the figure. done y y 2, and Equation 9.6 applied to points and 2 gives by these forces in same, however: As the speed of the gas increases, its pressure decreases. P 2 v 2 P v 2 [9.8] Because the water is not backing up in the pipe, its speed v 2 in the constricted region must be greater than its speed v in the region of greater diameter. From Equation 9.8, we see that P 2 must be less than P because v 2 v. This result is often expressed by the statement that swiftly moving fluids exert less pressure than do slowly moving fluids. This important fact enables us to understand a wide range of everyday phenomena. W fluid P V P 2 V P Figure 9.30 (a) The pressure P is greater than the pressure P 2, because v v 2. This device can be used to measure the speed of fluid flow. (b) A Venturi tube, located at the top of the photograph. The higher level of fluid in the middle column shows that the pressure at the top of the column, which is in the constricted region of the Venturi tube, is lower than the pressure elsewhere in the column. A (a) A 2 P2 v v 2 (b) Courtesy of Central Scientific Company Image not Available

17 K E 2 mv mv 2 nal PE potential mgy energy 2 mgy is e by W fluid the fluid KE on the PE segm P V P 2 V 2 mv mv 2 mg y 2 mg y P P 2 2 v v 2 gy 2 gy s as follows: P 2 v 2 gy P 2 2 v 2 2 gy 2 Bernoulli s equation states that the sum of the pressure P, the kinetic energy per unit volume, 2 v 2, and the potential energy per unit volume, gy, has the same value at all points along a streamline.

18 P 2 v 2 gy constant Bernoulli s equation states that the sum of the pressure P, the kinetic energy per unit volume, 2 v 2, and the potential energy per unit volume, gy, has the same value at all points along a streamline. P P 2 P 2 v 2 P 2 2 v 2 2 v v 2 TIP 9.4 Bernoulli s Principle for Gases Equation 9.6 isn t strictly true for gases because they aren t incompressible. The qualitative behavior is the same, however: As the speed of the gas increases, its pressure decreases. A A 2 swiftly moving fluids exert less pressure than do slowly moving fluids.

19 ognize that this so-called atomizer is used in perfume bottles and. The same principle is used in the carburetor of a gasoline engine. e low-pressure region in the carburetor is produced by air drawn in through 9.8 the air APPLICATIONS filter. The gasoline OF FLUID vaporizes, DYNAMICS mixes with the air, and nder of the engine for combustion. n with advanced arteriosclerosis, the Bernoulli effect produces a d vascular flutter. In this condition, the artery is constricted as a related plaque on its inner walls, as shown in Figure To maintain w rate, the blood must travel faster than normal through the cone speed of the blood is sufficiently high in the constricted region, the e is low, and the artery may collapse under external pressure, caustary interruption in blood flow. During the collapse there is no ct, so the vessel reopens under arterial pressure. As the blood rushes onstricted artery, the internal pressure drops and the artery closes riations in blood flow can be heard with a stethoscope. If the plaque dged and ends up in a smaller vessel that delivers blood to the heart, heart attack. m is a weakened spot on an artery where the artery walls have balrd. Blood flows more slowly though this region, as can be seen uation of continuity, resulting in an increase in pressure in the aneurysm relative to the pressure in other parts of the artery. n is dangerous because the excess pressure can cause the artery to Figure 9.34 Artery Plaque Figure 9.35 Blood must travel faster than normal through a constricted region of an artery. A P P L I C A T I O N Vascular A stream of Flutter air pass-and Aneurysms F Drag Lift an aircraft wing can also be explained in part by the Bernoulli effect. s are designed so that the air speed above the wing is greater than w. As a result, the air pressure above the wing is less than the presd there is a net upward force on the wing, called the lift. (There is tal component called the drag.) Another factor influencing the lift wn in Figure 9.36, is the slight upward tilt of the wing. This causes striking the bottom to be deflected downward, producing a reaction by Newton s third law. Finally, turbulence also has an effect. If the Figure 9.36 Streamline flow around an airplane wing. The pressure above is less than the pressure below, and there is a dynamic upward lift force.

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