# Chapter 9: Solids and Fluids

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1 Chapter 9: Solids and Fluids State of matters: Solid, Liquid, Gas and Plasma. Solids Has definite volume and shape Can be crystalline or amorphous Molecules are held in specific locations by electrical forces vibrate about equilibrium positions Can be modeled as springs connecting molecules External forces can be applied to the solid and compress the material In the model, the springs would be compressed When the force is removed, the solid returns to its original shape and size This property is called elasticity

2 Liquid Has a definite volume No definite shape Exists at a higher temperature than solids The molecules wander through the liquid in a random fashion The intermolecular forces are not strong enough to keep the molecules in a fixed position Gas Has no definite volume Has no definite shape Molecules are in constant random motion The molecules exert only weak forces on each other Average distance between molecules is large compared to the size of the molecules

3 Plasma Matter heated to a very high temperature Many of the electrons are freed from the nucleus Result is a collection of free, electrically charged ions Plasmas exist inside stars Elastic Deformation: Young s Modulus Stress is the force per unit area causing the deformation Strain is a measure of the amount of deformation The elastic modulus is the constant of proportionality between stress and strain tensile stress uniaxial strain F A ΔL = Y L o Young s modulus Young s modulus unit of modulus: N/m 2 (= pascal, Pa)

4 Elastic Deformation: Shear Modulus shear modulus shear stress F A ΔX = S h shear strain shear modulus Notes on Young s and Shear Modulus The elastic modulus can be thought of as the stiffness of the material It is possible to exceed the elastic limit of the material -- No longer directly proportional -- May not return to its original length -- Break Pressure and Bulk Modulus The pressure P is the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts P = F A Unit of Pressure: N/m 2 = pascal (Pa) ΔV ΔP = B V o bulk modulus Bulk modulus Solids have Young s, Bulk, and Shear moduli Liquids have only bulk moduli, they will not undergo a shearing or tensile stress The liquid would flow instead

5 Elastic Modulus of Some Materials Mass Density: The mass density ρ is the mass m of a substance divided by its volume V ρ = m / V Unit of Mass Density: kg/m 3 Specific Gravity: (for any substance) Density Density of substance Specific gravity = Density of water at 4 o C Unit of specific gravity: unit-less, number The densities of most liquids and solids vary slightly with changes in temperature and pressure Densities of gases vary greatly with changes in temperature and pressure

6 Pressure The pressure P has a general definition of (normal) force per unit area P = F A SI unit: pascal (N/m 2 ) In a static liquid (or gas), the pressure is isotropic (non-directional). For a gas (low mass density) the pressure in a small volume can be regarded as homogeneous (constant throughout space). For liquids (high mass density), the pressure depends on the depth of the liquid. Atmospheric pressure at sea level is Pa = 1 atm Exam Average: 51.6

7 Pressure and Depth in a Static Fluid PA = P 0 A + Mg P = P 0 + ρgh h: depth below the face of fluid Hoover Dam. Can we use a less massive structure to hold the water if the size (volume) of the reservoir (with same depth) is much narrower? Absolute and Relative (Gauge) Pressure relative pressure P 2 = P 1 + ρgh absolute pressure Pressure in different units: One atmosphere (1 atm) = 76.0 cm of mercury = 760 Torr = x 10 5 Pa = 14.7 lb/in 2 (psi) Barometer, invented by Torricelli in 1643.

8 Pascal s Principle Pascal s Principle Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to every point of the fluid and the enclosing walls. F2 A 2 = F1 A 1 Example: r 1 =5.00 cm, r 2 =30.0 cm. m car =4000 lb. Can you lift the car? A F 2 = F 2 1 A 1 Can we get something out of nothing? Of course not! Buoyant Forces and Archimedes Principle Magnitude of buoyant force = Weight of displaced fluid Archimedes Principle The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object The magnitude of the buoyant force always equals the weight of the displaced fluid B = ρ V g = w fluid fluid fluid The buoyant force is exerted by the fluid. The buoyant force is the same for a totally submerged object of any size, shape, or density

9 Totally Submerged Object The net force is B-mg=(ρ fluid -ρ obj )gv obj Will it float? If ρ obj < ρ fluid, float, If ρ obj > ρ fluid, sink. Floating Object The object is in static equilibrium The upward buoyant force is balanced by the downward force of gravity Volume of the fluid displaced is equal to the volume of the object beneath the fluid level How much volume will be submerged? V V submerged total V = displaced _ fluid V total = ρ ρ obj fluid Tip of the iceberg! ρ ice = g/cm 3, ρ water = g/cm 3 at 0

10 Problems On Static Fluid A light spring of constant k = 160 N/m rests vertically on the bottom of a large beaker of water. A 5.00-kg block of wood (density 650 kg/m 3 ) is connected to the spring, and the block-spring system is allowed to come to static equilibrium (Fig. P9.34b). What is the elongation ΔL of the spring? A bargain hunter purchases a gold crown at a flea market. After she gets home, she hangs it from a scale and finds its weight to be 7.84N. She then weights the crown while it is immersed in water of density 1000 kg/m 3, and now the scale reads 6.86N. Is the crown made of pure gold (density of gold is 19.3 x 10 3 kg/m 3 ). Fluids in Motion: Streamline Flow Streamline flow Every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier Streamline is the path Different streamlines cannot cross each other The streamline at any point coincides with the direction of fluid velocity at that point

11 Fluids in Motion: Turbulent Flow The flow becomes irregular exceeds a certain velocity any condition that causes abrupt changes in velocity Eddy currents are a characteristic of turbulent flow Characteristics of an Ideal Fluid The fluid is nonviscous There is no internal friction between adjacent layers The fluid is incompressible Its density is constant The fluid motion is steady Its velocity, density, and pressure at a certain spatial point do not change in time The fluid moves without turbulence No eddy currents are present The elements have zero angular velocity about its center

12 Equation of Continuity A 1 v 1 = A 2 v 2 A consequence of Conservation of Mass The product of the crosssectional area of a pipe and the fluid speed is a constant Speed is high where the pipe is narrow and speed is low where the pipe has a large diameter Av is called the flow rate Bernoulli s Equation In the steady flow of a nonviscous, incompressible fluid of density ρ, the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by P ρ v1 + ρgy1 = P2 + ρv2 + ρgy Bernoulli s equation is a consequence of Conservation of Energy applied to an ideal fluid

13 Application of Bernoulli s Equation At the same height: P ρv 2 1 = P ρv 2 2 Example Problems A siphon is a device that allows a fluid to seemingly defy gravity. The flow must be initiated by a partial vacuum in the tube, as in a drinking straw. (a) Show that the speed at which the water emerges from the siphon is given by will the siphon work? v = 2gh. (b) For what values of y The figure at right shows a water tank with a valve at the bottom. If this valve is opened, what is the maximum height attained by the water stream coming out of the right side of the tank? Assume that h = 10.0 m, L = 2.00 m, and θ = 30.0 and that the cross sectional area at A is very large compared with that at B.

14 Surface Tension Net force on molecule A is zero Pulled equally in all directions Net force on B is not zero No molecules above to act on it Pulled toward the center of the fluid The net effect of this pull on all the surface molecules is to make the surface of the liquid contract When liquid meets solid Cohesive forces are forces between like molecules Adhesive forces are forces between unlike molecules The shape of the surface depends upon the relative size of the cohesive and adhesive forces

15 Capillary Action Capillary action is the result of surface tension and adhesive forces The liquid rises in the tube when adhesive forces are greater than cohesive forces The liquid drops in the tube when cohesive forces are greater than adhesive forces Chapter 9 Summary Mass Density Specific Gravity Elastic modulus (bulk modulus, Young s modulus, shear modulus) Fluid pressure and depth Pascal s principle Archimedes principle on buoyant force Equation of continuity Bernoulli s equation

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