Solids and Fluids

1. Solids and Fluids

2. States of matter: • - solid (define volume and shape): crystalline solid or amorphous solid • - liquid (define volume but not shape) • -gas (no define volume or shape) • - plasma( highly ionized state of matter containing equal amounts of positive and negative charges)

3. A)The Deformation of Solids • If the external forces are removed and the object tends to return to its original shape and size is called elastic behavior. • The elastic proprieties of solids are discussed in terms of stress and strain • Stress- is the force per unit area causing a deformation • Strain – is a measure of the amount of the deformation

4. For sufficiently small stress, stress is proportional to strain, with the constant of proportionality (elastic modulus) depending on the material being deformed and on the nature of the deformation • Stress = elastic modulus x strain • F =k Δx

5. Young’s modulus: elasticity in length Consider a long bar (fig), when and external force is applied, an internal force in the bar resist the distortion Tensile stress- the ratio F/A SI unit: 1Pa = 1N / 1 m2 Tensile strain – ratio of ΔL / L0 The Young’s modulus: F/A = Y ΔL / L0 A material having a large Young’s modulus is difficult to stretch or compress

6. Shear Modulus: elasticity of Shape : an object is subjected to a force parallel to one its faces while the opposite face is held fixed by a second force The shear stress F/A (the ratio of the magnitude of the parallel force to the area of the face being sheared) The shear strainΔx /h (Δx- horizontal distance, h-height) • F/A =S Δx /h , S-shear modulus ( a material having a large shear modulus is difficult to bend)

7. Bulk modulus: volume elasticity ( the response of a substance to uniform squeezing), this occurs when an object is immersed in a fluid • The volume stress ΔP (pressure)- is the ratio of the magnitude of the change in the applied force to the surface area (ΔF/A) • The volume strain ΔV/ V • ΔP =-B ΔV/ V • A material having a large bulk modulus doesn’t compress easily • 1/B = compressibility • Both solids and liquids have bulk moduli !!!

8. B)Density and Pressure • Density ρ of an object having uniform composition is defined as its mass m divided by its volume V: • ρ= M/V ; Si unit kg/ m3 • If F is a magnitude of a force exerted perpendicular to a given surface of area A, then the pressure P is: • P =F/A; SI unit Pa (Pascal) 1Pa = 1N / 1 m2

9. C)Variation of pressure with depth: all portions of the fluid must be in static equilibrium, all points at the same depth must be at the same pressure • P2A -P1A -Mg = 0; M =ρV = ρA(y1-y2) • P2 = P1 + ρg(y1-y2) • P = P0 + ρgh • 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

10. D)Pascal 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 • An application of Pascal Principle is the hydraulic lift P1=P2 • F1/A1 = F2/A2

11. E)Pressure measurements: • Open-tube manometer- one end of a U tube containing a liquid is open to the atmosphere and the other end is connected to a system of unknown pressure P: P = P0 + ρgh P-absolute pressure; P- P0 – gauge pressure • The Barometer- a long tube closed at one end is filled with mercury and then is inverted into a dish with mercury(P vacuum =0) P0 =

12. One atmosphere is the pressure equivalent of a column of mercury that is exactly 0.76m in height at 0oC with g=9.806 m/s2. At this temperature, mercury ha s a density of 13.593 x 103 kg/m3 • P0=ρgh = 1.013 x105 Pa= 1 atm

13. F)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 weight of the fluid displaced by the object. • B = Mg =P2A-P1A; B=ρfluid Vfluid g • 1.A totally submerge object when an object is totally submerge in a fluid of density ρfluid, the Buoyant force acting on the oject has a magnitude of B=ρfluid Vobj g; w= mg=ρobj Vobj g • B-w =(ρfluid -ρobj )Vobj g

14. If Fnet>0(accelerates upward); Fnet<0(accelerates downwards) • 2. A Floating object – a partially submerged object in static equilibrium floating in a fluid • B=ρfluid Vfluid g; w= B=ρobj Vobj g; B=w • ρfluid Vfluid g =ρobj Vobj g • ρobj/ρfluid = Vfluid / Vobj

15. G) Fluids in motion • Viscosity- internal friction in the fluid – resistance between two adjectent layers of the fluid moving relative to each other • Ideal Fluid- satisfies the flowing conditions: 1. is nonviscous (no internal friction) 2. is incompressible (density is constant) 3. Fluid motion is steady (v,ρ,P don’t change with time) 4. Fluid moves without turbulence (zero angular velocity about its center= no currents)

16. Equation of continuity Δx1=v1Δt ΔM1=ρ1A1Δx1=ρ1A1v1Δt ΔM2=ρ2A2v2Δt ΔM1=ΔM2 ρ1A1v1=ρ2A2v2 If incompressible fluid ρ1=ρ2 A1v1=A2v2 –equation of continuity The product of the cross-sectional area of the pipe and the fluid speed at that cross section is a constant • Flow rate Av= constant is equivalent to the fact that the volume of the fluid that enters one end of the tube is equals the volume of fluid leaving the tube, in a same time interval (fluid is incompressible and are no leaks)

17. Bernoulli’s Equation – is a consequence of energy conservation as applied to an ideal fluid W1=F1Δx1 =P1A1Δx1 =P1V W2= -P2A2Δx2 = -P2V The net work done W=P1V -P2V=ΔKE +ΔPE P1V - P2V=1/2mv22-1/2mv12+ mgy2-mgy1 P1 - P2=1/2ρv22-1/2ρv12+ ρgy2-ρgy1 P1+1/2ρv12 +ρgy1 =P2+1/2ρv22+ ρgy2 Bernoulli equation: P+1/2ρv2 +ρgy =constant The sum of the pressure, the KE per unit volume, and PE per unit volume, has the same value at all points along a stremline

18. H) Surface tension, Capillarity action and viscous fluid flow The Fnet on a molecule at A is 0, the molecule is surrounded by ither molecules, The Fnet in B is downward because it isn’t completely surrounded by other molecules The surface tensionγ is the magnitude of the face tension force divided by the length along which force acts γ= F/L; SI unit: N/m=Nm/m2=J/m2

19. The surface of liquid (considerate the forces between the molecules) Forces between like molecules, such as the forces between the water molecule are called cohesive forces, and forces between unlike molecules, water/glass, are called adhesive forces • θ- contact angle

20. Capillary Action(in capillary tubes, the diameter of the opening is very small) • If we inserted into a fluid, and adhesive forces dominate over the cohesive forces, the liquid is rises in the tube F=γL =γ(2πr); Fvertical= γ(2πr)(cosθ) w= Mg= ρVg =ρgπr2h γ(2πr)(cosθ) =ρgπr2h h= (2γ/ρgr) (cosθ)

21. If a capillary tube is inserted into a liquid in which cohesive forces dominate over adhesive forces, the level of the liquid in the capillary tube will be below the surface of the surrounding fluid

22. Viscous Fluid Flow • Viscosity refers to the internal friction of a fluid • The force required to move the upper plate at a fixed speed v is F=η Av/d η(eta)- coefficient of viscosity, SI units N s/m2 1poise = 10-1 N s/m2 Poiseuille’s Law Rate of flow=ΔV/Δt=πR4(P1-P2)/8ηL Reynolds number: RN =ρvd/η

23. Motion through a viscous medium Stoke’s Law: the magnitude of the resistive force on a very small spherical object of radius r falling slowly through a fluid of viscosity η with speed v: Fr =6πηrv 6πηrv w= ρgV = ρg(4/3πr3) B = ρfgV= ρfg(4/3πr3) Fr +B=w 6πηrv + ρfg(4/3πr3)=ρg(4/3πr3) v=2r2g / 9η (ρ-ρf) the speed when the net force goes to zero