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Lecture 19: Introduction to Solids & Fluids Questions of Yesterday 1) A solid sphere and a hoop of equal radius and mass are both rolled up an incline with the same initial velocity. Which object will travel farthest up the inclined plane? a) the sphere b) the hoop c) they’ll both travel the same distance up the plane d) it depends on the angle of the incline 2) If an acrobat rotates once each second while sailing through the air, and then contracts to reduce her moment of inertia to 1/3 of what is was, how many rotations per second will result? a) once each second b) 3 times each second c) 1/3 times each second d) 9 times each second Practice Problem A 10.00-kg cylindrical reel with the radius of 0.500 m and a frictionless axle starts from rest and speeds up uniformly as a 5.00 kg bucket falls into a well, making a light rope unwind from the reel. The bucket starts from rest and falls for 5.00 s. 10.0 kg What is the linear acceleration of the falling bucket? 0.500 m How far does it drop? What is the angular acceleration of the reel? Use energy conservation principles to determine 5.00 kg the speed of the spool after the bucket has fallen 5.00 m 4 States of Matter Solid Fluid Gas Plasma Definite Expands to fill Expands to fill Definite Shape & any volume any volume Volume Volume Takes shape of Takes shape of Takes shape of Molecules Container Container container close together Molecules even Made up of ions Molecules & slow farther apart Fastest of all farther apart & even faster matter states & faster Density Distance between molecules Density M Density r= The amount of matter (mass) V in a given volume Solids Applying force can change shape & size (deform) When force is removed -> original shape & size Remind you of anything? Definite x=0 Shape & SOLIDS are ELASTIC Volume FS = -kx STRESS = ELASTIC MODULUS x STRAIN Force per Measure of Elasticity of Unit Area Deformation material Length Elasticity Elastic Modulus and Induced strain depends on type of stress F F A L0 L0 F DL F L0 DL L0 STRESS = ELASTIC MODULUS x STRAIN F = Y DL Relative SI Units = Length Change N/m2 = A L0 Pascal (Pa) Young’s Modulus LENGTH Volume Elasticity F F DV V0 V0 STRESS = ELASTIC MODULUS x STRAIN F = -B DV Relative SI Units = Volume Change N/m2 = A V0 Pascal (Pa) Bulk Modulus VOLUME Pressure Uniform force F is acting over entire surface area A in a direction perpendicular to surface area F PRESSURE (P) Perpendicular Force per unit area A P= F A When would this situation occur? Pressure Uniform force F is acting over entire surface area A in a direction perpendicular to surface area F PRESSURE (P) Perpendicular Force per unit area A P= F A Fluids are NOT elastic -> do not return to initial state after being deformed But… Fluids do exerted force Pressure Force exerted by fluid on a submerged object is always PERPENDICULAR to surface of object F Fluids exert PRESSURE on submerged objects and the A walls of their container Fluids are NOT elastic -> do not return to initial state after being deformed But… Fluids do exerted force Pressure If a fluid is at rest in a container what y do we know about it? 0 It is in EQUILIBRIUM, so… The net FORCE acting on any F1 F2 portion of fluid is ZERO y1 F1(y1) = -F2(y1) y2 all points at the same DEPTH must be at the same PRESSURE P1(y1) = P2(y1) Pressure The net FORCE acting on any y portion of fluid is ZERO 0 all points at the same DEPTH P1A must be at the same PRESSURE y1 ∑Fy = P2A - P1A - Mg = 0 y2 r= M V Mg P2A P2 = P1 + rg(y1 - y2) Pressure What is the pressure at the surface of the fluid (open to the air)? P0A y Gas making up atmosphere 0 exerts pressure on fluid h P2 = P1 + rg(y1 - y2) y1 PA P = P0 + rgh Pressure P at depth h below the surface of a liquid open to the atmosphere is greater than atmosphere pressure (P0 = 1.013*105 Pa) by the amount rgh Pressure What if you change the pressure exerted at the surface? F P = P0 + rgh PASCAL’S PRINCIPLE A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the container Pressure PASCAL’S PRINCIPLE A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the container What happens if you apply a force F1 to one side of this apparatus? F1 A1 A2 F2 Pressure PASCAL’S PRINCIPLE A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the container F1/A1 = F2/A2 F1 Dx1 Dx2 F2 Pressure F1/A1 = F2/A2 How does Dx1 compare to Dx2? Fluids have a definite volume (incompressible) -> DV = 0 F1Dx1 = F2Dx2 What does this tell you about the work done F1 Dx1 Dx2 on the fluid? F2 Buoyancy What allows an object to float in a fluid? Is the object in equilibrium? M V Buoyancy What allows an object to float in a fluid? B Is the object in equilibrium? m ∑Fy = B - mobjg = 0 V mg Buoyancy What allows an object to float in a fluid? P1A Is the object in equilibrium? M ∑Fy = B - mobjg = 0 V Mg P2A ∑Fy = P2A - P1A - Mfluidg = 0 B = P2A - P1A = rfluidVfluidg BUOYANT FORCE Buoyancy What allows an object to float in a fluid? B Is the object in equilibrium? m ∑Fy = B - mobjg = 0 V mg ∑Fy = P2A - P1A - Mfluidg = 0 If the object is in B = P2A - P1A = rfluidVfluidg equilibrium with the fluid… BUOYANT robj Vfluid = FORCE rfluid Vobj Buoyancy B = rfluidVfluidg What if the object is rising or sinking? B B B a a m m m V V V mg a = 0 mg mg B = mobjg B - mobjg > 0 B - mobjg < 0 robj Vfluid = (rfluid-robj)Vobjg > 0 (rfluid-robj)Vobjg < 0 rfluid Vobj Properties of an Ideal Fluid An Ideal Fluid is… NONVISCOUS no internal friction between adjacent layers INCOMPRESSIBLE constant density Ideal Fluid Motion is… STEADY velocity, density, pressure at each point is constant in time WITHOUT TURBULENCE Angular velocity about center of each element is zero All points can translate but not rotate Ideal Fluid Motion How does v1 compare to v2? A2 v2 Dx2 Mass is conserved A1 v1 DM1 = DM2 Dx1 = v1Dt r1A1v1 = r2A2v2 Fluid is incompressible Volume of fluid leaving 1 = Volume of fluid entering 2 A1v1 = A2v2 in the same time interval Equation of Continuity Ideal Fluid Motion P2A2 Is energy conserved in an ideal fluid? v2 Dx2 P1A1 Dy2 v1 DM1 = DM2 Dx1 Dy1 What is the work done on the fluid? W1 = P1A1Dx1 W2 = -P2A2Dx2 W = P1V - P2V Ideal Fluid Motion P2A2 W = P1V - P2V A2 v2 Dx2 P1A1 Dy2 v1 Dx1 Dy1 Is the energy of the fluid changing? What types of energy are present? Wfluid = DKE + DPE P1 + (1/2)rv12 + rgy1 = P2 + (1/2)rv22 + rgy2 Ideal Fluid Motion P2A2 W = P1V - P2V A2 v2 Dx2 P1A1 Dy2 v1 Dx1 Dy1 BERNOULLI’S EQUATION The sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is equal at all points along the streamline P1 + (1/2)rv12 + rgy1 = P2 + (1/2)rv22 + rgy2 Questions of the Day 1) Two women of equal mass are standing on the same hard wood floor. One is wearing high heels and the other is wearing tennis shoes. Which statement is NOT true? a) both women exert the same force on the floor b) both women exert the same pressure on the floor c) the normal force that the floor exerts is the same for both women 2) A boulder is thrown into a deep lake. As the rock sinks deeper and deeper into the water what happens to the buoyant force? a) it increases b) it decreases c) it stays the same