Physics 106 Final Exam

1. Physics 106 Final Exam • When? • May. 10 Tuesday, 2:30 — 5:00 pm • Duration: 2.5 hours • Where? • ECEC-100 (SEC-008, 10) • What? • Phys-106 (75%), Phys-105 (25%) • How? • Review sessions (today’s lecture and next week’s recitation) • Equation sheet • Prof. Janow review session on Monday May 9, 3:00-5:00 pm in THL-2 • Sample exams on my website: web.njit.edu/~cao/106 • What if? • 28 multiple choice problems (2.5hr/28 ~ 5 min/prob) • 24 correct answers yields a score of 100% • Today follows Thursday schedule: Chap. 14.1-14.5 • HW13 due by 11:00 pm on May 10

2. Physics 106 Lecture 13Fluid MechanicsSJ 8th Ed.: Chap 14.1 to 14.5 • What is a fluid? • Pressure • Pressure varies with depth • Pascal’s principle • Methods for measuring pressure • Buoyant forces • Archimedes principle • Fluid dynamics assumptions • An ideal fluid • Continuity Equation • Bernoulli’s Equation

3. What is a fluid? Solids: strong intermolecular forces • definite volume and shape • rigid crystal lattices, as if atoms on stiff springs • deforms elastically (strain) due to moderate stress (pressure) in any direction Fluids: substances that can “flow” • no definite shape • molecules are randomly arranged, held by weak cohesive intermolecular forces and by the walls of a container • liquids and gases are both fluids Liquids: definite volume but no definite shape • often almost incompressible under pressure (from all sides) • can not resist tension or shearing (crosswise) stress • no long range ordering but near neighbor molecules can be held weakly together Gases: neither volume nor shape are fixed • molecules move independently of each other • comparatively easy to compress: density depends on temperature and pressure Fluid Statics - fluids at rest (mechanical equilibrium) Fluid Dynamics – fluid flow (continuity, energy conservation)

4. Mass and Density • Density is mass per unit volume at a point: • scalar • units are kg/m3, gm/cm3.. • rwater= 1000 kg/m3= 1.0 gm/cm3 • Volume and density vary with temperature - slightly in liquids • The average molecular spacing in gases is much greater than in liquids.

5. PDA h Force & Pressure • The pressure P on a “small” area DA is the ratio of the magnitude of the net force to the area • Pressure is a scalar while force is a vector • The direction of the force producing a pressure is perpendicular to some area of interest • At a point in a fluid (in mechanical equilibrium) the pressure is the same in any direction Pressure units: • 1 Pascal (Pa) = 1 Newton/m2 (SI) • 1 PSI (Pound/sq. in) = 6894 Pa. • 1 milli-bar = 100 Pa.

6. Forces/Stresses in Fluids • Fluids do not allow shearing stresses or tensile stresses. Tension Shear Compression • The only stress that can be exerted on an object submerged in a static fluid is one that tends to compress the object from all sides • The force exerted by a static fluid on an object is always perpendicular to the surfaces of the object Question: Why can you push a pin easily into a potato, say, using very little force, but your finger alone can not push into the skin even if you push very hard?

7. y=0 y1 F1 P1 h F2 P2 y2 Mg The extra pressure at extra depth h is: • In terms of density, the mass of the shaded fluid is: Pressure in a fluid varies with depth Fluid is in static equilibrium The net force on the shaded volume = 0 • Incompressible liquid - constant density r • Horizontal surface areas = A • Forces on the shaded region: • Weight of shaded fluid: Mg • Downward force on top:F1 =P1A • Upward force on bottom:F2 = P2A

8. air P0 h Ph liquid P0 Ph Pressure relative to the surface of a liquid Example: The pressure at depth h is: • All points at the same depth are at the same pressure; otherwise, the fluid could not be in equilibrium • The pressure at depth h does not depend on the shape of the container holding the fluid • P0 is the local atmospheric (or ambient) pressure • Ph is the absolute pressure at depth h • The difference is called the gauge pressure Preceding equations also hold approximately for gases such as air if the density does not vary much across h

9. Atmospheric pressure and units conversions • P0 is the atmospheric pressure if the liquid is open to the atmosphere. • Atmospheric pressure varies locally due to altitude, temperature, motion of air masses, other factors. • Sea level atmospheric pressure P0 = 1.00 atm = 1.01325 x 105 Pa = 101.325 kPa = 1013.25 mb (millibars) = 29.9213” Hg = 760.00 mmHg ~ 760.00 Torr = 14.696 psi (pounds per square inch)

10. near-vacuum Mercury (Hg) How high is the Mercury column? One 1 atm = 760 mm of Hg = 29.92 inches of Hg How high would a water column be? Height limit for a suction pump Pressure Measurement: Barometer • Invented by Torricelli (1608-47) • Measures atmospheric pressure P0 as it varies with the weather • The closed end is nearly a vacuum (P = 0) • One standard atm = 1.013 x 105 Pa.

11. DF p0 ph • Add piston of area A with lead balls on it & weight W. Pressure at surface increases by DP = W/A • Pressure at every other point in the fluid (Pascal’s law), increases by the same amount, including all locations at depth h. Ph Pascal’s Principle A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and to the walls of the container. • The pressure in a fluid depends on depth h and on the value of P0 at the surface • All points at the same depth have the same pressure. DP Example: open container

12. Pascal’s Law Device - Hydraulic press • Assume the working fluid is incompressible • Neglect the (small here) effect of height on pressure A small input force generates a large output force • The volume of liquid pushed down on the left equals the volume pushed up on the right, so: • Assume no loss of energy in the fluid, no friction, etc. Other hydraulic lever devices using Pascal’s Law: • Squeezing a toothpaste tube • Hydraulic brakes • Hydraulic jacks • Forklifts, backhoes mechanical advantage

13. Greek mathematician, physicist and engineer Computed p and volumes of solids Inventor of catapults, levers, screws, etc. Discovered nature of buoyant force – Eureka! hollow ball same upward force ball of liquid in equilibrium Archimedes Principle C. 287 – 212 BC Why do ships float and sometimes sink? Why do objects weigh less when submerged in a fluid? identical pressures at every point • An object immersed in a fluid feels an upward buoyant force that equals the weight of the fluid displaced by the object. Archimedes’s Principle • The fluid pressure increases with depth and exerts forces that are the same whether the submerged object is there or not. • Buoyant forces do not depend on the composition of submerged objects. • Buoyant forces depend on the density of the liquid and g.

14. The cube rises if B > Fg (rfl > rcube) Similarly for irregularly shaped objects rcube is the average density The cube sinks if Fg>B (rcube > rfl) Archimedes’s principle - submerged cube A cube that may be hollow or made of some material is submerged in a fluid Does it float up or sink down in the liquid? • The pressure at the top of the cube causes a downward force of Ptop A • The larger pressure at the bottom of the cube causes an upward force of Pbot A • The upward buoyant force B is the weight of the fluid displaced by the cube: • The extra pressure at the lower surface compared to the top is: • The weight of the actual cube is:

15. The direction of the motion of an object in a fluid is determined only by the densities of the fluid and the object • If the density of the object is less than the density of the fluid, the unsupported object accelerates upward • If the density of the object is more than the density of the fluid, the unsupported object sinks The apparent weight is the external force needed to restore equilibrium, i.e. Archimedes's Principle: totally submerged object The upward buoyant force is the weight of displaced fluid: Object – any shape - is totally submerged in a fluid of density rfluid The downward gravitational force on the object is: The volume of fluid displaced and the object’s volume are equal for a totally submerged object. The net force is:

16. Archimedes’s Principle: floating object An object sinks or rises in the fluid until it reaches equilibrium. The fluid displaced is a fraction of the object’s volume. At equilibrium the upward buoyant force is balanced by the downward weight of the object: The volume of fluid displacedVfluidcorresponds to the portion of the object’s volume below the fluid level and is always less than the object’s volume. Equate: Solving: • Objects float when their average density is less than the density of the fluid they are in. • The ratio of densities equals the fraction of the object’s volume that is below the surface

17. Apply Archimedes’ Principle displaced seawater glacial fresh water ice From table: Water expands when it freezes. If not.... ...ponds, lakes, seas freeze to the bottom in winter What if iceberg is in a freshwater lake? Floating objects are more buoyant in saltwater Freshwater tends to float on top of seawater... Example: What fraction of an iceberg is underwater?

18. Low viscosity • gases • Medium viscosity • water • other fluids that pour and flow easily • High viscosity • honey • oil and grease • glass Ideal Fluids – four approximations to simplify the analysis of fluid flow: • The fluid is nonviscous – internal friction is neglected • The flow is laminar (steady, streamline flow) – all particles passing through a point have the same velocity at any time. • The fluid is incompressible – the density remains constant • The flow is irrotational – the fluid has no angular momentum about any point. A small paddle wheel placed anywhere does not feel a torque and rotate Fluids’ Flow is affected by their viscosity • Viscosity measures the internal friction in a fluid. • Viscous forces depend on the resistance that two adjacent layers of fluid have to relative motion. • Part of the kinetic energy of a fluid is converted to internal energy, analogous to friction for sliding surfaces

19. cross-section area A velocity v length dx Flow of an ideal fluid through a short section of pipe Constant density and velocity within volume element dV Incompressible fluid means dr/dt = 0 Mass flow rate = amount of mass crossing area A per unit time = a “current”sometimes called a “mass flux”

20. The fluid is incompressible so: r2 • This is called the equation of continuity for an incompressible fluid • The product of the area and the fluid speed (volume flux) at all points along a pipe is constant. r1 Equation of Continuity: conservation of mass • An ideal fluid is moving through a pipe of nonuniform diameter • The particles move along streamlines in steady-state flow • The mass entering at point 1 cannot disappear or collect in the pipe • The mass that crosses A1 in some time interval is the same as the mass that crosses A2 in the same time interval. The rate of fluid volume entering one end equals the volume leaving at the other end Where the pipe narrows (constriction), the fluid moves faster, and vice versa