HW #10 posted, due Thursday, Dec 2, 11:59 p.m. (last HW that contributes to the final grade) Last Lecture Class: States/Phases of Matter, Deformation of Solids, Density, Pressure Today: Pressure vs. Depth, Buoyant Forces and Archimedes Principle Office hours today cancelled 1
Pressure vs. Depth in the Ocean Deep sea divers have to be careful not to surface too quickly so as not to develop problems due to the bends. Submarines must be engineered to withstand enormous pressures (to prevent collapse) at great depths in the ocean. In general, we know from our wealth of common knowledge that the pressure in the ocean tends to increase with depth. What is the physics of this? 2
Recall From Last Lecture: Pressure Fluids (liquids, gases) cannot sustain shearing stresses. The only stress that a fluid can exert on a submerged object is one that tends to compress it. Force exerted by fluid on object ALWAYS PERPENDICULARto the object s surfaces. If F is the magnitude of a force exerted perpendicular to a surface of area A, the pressure P is : P F A F : Newtons A : m 2 P : Pascals 3
Variation of Pressure With Depth All portions of the fluid must be in F 1 F 2 static equilibrium. Thus, at some given depth, all points must be at the same pressure. cross sectional area A mg P 1 A P 2 A 2 External forces: Downward force of gravity, mg Upward force exerted by liquid below due to pressure = P 2 A Downward force exerted by liquid above due to pressure = P 1 A 4
Variation of Pressure With Depth cross sectional area A mg P 1 A P 2 A y 1 y 2 Balancing forces: P A Mass: m 2 P1 A mg 0 V A( y 1 y2) Substituting: P P g( y ) As (y 1 y 2 ) becomes larger, P 2 increases!! 2 1 ( 1 y2 LtP Let 0 = pressure at surface, and let lth = depth thbl below surface : P P 0 g gh The pressure P at a depth h below the surface is greater than that at the surface by the amount ρgh. 5
Example: Force on Your Ear Underwater Beijing Olympic Pool If you swim underwater, you may feel pain in your ears (i.e., they might pop ). Assuming your ear drums have an area of 1 cm 2, calculate the force on your ear drums at a depth of 5 m. [Note: Air inside ear is normally at atmospheric pressure.] 6
Example: Multiple Fluid Layers 30 cm 20 cm A container is filled to a depth of 20 cm with water. On top of the water floats a 30 cm thick layer of oil, with specific gravity 0.700. What is the pressure at the bottom? Specific Gravity: Ratio of substance s density to the density of water 7
Pascal s Principle piston Suppose a piston compresses an enclosed fluid, which has a surface pressure of P 0 prior to compression. We know the pressure depends on depth. How is the increase in pressure on the surface transmitted throughout the rest of the fluid? 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. 8
Application: Hydraulic Lifts In a hydraulic lift, a small applied force is transformed into a much larger force. How do these work? 9
F 1 piston area area A 1 A 2 F 2 hydraulic fluid By Pascal s Principle: Pressure at Piston = Pressure at Car Lift F A 1 F2 A2 F2 F1 1 A2 A1 A 2 > A 1 : So force applied by piston it is amplified!! 10
F 1 piston area area A 1 A 2 F 2 hydraulic fluid But Energy Must Be Conserved: Work Done by Piston = Work Done by Car Lift F d 1 1 F d 2 2 d 2 d 1 F 1 F 2 F 1 < F 2 : So car moves up less than the piston it moves down!! 11
How Can These Massive Ships Float? U.S. s Nimitz class aircraft carriers are largest ships in the world. ~1100 feet long Displacements ~ 100,000 tons Can transport up to ~90 F 18 aircraft 12
Archimedes Principle Archimedes of Syracuse, 287 212 B.C. Greek mathematician, physicist, engineer, astronomer, Perhaps greatest mathematician i of antiquity, and of all time weight of displaced fluid mg Archimedes Principle : An object completely or partially submerged in a liquid is buoyed by a force with magnitude equal to the weight of the fluid displaced d by the object. 13
Buoyant Forces buoyant force Upward Buoyant Force = Mass of Fluid Displaced by Object x g mg Mass of Fluid Displacedby Object = ρ fluid x V object The upward buoyant force is why it is easier to lift something in a swimming pool, as compared to on dry land! 14
Buoyant Forces buoyant force Upward Buoyant Force = Mass of Fluid Displaced by Object x g mg Mass of Fluid Displacedby Object = ρ fluid x V object If object is totally t submerged in a fluid (as shown here) : Net Upward Force = (ρ fluid x V object x g) (ρ object x V object x g) If > 0: Rises!! If < 0: Sinks!! upward buoyant force weight of object 15
buoyant force Buoyant Forces mg If object is floating on the surface (i.e., partially submerged) : Upward buoyant force is balanced by downward gravity force : ρ fluid x V submerged x g = ρ object x V object x g upward buoyant force (only due to volume of object that is submerged!!) weight of object 16
It s Only the Tip of the Iceberg The density of water is 1000 kg/m 3, and the density of ice is 917 kg/m 3. Calculatethe the fraction of aniceberg s volume that is submerged underwater. Visualization of what an iceberg may look like (from wikipedia). 17
Conceptual Question The density of lead is greater than iron, and the density of iron is greater than water. If two solid objects of lead and iron, both with identical dimensions, are fully submerged under water, is the buoyant force on the lead object (a) greater than (b) less than (c) equal to the buoyant force on the iron object? 18
Is It Gold? Example 9.8, p. 287 A bargain hunter purchases a gold crown at a flea market. When she gets home, she hangs it from a scale, and finds its weight to be 7.84 N. She then weighs the crown immersed in water, and now the scale reads 6.86 686N. Is it gold??? 19
Next Class 9.7 9.8 : Fluid Motion, Bernoulli s Equation 20