HW #10 posted, due Thursday, Dec 2, 11:59 p.m. (last HW that contributes to the final grade)

1. 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

2. 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 ?

3. 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 : F : Newtons A : m2 P : Pascals

4. Variation of Pressure With Depth All portions of the fluid must be in static equilibrium. F2 F1 Thus, at some given depth, all points must be at the same pressure. cross-sectional area A External forces: Downward force of gravity, mg P1A Upward force exerted by liquid below due to pressure = P2A mg P2A Downward force exerted by liquid above due to pressure = P1A

5. Variation of Pressure With Depth cross-sectional area A Balancing forces: P1A y1 Mass: y2 mg P2A Substituting: As (y1 – y2) becomes larger, P2 increases !! Let P0 = pressure at surface, and let h = depth below surface : The pressure P at a depth h below the surface is greater than that at the surface by the amount ρgh.

6. 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 cm2, calculate the force on your ear drums at a depth of 5 m. [Note: Air inside ear is normally at atmospheric pressure.]

7. 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. Specific Gravity: Ratio of substance’s density to the density of water What is the pressure at the bottom ?

8. Pascal’s Principle piston Suppose a piston compresses an enclosed fluid, which has a surface pressure of P0 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.

9. Application: Hydraulic Lifts In a hydraulic lift, a small applied force is “transformed” into a much larger force. How do these work ?

10. F1 piston area A1 area A2 F2 hydraulic fluid By Pascal’s Principle: Pressure at Piston = Pressure at Car Lift A2 > A1 : So force applied by piston is amplified !!

11. F1 piston area A1 area A2 F2 hydraulic fluid But … Energy Must Be Conserved: Work Done by Piston = Work Done by Car Lift F1 < F2 : So car moves up less than the piston moves down !!

12. 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

13. Archimedes’ Principle Archimedes of Syracuse, 287 –212 B.C. • Greek mathematician, physicist, engineer, astronomer, … • Perhaps greatest mathematician of antiquity, and of all time weight of displaced fluid 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 by the object. mg

14. Buoyant Forces buoyant force Upward Buoyant Force Mass of Fluid Displaced by Object x g = Mass of Fluid Displaced by Object mg = ρfluid x Vobject The upward buoyant force is why it is easier to lift something in a swimming pool, as compared to on dry land !

15. Buoyant Forces buoyant force Upward Buoyant Force Mass of Fluid Displaced by Object x g = Mass of Fluid Displaced by Object mg = ρfluid x Vobject If object is totally submerged in a fluid (as shown here) : Net Upward Force = (ρfluid x Vobject x g) – (ρobject x Vobject x g) weight of object upward buoyant force If > 0: Rises !! If < 0: Sinks !!

16. Buoyant Forces buoyant force If object is floating on the surface (i.e., partially submerged) : mg Upward buoyant force is balanced by downward gravity force : ρfluid x Vsubmerged x g = ρobject x Vobject x g upward buoyant force weight of object (only due to volume of object that is submerged !!)

17. “It’s Only the Tip of the Iceberg” The density of water is 1000 kg/m3, and the density of ice is 917 kg/m3. Calculate the fraction of an iceberg’s volume that is submerged underwater. Visualization of what an iceberg may look like (from wikipedia).

18. 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 ?

19. “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 N. Is it gold ???

20. Next Class • 9.7 – 9.8 : Fluid Motion, Bernoulli’s Equation