Phsics 0 Lecture 6 Lecture 6 Goals: Understand pressure in s v Use rchimedes principle to understand buoanc v Understand the equation of continuit v Use an ideal-fluid model to stud fluid flow. luids l t ordinar temperatures, matter exists in one of three states v Solid: Has a shape and forms a surface v Liquid: Has no shape but forms a surface v Gas : No shape and does not form a surface solid gas Phsics 0: Lecture 6, Pg Increasing temperature Phsics 0: Lecture 6, Pg luids l luids: Liquids or Gases v Shape determined b container v Close contact at boundaries. No macroscopic gaps v orces exerted b fluids are alwas normal to the surface. v luids can flow l luids: v The have mass v The have weight v The carr energ v The have velocit when flow v Newton s Laws appl luids l n intrinsic parameter of a fluid v Densit m = units : kg/m = 0 - g/cm (water) =.000 x 0 kg/m =.000 g/cm (ice) = 0.97 x 0 kg/m = 0.97 g/cm (air) =.9 kg/m =.9 x 0 - g/cm (Hg) =.6 x0 kg/m =.6 g/cm (W or u) = 9. x0 kg/m = 9. g/cm Phsics 0: Lecture 6, Pg Phsics 0: Lecture 6, Pg 4 r luids l nother parameter: Pressure v luid exerts force on the s it contacts v Molecules constantl bounces off the surface with = ( mv/ t) (momentum transfer / time) l n force exerted b a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface. v orce (a vector) in a fluid can be expressed in terms of pressure (a scalar) as: = Pnˆ Do not confuse pressure with momentum nˆ P = Phsics 0: Lecture 6, Pg 5 What is the SI unit of pressure?. Pascal B. tmosphere C. Bernoulli D. Young E. p.s.i. Units : N/m = Pa (Pascal) bar = 0 5 Pa mbar = 0 Pa torr =. Pa atm =.0 x0 5 Pa = 0 mbar = 760 Torr = 4.7 lb/ in (=PSI) Phsics 0: Lecture 6, Pg 6 Page
Phsics 0 Lecture 6 bed of nails l Distributing the force over a large area. Ping pong ball bazooka l What is the initial force on the ping pong ball? l ssuming this force is constant, how fast is the ping pong ball travelling when it leaves the tube? 5 = P = atm π r = 0 π (0.00) = 0 N N Phsics 0: Lecture 6, Pg 7 W = d = 0 N m = 60 J W = mv = 60 J v = 440 m/s This is a crude model.the speed of sound in air is 0 m/s Phsics 0: Lecture 6, Pg 8 Pressure vs. Depth Incompressible luids (s) l When the pressure is small, P 0 relative to the bulk modulus of the fluid, we can treat the densit as constant independent of pressure: P incompressible fluid l or an incompressible fluid, the P densit is the same everwhere, mg but the pressure is NOT! l P() = P 0 - g l Gauge pressure (subtract P 0 ) = + m g l P Gauge = P() - P 0 = + g / = / + g/ P = P g Phsics 0: Lecture 6, Pg 9 P = P Pressure vs. Depth in water g water =.0 gm/cm = g =0m/s 0 5 = P 0 N/m Ever 0 meters the pressure increases b atm kg/m = 0m mg P + P P 0 P Phsics 0: Lecture 6, Pg 0 Pascal s Principle l or a uniform fluid in an open container pressure same at a given depth independent of the container l Does P =P B? p() l luid level is the same everwhere in a connected container, assuming no surface forces B Phsics 0: Lecture 6, Pg Phsics 0: Lecture 6, Pg Page
Phsics 0 Lecture 6 Pressure l What happens with two fluids?? l Consider a U tube containing s of densit and as shown: d I Pressure Measurements: Barometer l Invented b Torricelli l long closed tube is filled with mercur and inverted in a dish of mercur v The closed end is nearl a vacuum v P =P B Compare the densities of the s: () < (B) = (C) > l Measures atmospheric pressure as atm = 0.760 m (of Hg) Phsics 0: Lecture 6, Pg Phsics 0: Lecture 6, Pg 4 Pascal s Principle in action: Hdraulics, a force amplifier l Consider the sstem shown: v downward force is applied to the piston of area. v This force is transmitted through the to create an upward force. v Pascal s Principle sas that increased pressure from ( / ) is transmitted throughout the. = d P P P = P / = / / = / Phsics 0: Lecture 6, Pg 5 d Pascal s Principle: Example l Now consider the set up shown on right. v Mass M is placed on the larger bore piston, 0. v If =, how do d and d B compare? v This results in a difference d in the levels on the left. v Equilibrium occurs when the pressures at P (left & right) are equal. P = P / = / g d / = g d / d = d B Case Case d d B P P M 0 M 0 Phsics 0: Lecture 6, Pg 6 rchimedes Principle: Eureka Moment l Suppose we weigh an in air () and in water (). How do these weights compare? W W < W W = W W > W W? l Buoant force = Weight of the fluid displaced Phsics 0: Lecture 6, Pg 7 The Golden Crown l In the first centur BC the Roman architect itruvius related a stor of how rchimedes uncovered a fraud in the manufacture of a golden crown commissioned b Hiero II, the king of Sracuse. The crown (corona in itruvius s Latin) would have been in the form of a wreath, such as one of the three pictured from grave sites in Macedonia and the Dardanelles. Hiero would have placed such a wreath on the statue of a god or goddess. Suspecting that the goldsmith might have replaced some of the gold given to him b an equal weight of silver, Hiero asked rchimedes to determine whether the wreath was pure gold. nd because the wreath was a hol dedicated to the gods, he could not disturb the wreath in an wa. (In modern terms, he was to perform nondestructive testing). rchimedes solution to the problem, as described b itruvius, is neatl summarized in the following excerpt from an advertisement: l The solution which occurred when he stepped into his bath and caused it to overflow was to put a weight of gold equal to the crown, and known to be pure, into a bowl which was filled with water to the brim. Then the gold would be removed and the king s crown put in, in its place. n allo of lighter silver would increase the bulk of the crown and cause the bowl to overflow. l rom http://www.math.nu.edu/~crorres/rchimedes/crown/crownintro.html Phsics 0: Lecture 6, Pg 8 Page
Phsics 0 Lecture 6 Sink or loat? l The buoant force is equal to the weight of the that is displaced. l If the buoant force is larger than the weight of the, it will float; otherwise it will sink. B mg Bar Trick What happens to the water level when the ice melts? Expt. Expt. piece of rock on top of ice l We can calculate how much of a floating will be submerged in the : v Object is in equilibrium g B = mg = g = Phsics 0: Lecture 6, Pg 9. It rises B. It stas the same C. It drops Phsics 0: Lecture 6, Pg 0 = = = 4 = 5 m < m < m < m 4 < m 5 What is the final position of each block? = = = 4 = 5 m < m < m < m 4 < m 5 What is the final position of each block? Not this But this Phsics 0: Lecture 6, Pg Phsics 0: Lecture 6, Pg Home Buoanc Buoanc l small lead weight is fastened to a large block and the combination floats on water with the water level with the top of the block as shown. l small lead weight is fastened to a large block and the combination floats on water with the water level with the top of the block as shown. v If ou turn the + upside-down, What happens? v If ou turn the + upside-down, What happens (assuming densit of > water)? () It sinks (B) (C) (D) () It sinks (B) (C) (D) ctive igure ctive igure Phsics 0: Lecture 6, Pg Phsics 0: Lecture 6, Pg 4 Page 4
Phsics 0 Lecture 6 Home More Buoanc l Two identical cups are filled to the same level with water. One of the two cups has plastic balls floating in it. v Which cup weighs more? Cup I Cup II More Buoanc l Two identical cups are filled to the same level with water. One of the two cups has plastic balls floating in it. l v Which cup weighs more? Cup I Cup II () Cup I (B) Cup II (C) the same (D) can t tell () Cup I (B) Cup II (C) the same (D) can t tell Phsics 0: Lecture 6, Pg 5 Phsics 0: Lecture 6, Pg 6 Home Even More Buoanc l plastic ball floats in a cup of water with half of its volume submerged. Next some oil ( oil < ball < water ) is slowl added to the container until it just covers the ball. oil l ll of chapter 4 or Tuesda v Relative to the water level, the ball will: water Hint : What is the buoant force of the part in the oil as compared to the air? () move up (B) move down (C) sta in same place Phsics 0: Lecture 6, Pg 7 Phsics 0: Lecture 6, Pg 8 Page 5