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Announcements Physics 131: Lecture 23 Today’s Agenda Description of Fluids at Rest Pressure vs Depth Pascal’s Principle: hydraulic forces Archimedes Principle: objects in a fluid Bernoulli’s equation Today: Last new material Wednesday in class (Dr. Harlow and I will review the semester) Final Exam: Thursday 12/13 at 2:00pm Review Session: Thursday 12/6 10:00am 10:00am-12pm 12pm in SS2118 (Dr. Harlow and I) Exam Jam My extra office hours: 12/11 9am – 12 and 2 – 5pm 12/12 9am – 12 and 2 – 5pm Drop in center: Check link on portal Conceptual Questions (mastering physics) Physics 201: Lecture 1, Pg 1 Physics 201: Lecture 1, Pg 2 Fluids States of Matter What parameters do we use to describe fluids? Fluids Solid Hold Volume Hold Shape Liquid Hold H ld V Volume l Adapt Shape Gas Adapt Volume Adapt Shape Density Pressure m V P F A These letters look similar! Be careful. Physics 201: Lecture 1, Pg 3 Physics 201: Lecture 1, Pg 4 You are my density Pressure The density of a material is its mass per unit volume: Physics 201: Lecture 1, Pg 5 Pressure = Force/Area Units SI Pascal = Newtons/m2 If I apply 30 Pa of pressure over an area of 20 m2, how much force do I apply? F = P A = 30 N/m2 20 m2 = 600 Newtons Physics 201: Lecture 1, Pg 6 Page 1 Pressure Pressure The same force applied over a smaller area results in greater pressure – think of poking a balloon with your finger and then with a needle. Physics 201: Lecture 1, Pg 7 Physics 201: Lecture 1, Pg 8 Pressure vs. Depth Atmospheric Pressure Basically weight of atmosphere! Air molecules are colliding with you right now! Pressure = 1x105 N/m2 = 14.7 lbs/in2! Example: Magdeburg sphere w/ r = 0.1 m A = 4 p r2 = 0.125 m2 F = 12,000 Newtons (over 2,500 lbs)! Physics 201: Lecture 1, Pg 9 Physics 201: Lecture 1, Pg 10 Gauge Pressure Clicker Question 1: Many pressure gauges, such as tire gauges and the gauges on air tanks, measure not the actual or absolute pressure p but what is called gauge pressure pg. What can you say about the pressures at points 1, 2, and 3? A. p1 = p2 = p3. B. p1 = p2 > p3. C p3 > p1 = p2. C. D. p3 > p1 > p2. E. p1 = p3 > p2. where 1 atm = 101.3 kPa. Physics 201: Lecture 1, Pg 11 Physics 201: Lecture 1, Pg 12 Page 2 Pascal’s Principle Pascal’s Principle Pascal’s principle: An external pressure applied to an enclosed fluid is transmitted unchanged to every point within the fluid. Consider the system shown: A downward force F1 is applied to the piston of area A1. This force is transmitted through the liquid to create an upward force F2. Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid. F1 F 2 A1 A 2 F1 F2 d2 d1 A1 A2 A F2 F1 2 A1 Check that F•d is the same on both sides. Energy is conserved! Physics 201: Lecture 1, Pg 13 Physics 201: Lecture 1, Pg 14 Clicker Question 2: Clicker Question 3: The small piston of a hydraulic lift has a cross sectional area of 3 cm2 . You need to lift a 15,000 N weight by applying a force of only 200 N. What is the area of the large piston, assuming both pistons are at the same height, as shown in the drawing? If you push the small piston down a distance of 25 cm. How far up does the weight move? (a) 0.3 cm (b) 8.2 cm (c) 25 cm (a) 75 cm2 (b) 125 cm2 (c) 225 cm2 Physics 201: Lecture 1, Pg 15 Physics 201: Lecture 1, Pg 16 When the cylinder is lowered into the water, how will the scale readings for the cylinder and the water change? Suppose we weigh an object in air and in water. Since the pressure at the bottom of the object is greater than that at the top of the object, the water exerts a net upward force, the buoyant force, on the object. The buoyant force is equal to the difference in th pressures times the ti the th area. a) b) c) d) e) W2? W1 Archimedes’ Principle Clicker Question 4: cylinder’s scale will decrease, but water’s scale will increase cylinder’s scale will increase, but water’s scale will decrease cylinder’s scale will decrease, but water’s scale will not change both scales will decrease W1 W2? both scales will increase FB F2 F1 FB ( p2 p1 ) A ghA FB liquid gVliquid M liquid g Wliquid y1 F 1 y2 p 1 A p 2 F 2 Therefore, the buoyant force is equal to the weight of the fluid displaced. Physics 201: Lecture 1, Pg 17 Physics 201: Lecture 1, Pg 18 Page 3 Clicker Question 5: Archimedes’ Principle Imagine holding two identical bricks in place underwater. Brick 1 is just beneath the surface of the water, and brick 2 is held about 2 feet down. The force needed to hold brick 2 in place is: So the force the water pushes up with is equal to weight of the water displaced by the object. FB = Wfluid FB = mfluidg = (Vg)fluid a) greater b) the same 1 c) smaller 2 Physics 201: Lecture 1, Pg 19 Physics 201: Lecture 1, Pg 20 Clicker Question 5.5: Will it Float? Object is in equilibrium Two blocks are of identical size. One is made of lead and sits on the bottom of a pond; the other is of wood and floats on top. Upon which is the buoyant force greater? y FB (mg ) object FBmg liquid g Vdispl. object g Vobject Vdispl. displ Vobject A. On the lead block. B. On the wood block. C. They both experience the same buoyant force. object liquid The Tip of The Iceberg: What fraction of an iceberg is submerged? Vwater displ. Vice Physics 201: Lecture 1, Pg 21 ice 917 kg/m3 90% water 1024 kg/m3 Physics 201: Lecture 1, Pg 22 Archimedes’ Principle Clicker Question 6: An object made of material that is denser than water can float only if it has indentations or pockets of air that make its average density less than that of water. A plastic cube (ρ = 780 kg/m3, V = 0.71 m3) is suspended by a string such that half the volume is submersed in water (ρ = 1000 kg/m3) as shown in the figure. What is the tension T in the string? (a) (b) (c) (d) (e) Physics 201: Lecture 1, Pg 23 372 N 428 N 931 N 1950 N 2380 N Physics 201: Lecture 1, Pg 24 Page 4 Clicker Question 7: Clicker Question 8: Suppose you float a large ice-cube in a glass of water, and that after you place the ice in the glass the level of the water is at the very brim. When the ice melts, the level of the water in the glass will: Which weighs more: A. A large bathtub filled to the brim with water. B. A large bathtub filled to the brim with water with a battle-ship floating in it. C. They will weigh the same. A. Go up, A up causing the water to spill out of the glass glass. B. Go down. C. Stay the same. Tub of water Overflowed water Physics 201: Lecture 1, Pg 25 Physics 201: Lecture 1, Pg 26 Fluids in motion Example Water comes out of a hose which has a radius of 1 cm. You are holding your finger over the hose and the area the hose can flow through is 2.01 10-4 m2. If water in hose has speed 1m/s, what is speed p when water p passes yyou finger? g Assume the water is incompressible. 1A1v1 = 2A2v2 1 = 2 A1v1 = A2v2 (.01 m)2(1 m/s) = (2.01 10-4 m2) v2 v2 = 1.56 m/s Physics 201: Lecture 1, Pg 27 Physics 201: Lecture 1, Pg 28 Bernoulli’s equation Bernoulli’s equation When a fluid moves from a wider area of a pipe to a narrower one, its speed increases; therefore, work has been done on it. W = Fd = P(Ad) = (P)V W =½mvf2 - ½mvi2 W = Fd = P(Ad) = (P)V So: ((P2 – P1))V = ½mvf2 - ½mvi2 m= V (P2 – P1)V = ½ Vvf2 - ½ Vvi2 (P2 – P1) = ½ v12 - ½ v22 P2 + ½ v22 = P1 + ½ v12 P2+gy2 + ½v22 = P1+gy1 + ½ v12 Physics 201: Lecture 1, Pg 29 Physics 201: Lecture 1, Pg 30 Page 5 Bernoulli’s equation Bernoulli’s equation The general case, where both height and speed may change, is described by Bernoulli’s equation: The kinetic energy of a fluid element is: Equating the work done to the increase in kinetic energy gives: This equation is essentially a statement of conservation of energy in a fluid. Physics 201: Lecture 1, Pg 31 Physics 201: Lecture 1, Pg 32 Applications of Bernoulli’s equation Bernoulli’s equation (at constant height) P2 + ½ v22 = P1 + ½ v12 P + ½ v2 This quantity must always remain the same So fluids with faster velocity will exert less pressure! This has many interesting applications! The Bernoulli effect is simple to demonstrate – all you need is a sheet of paper. Hold it by its end, so that it would be horizontal if it were stiff, and blow across the top. The paper will rise, due to the higher speed, and therefore lower pressure, above the sheet. Demos! Physics 201: Lecture 1, Pg 33 Physics 201: Lecture 1, Pg 34 Applications of Bernoulli’s equation Applications of Bernoulli’s equation: Airplane wing Physics 201: Lecture 1, Pg 35 This lower pressure at high speeds is what rips roofs off houses in hurricanes and tornadoes, and causes the roof of a convertible to expand upward. It even helps prairie dogs with air circulation! Physics 201: Lecture 1, Pg 36 Page 6 Clicker Question 9: Oil (ρo = 920 kg/m3) flows through a horizontal cylindrical pipe of radius R1 = 12 cm with a speed of v1 = 2 m/s. Along the flow direction the pipe widens to a radius R2 = 24 cm (as shown in the figure), the speed of the oil is then equal to v2. What is the pressure difference ∆P = P2 – P1 between the wide and narrow parts of the pipe? (a) ∆P = 660 Pa (b) ∆P = 1185 Pa (c) ∆P = 1520 Pa (d) ∆P = 1725 Pa (e) ∆P = 1950 Pa Physics 201: Lecture 1, Pg 37 Page 7