Purdue University, Physics 220 Lecture 18 Fluid Dymanics Lecture 17 Purdue University, Physics 220

Archimedes’ Principle Buoyant Force FB Weight of fluid displaced FB = fluid Vdisp g Fg = mg = object Vobject g Object sinks if object > fluid Object floats if object < fluid If object floats… FB = Fg Therefore fluid Vdisp g = object Vobject g Therefore Vdisp/ Vobject = object/fluid Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Exercise 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: A) Go up, causing the water to spill out of the glass B) Go down C) Stay the same CORRECT Wice = FB = W g Vdisplaced Wwater = W g Vwater Wice = Wwater Vdisplaced = Vwater Lecture 17 Purdue University, Physics 220

Exercise 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 weigh the same Tub of water Tub of water + ship Overflowed water CORRECT Weight of ship = Buoyant force = Weight of displaced water Lecture 17 Purdue University, Physics 220

iClicker 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 sinking to the bottom C) They will weigh the same CORRECT Tub of water Tub of water + ship Overflowed water Weight of ship > Buoyant force = Weight of displaced water Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Fluid Flow Concepts Volume flow rate: V/t = Av (m3/s) Mass flow rate: m/t = Av (kg/s) Continuity: A1 v1 = A2 v2 mass flow rate the same everywhere Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Continuity A four-lane highway merges down to a two-lane highway. The officer in the police car observes 8 cars passing every second, at 30 mph. How many cars does the officer on the motorcycle observe passing every second? A) 4 B) 8 C) 16 How fast must the cars in the two-lane section be going? A) 15 mph B) 30 mph C) 60 mph New slide Fall 2005 not in handouts. All cars stay on the road… Must pass motorcycle too Must go faster, else pile up! Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Faucet A stream of water gets narrower as it falls from a faucet (try it & see). Explain this phenomenon using the equation of continuity. A1 A2 V1 V2 From the continuity equation, we know that A1V1=A2V2. Therefore, as the velocity increases, the cross-sectional area has to decrease. Since the water is in free fall gravity causes the speed to increase as it goes down. Because the speed is increasing the cross sectional area has to decrease in order to keep the product of the two equal. Lecture 17 Purdue University, Physics 220

Flow Speed and Pressure Continuity Equation says fluid speeds up going to smaller opening, slows down going to larger opening Acceleration due to change in pressure: P1 > P2 Smaller tube has FASTER flow and LOWER pressure Where the velocity is high, the pressure is low and where the velocity is low the pressure is high Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Bernoulli’s Equation Consider tube where both area and height change W = DK + DU (P1-P2) V = ½ m (v22 – v12) + mg(y2-y1) (P1-P2) V = ½ rV (v22 – v12) + rVg(y2-y1) P1+rgy1 + ½ rv12 = P2+rgy2 + ½rv22 Lecture 17 Purdue University, Physics 220

Dynamics of Ideal Fluids Conservation of mass  Continuity equation The same volume of fluid passes per unit time Conservation of energy  Bernoulli’s equation which relates height, velocity and pressure of the fluid Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Question What will happen when I “blow” air between the two plates? A) Move Apart B) Come Together C) Nothing Several Demos here include cans on straws. There is air pushing on both sides of plates. If we get rid of the air in the middle, then just have air on the outside pushing them together. Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Exercise Through which hole will the water come out fastest? P1+rgy1 + ½ rv12 = P2+rgy2 + ½rv22 A Note: All three holes have same pressure P=1 Atmosphere B rgy1 + ½ rv12 = rgy2 + ½rv22 C gy1 + ½ v12 = gy2 + ½v22 Smaller y gives larger v. Hole C is fastest Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 iClicker A large bucket full of water has two drains. One is a hole in the side of the bucket at the bottom, and the other is a pipe coming out of the bucket near the top, which bent is downward such that the bottom of this pipe even with the other hole, like in the picture below: Through which drain is the water spraying out with a larger speed? A) The hole B) Same C) The pipe The speed of the water depends on the pressure at the opening through which the water flows, and on the depth of the opening from the surface of the water. Since both the hole and the opening of the downspout are at [the same] depth below the surface, water from the two will be discharged with the same speeds. Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Example A garden hose with inner diameter 2 cm, carries water at 2.0 m/s. To spray your friend, you place your thumb over the nozzle giving an effective opening diameter of 0.5 cm. What is the speed of the water exiting the hose? What is the pressure difference between inside the hose and outside? Continuity Equation A1 v1 = A2 v2 v2 = v1 ( A1/A2) = v1 ( r12 / r22) = 2 m/s x 16 = 32 m/s Lift on house if v=30 m/s. Density of air = 1.27 kg / m^3 Bernoulli Equation P1+gy1 + ½ v12 = P2+gy2 + ½v22 P1 – P2 = ½  (v22 – v12) = ½ x (1000 kg/m3) (1024-4 m2/s2) = 5.1x105 Pa Lecture 17 Purdue University, Physics 220

Purdue University, Physics 220 Lift a House Calculate the lift force on a 15 m x 15 m house when a 30 m/s wind (1.29 kg/m3) blows over the top. P1+rgy1 + ½ rv12 = P2+rgy2 + ½rv22 P1 – P2 = ½ r (v22 – v12) = ½ r (v22 – v12) = ½ (1.29) (302) N / m2 = 581 N/ m2 F = P A = 581 N/ m2 (15 m)(15 m) = 131,000 N = 29,000 pounds! Lecture 17 Purdue University, Physics 220