Physics 201 : Lecture 24 Fluid Statics Pascal’s Principle Archimedes Principle (Buoyancy) Fluid Dynamics Continuity Equation Bernoulli Equation 4/10/2018 Physics 201, UW-Madison

Pressure variation with depth Fluids Pressure (P) P = Force/Area [N/m2] 1 N/m2 = 1 Pascal (Pa) Density = Mass/Volume  = M / V units = kg/m3 Pressure variation with depth P =  g h Atmospheric Pressure Even when there is no breeze air molecules are continuously bombarding everything around - results in pressure normal atmospheric pressure = 1.01 x 105 Pa (14.7 lb/in2) 4/10/2018 Physics 201, UW-Madison

Compressiblity         Density & Pressure are related by the Bulk Modulus LIQUID: incompressible (density almost constant) GAS: compressible (density depends a lot on pressure)         Bulk modulus (Pa=N/m2) H2O Steel Gas (STP) Pb 4/10/2018 Physics 201, UW-Madison

Variation of pressure with depth True for all shapes of containers 4/10/2018 Physics 201, UW-Madison

Pascal’s Principle A change in pressure in an enclosed fluid is transmitted undiminished to all the fluid and to its container. This principle is used in hydraulic system P1 = P2 (F1 / A1) = (F2 / A2) Can be used to achieve a mechanical advantage F2 = F1 (A2 / A1) Work done is the same: height by which the surface A2 rises is smaller than the change in the height of surface with area A1. F1 A1 F2 A2 4/10/2018 Physics 201, UW-Madison

Using Fluids to Measure Pressure Use Barometer to measure Absolute Pressure Barometer Top of tube evacuated (p=0) Bottom of tube submerged into pool of mercury open to atmosphere (p=p0) Pressure dependence on depth: Use Manometer to measure Gauge Pressure p0 Dh Manometer p1 Measure pressure of volume (p1) relative to the atmospheric pressure (º gauge pressure ) The height difference (Dh) measures the gauge pressure: 1 atm = 760 mm (29.9 in) Hg = 10.3 m (33.8 ft) H20 4/10/2018 Physics 201, UW-Madison

Measurement of Pressure Manometer If both sides of an U-tube are open to atmosphere the levels of the fluid are the same on both sides If one side is connected to a “pressurized side” the level difference between the two sides can be used to measure pressure. 4/10/2018 Physics 201, UW-Madison

Measuring Blood Pressure Blood pressure is quite high, 120/80 mm of Hg Use higher density fluid in a manometer: Mercury 4/10/2018 Physics 201, UW-Madison

Archimedes FB = (mg)disp F = g A h Object immersed in a fluid is subject to a “buoyant force”. Force on sides cancel Force on top Ft = ghT A Force on bottom Fb = ghB A F = g A h FB = (mg)disp 4/10/2018 Physics 201, UW-Madison

Displace just enough fluid such that forces = 0! Float Weight of object = 0gV Buoyant force is the weight of the displaced fluid Weight of fluid = fgV Displace just enough fluid such that forces = 0! 4/10/2018 Physics 201, UW-Madison

Archimedes Principle Buoyant Force (B) weight of fluid displaced (P=F/A, P=rgh) B = fluid g Vdisplaced W = object g Vobject object sinks if object > fluid object floats if object < fluid Eureka! If object floats…. B=W Therefore fluid g Vdisplaced = object g Vobject Therefore Vdisplaced/Vobject = object / fluid 4/10/2018 Physics 201, UW-Madison 1

Archimedes Principle The weight of a glass filled to the brim with water is Wb. A cube of ice is placed in it, causing some water to spill. After the spilled water is cleaned up, the weight of the glass with ice cube is Wa. How do the weights compare: 1. Wb > Wa. 2. Wb < Wa. 3. Wb = Wa. Archimedes’ Principle: The buoyant force on an object equals the weight of the fluid it displaces. Weight of water displaced = Buoyant force = Weight of ice 4/10/2018 Physics 201, UW-Madison

Question 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: 1. Go up causing the water to spill. 2. Go down. 3. Stay the same. Archimedes’ Principle: The buoyant force on an object equals the weight of the fluid it displaces. Weight of water displaced = Buoyant force = Weight of ice When ice melts it will turn into water of same volume 4/10/2018 Physics 201, UW-Madison

Question An oil tanker is floating in a port. The oil tanker is loaded with oil. The density of oil is less than that of water. The waterline (i.e., a line marked on the outside of a ship) of a loaded tanker compared to that of the empty tanker is: 1. lower. 2. the same. 3. higher. 4/10/2018 Physics 201, UW-Madison

Buoyancy Cup I Cup II Both the same Two cups hold water at the same level. One of the two cups has plastic balls (projecting above the water surface) floating in it. Which cup weighs more? Cup I Cup II Both the same Archimedes principle tells us that the cups weigh the same. Each plastic ball displaces an amount of water that is exactly equal to its own weight. 4/10/2018 Physics 201, UW-Madison

Sunken Balls Two identical glasses are filled to the same level with water. Solid steel balls are at the bottom in one of the glasses. Which of the two glasses weighs more? 1. The glass without steel balls 2. The glass with steel balls 3. Both glasses weigh the same The steel balls sink. The buoyant force equal to the weight of the displaced water is not sufficient to counter the weight of the steel balls. Therefore, the glass with steel balls weighs more. 4/10/2018 Physics 201, UW-Madison

Archimedes Principle The buoyant force on an immersed body has the same magnitude as 1. The weight of the body. 2. The weight of the fluid displaced by the body 3. The difference between the weights of the body and the displaced fluid. 4. The average pressure of the fluid times the surface area of the body. 4/10/2018 Physics 201, UW-Madison

Buoyant force and depth Imagine holding two identical bricks under water. Brick A is just beneath the surface of the water, while brick B is at a greater depth. The force needed to hold brick B in place is: 1. larger 2. the same as 3. smaller than the force required to hold brick A in place. The buoyant force on each brick is equal to the weight of the water it displaces and does not depend on depth. 4/10/2018 Physics 201, UW-Madison

1. Yes, as long as the water gets up to the ship’s waterline. A 200-ton ship enters the lock of a canal. The fit between the sides of the lock and the ship is so tight that the weight of the water left in the lock after it closes is much less than 200 tons. Can the ship still float if the quantity of water left in the lock is much less than the ship’s weight? 1. Yes, as long as the water gets up to the ship’s waterline. 2. No, the ship touches bottom because it weighs more than the water in the lock. What matters is not the weight of the water left in the lock, but the weight of the water forced out of the lock by the ship. As long as the density of the ship is less than that of water, and the water gets to the waterline, it floats. 4/10/2018 Physics 201, UW-Madison

Fluid Flow Fluid flow without friction Volume flow rate: V/t = A d/t = Av (m3/s) Continuity: A1 v1 = A2 v2 i.e., flow rate the same everywhere e.g., flow of river 4/10/2018 Physics 201, UW-Madison

Problem Two hoses, one of 20-mm diameter, the other of 15-mm diameter are connected one behind the other to a faucet. At the open end of the hose, the flow of water measures 10 liters per minute. Through which pipe does the water flow faster? 1. The 20-mm hose 2. The 15-mm hose 3. Water flows at the same speed in both cases 4. The answer depends on which of the two hoses comes first in the flow When a tube narrows, the same volume occupies a greater length. For the same volume to pass through points 1 and 2 in a given time, the velocity must be greater at point 2. The process is reversible. 4/10/2018 Physics 201, UW-Madison

Faucet A1 A2 V1 V2 A stream of water gets narrower as it falls from a faucet (try it & see). The velocity of the liquid increases as the water falls due to gravity. If the volume flow rate is conserved, them the cross-sectional area must decrease in order to compensate The density of the water is the same no matter where it is in space and time, so as it falls down and accelerates because of gravity,the water is in a sense stretched, so it thins out at the end. 4/10/2018 Physics 201, UW-Madison

Bernoulli’s Equation Pressure drops in a rapidly moving fluid whether or not the fluid is confined to a tube For incompressible, frictionless fluid: 4/10/2018 Physics 201, UW-Madison