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INTRO TO DENSITY LIQUIDS and GASES are types of. So we use the term to refer generally to both Liquids AND Gases. The DENSITY of a material is a measure of how tight the molecules are packed - Density ρ = / ( / ). Volume V = ( ). - If given density & dimensions, you can find mass m =. - Objects of the same material have the same. - Two liquids in a container: HIGHER density will be at the TOP / BOTTOM EXAMPLE 1: What is the total weight of air molecules inside a large warehouse that is 100 m wide, 100 m deep, 10 m high? (The density of air is 1.225 kg/m 3 ) EXAMPLE 2: The density of whole blood is nearly 1.06 kg/l. How many kilograms are in a pint (473 ml) of whole blood? Page 2
DENSITY VALUES AND SPECIFIC GRAVITY Density VALUES and UNITS to know: - Fresh Water 1,000 kg/m 3 = 1.0 kg/l = 1.0 g/cm 3 (lake, river, house water, etc) - Salt Water ~1,030 kg/m 3 (seawater, ocean, etc) - Whole Blood ~1,060 kg/m 3 - Air (at sea level) ~1.2 kg/m 3 (~800x lighter than water) ALSO: 1 cm 3 = 1 ml, 1m 3 = 1,000 L - Oil & Wood are usually slightly lighter than water. EXAMPLE 1: How much does 500 ml of a 2.2 g/cm 3 liquid weigh? (Use g = 10m/s 2 ) Specific Gravity (SG) is a term related to density: - SG X = / (no units!) - SG of Fresh Water = EXAMPLE 2: What is the volume of wooden cube with specific gravity 0.8 that weighs 16,000 N? (Use g = 10m/s 2 ) Page 3
PRACTICE: DENSITY / WOODEN DOOR PRACTICE: A wooden door is 1 m wide, 2.5 m tall, 6 cm thick, and weighs 400 N. What is the density of the wood in g/cm 3? (use g = 10 m/s 2 ) Page 4
PRACTICE: DENSITY / MASS OF BLOOD IN BODY PRACTICE: Suppose an 80 kg (176 lb) person has 5.5 L of blood (1,060 kg/m 3 ) in their body. How much of this person s total mass consists of blood? (BONUS: What percentage of the person s total mass is blood?) Page 5
PRACTICE: DENSITY / IS CROWN MADE OF GOLD PRACTICE: You want to verify if a 70-g crown is in fact made of pure gold (19.32 g/cm 3 ), so you lower it by a string into a deep bucket of water that is filled to the top. When the crown is completely submerged, you measure that 3.62 ml of water has overflown. Is the crown made of pure gold? Page 6
PRESSURE AND ATMOSPHERIC PRESSURE Pressure P = ( / aka ( )). EXAMPLE 1: Two identical wood blocks (800 kg/m 3, and 0.2 m x 0.2 m x 1.0 m) are placed on outdoors, horizontal surfaces as shown. Calculate the pressure of each block on the surfaces they each sit on. (a) (b) Just like how the objects above push down against the floor, the AIR above the objects pushes down against them. - This is called, and it has a standard value at SEA LEVEL: PAIR = 1.01 x 10 5 Pa = 1 atm = 14.7 lb/in 2 = 760 mmhg (assume this, unless otherwise stated) EXAMPLE 2: For the blocks above, calculate the force applied by the air above them to their top surfaces. (a) (b) Page 7
PRACTICE: WEIGHT AND PRESSURE OF AIR PRACTICE: A large warehouse is 100 m wide, 100 m deep, 10 m high: (a) What is the total weight of the air inside the warehouse? (b) How much pressure does the weight of the air apply on the floor? Page 8
PRESSURE IN AIR AND IN LIQUIDS (INTRO) If you are out in the open, as you go UP in height, there is LESS air above you, so: (1) Air Pressure P,AIR (less weight pushing down) (2) Air Density ρ,air (molecules more spread out) - Because the density of AIR is low, both changes are only significant over LARGE distances. EXAMPLE 1: Which of the following is the best approximation for the atmospheric pressure at 100 m above sea level? (a) 0.50 x 10 5 Pa (b) 1.00 x 10 5 Pa (c) 2.00 x 10 5 Pa If under any LIQUID, as you go DOWN in height/depth (go deeper), there is MORE liquid above you, so: (1) Water Pressure (more weight pushing down) - Changes are significant even for small distances (2) Water Density does NOT change much (assume constant) - Changes are insignificant even for large distances - The Pressure in a liquid changes with depth according to this equation: PBOT = PTOP + ρ g h Absolute Pressure Relative Pressure Gauge Pressure EXAMPLE 2: Suppose you are 1.8 m tall, your heart is located 1.4 m from your feet, and the blood pressure near your heart is 1.3 x 10 4 Pa. Calculate the blood (1,060 kg/m 3 ) pressure: (a) at the top of your head; (b) at the bottom of your feet. This equation also works for air pressure, but we usually ignore changes in air pressure since they require long distances. Page 9
CALCULATING PRESSURE IN LIQUIDS Remember that pressure in a liquid changes with depth PBOT = PTOP + ρ g h The point / line where two materials touch is called an - At these points, the pressure of both materials is the same. - So, for example, everywhere a liquid touches air PLIQUID = PAIR EXAMPLE 1: Suppose the density of a particular beer is 1,050 kg/m 3. What is the gauge pressure at the bottom of a SOLO red cup (12 cm tall) if it is filled with this beer to its very top? EXAMPLE 2: You pour a 6 cm column of 1,200 kg/m 3 blue liquid, and a 4 cm column of 800 kg/m 3 yellow liquid, on a 12 cm-tall beaker, as shown above. The liquids do not mix. Calculate: a) The absolute pressure at the blue/yellow interface b) The gauge pressure at the blue/yellow interface c) The absolute pressure at the bottom of the blue liquid Page 10
PRACTICE: PRESSURE / MARIANA TRENCH PRACTICE: The deepest known point on Earth is called the Mariana Trench, at ~11,000 m (~36,000 ft). If the surface area of the average human ear is 20 cm 2, how much average force would be exerted on your ear at that depth? Page 11
PRACTICE: PRESSURE / TWO-LIQUID BEAKER PRACTICE: A tall cylindrical beaker 10 cm in radius is placed on a picnic table outside. You pour 5 L of an 8,000 kg/m 3 liquid and 10 L of a 6,000 kg/m 3 liquid into. Calculate the total pressure at the bottom of the beaker. Page 12
PRACTICE: PRESSURE / FORCES UNDERWATER PRACTICE: A wooden cube, 1 m on all sides and having density 800 kg/m 3, is held under water in a large container by a string, as shown below. The top of the cube is exactly 2 m below the water line. Calculate the difference between the force applied by water to the top and to the bottom faces of the cube (Hint: calculate the two forces, then subtract). Page 13
PASCAL S LAW AND THE HYDRAULIC LIFT (1) PASCAL S LAW: Pressure in a confined fluid is transmitted equally throughout the fluid. ALSO: (2) Pressure in a liquid DOES / DOES NOT depend on the shape of the container. (3) In connected columns, liquid HEIGHT if is the same ( water seeks its own level ) HYDRAULIC LIFT: Pressure within the SAME liquid is equal at the SAME. (1) No pistons (caps) (2) Pistons of same thickness (3) Push down on left piston EXAMPLE: A hydraulic lift is designed with columns having areas 1 m 2 and 4 m 2. Pistons of equal thickness are placed on top of each column of water. If you push down with 10 N on the smaller column, causing it to lower by 20 cm: (a) How much force acts on the right piston? (b) How much does the right piston rise by? Hydraulic Lifts MULTIPLY by a factor of. This factor is called Mechanical Advantage. - Hydraulic Lifts also REDUCE by the same factor. - MOST Hydraulic Lift problems have cylindrical columns (Area =, Volume = = ): Page 14
EXAMPLE: HYDRAULICT LIFT / PROPORTIONAL REASONING EXAMPLE: A hydraulic lift is designed with cylindrical columns, one having double the radius as the other. Both columns are capped with pistons of the same density and thickness: (i) If you push on the piston on the thinner column with force F, how much force will act on the other piston? (a) F/4 (b) F/2 (c) F (d) 2F (e) 4F (ii) If you cause the piston on the thinner column to move a distance H, how much will the other piston move? (a) H/4 (b) H/2 (c) H (d) 2H (e) 4H Page 15
EXAMPLE: HYDRAULIC LIFT / FORCE TO LIFT CAR EXAMPLE: A hydraulic lift is designed with cylindrical columns having radii 20 cm and 2.0 m. What minimum downward force is needed on the smaller piston to lift an 800-kg car on the larger piston? Page 16
PRESSURE GAUGES: BAROMETER Pressure Gauges use height differences to calculate pressure PBOT = PTOP + ρgh - Torricelli s classic barometer uses Mercury (Hg) because it is 13.6 times denser than water! EXAMPLE: A classic barometer (as shown above) is built with a 1.0-m tall glass tube and filled with an unknown liquid. (a) The liquid goes 76 cm up the glass tube when the barometer is exposed to standard atmospheric pressure. What is the density of the unknown liquid? (b) When the same barometer, with the same liquid, is taken to a different location, the liquid goes 84 cm up the glass. What is the atmospheric pressure at this second location? Page 17
PRACTICE: BAROMETER / FIND ATMOSPHERIC PRESSURE PRACTICE: A classic barometer (shown below) is built with a 1.0-m tall glass tube and filled with mercury (13,600 kg/m 3 ). Calculate the atmospheric pressure, in ATM, surrounding the barometer if the column of liquid is 76 cm high. Page 18
PRESSURE GAUGES: MANOMETER Pressure Gauges use height differences to calculate pressure PBOT = PTOP + ρgh (1) Equal Pressure Equal Height: (2) Different Pressures: EXAMPLE: (a) To find the density of an unknown liquid, you pour some of it into a classic manometer (as shown above). The manometer is closed at one of its ends (vacuum), and has 1 atm air at the other end (at the bulb side). You measure the height difference between the two sides of the liquid to be 60 cm. Calculate the density of the liquid. (b) When you replace the 1 atm air from the end of the manometer with an unknown gas, the height difference for the same liquid is now 80 cm. Calculate the pressure of the unknown gas. Page 19
PRACTICE: MANOMETER / FIND ATMOSPHERIC PRESSURE PRACTICE: A classic manometer (as shown below) has one of its ends open, and a 2 atm gas on the other. When mercury (13,600 kg/m 3 ) is added to the manometer, you measure the top of the mercury column on the left to be 40 cm higher than the mercury column on the right. Calculate the atmospheric pressure that the manometer is exposed to. Page 20
PRESSURE GAUGES: TWO LIQUID U-TUBE Pressure Gauges use height differences to calculate pressure PBOT = PTOP + ρgh (1) 1 liquid, 2 open: (2) 1 liquid, 1 vacuum: (3) Most common: 2 liquids, 2 sides open: EXAMPLE: A U-tube (as above) has two long sides, both open at their ends. You first pour enough water so that the height of the water columns on both sides is 20 cm. You then pour enough of a particular oil (800 kg/m 3 ) on the right side so the column of oil above the water is 5 cm tall. Assuming the liquids don t mix: (a) What is the gauge pressure at the water-oil interface? (b) What is the difference in height (in cm) between top of water on the left, and the top of oil on the right? Page 21
PRACTICE: U-SHAPED TUBE / FIND DENSITY PRACTICE: Water and oil are poured into a u-shape tube, as shown below. The column of oil, on the right side, is 25 cm tall, and the distance between the top of the two columns is 9 cm. Calculate the density of the oil. Page 22
INTRO TO BUOYANCY & BUOYANT FORCE Objects immersed in liquids are pushed UP by a force called Buoyant Force (FB). - This happens due to a pressure difference between the Top & Bottom - ARCHIMEDES PRINCIPLE: The magnitude of Buoyant Force is the same as the weight of the liquid displaced FB = - Remember: ρ,obj = - The density that matters: density of, NOT of object. - The volume that matters: volume, not necessarily the entire volume. - MOST Buoyancy problems are FORCE problems ( F = ma). Most the time, objects are at rest AND equilibrium. EXAMPLE: When an object of unknown mass and volume is fully immersed in a large water tank and released from rest, it accelerates up. Once the object reaches equilibrium, 30% of its volume is above water. Calculate the density of this object. Page 23
EXAMPLE: BUOYANCY / COMPARE BUOYANT FORCES EXAMPLE: A 50 cm 3 piece of wood and a 50 cm 3 piece of metal are placed in a large water tank. The wood floats above the water line, while the metal sinks to the bottom of the tank. Which has greater buoyant force acting on it? a) Wood b) Metal c) Same buoyant force d) Not enough info to determine Page 24
PRACTICE: BUOYANCY / DENSITY OF AN OBJECT FLOATING UNDER WATER PRACTICE: When an object of unknown mass and volume is fully immersed in large oil (800 kg/m 3 ) container and released from rest, it stays at rest. Calculate the density of this object. Page 25
BUOYANCY / THE THREE COMMON CASES An object floats or sinks depending on its DENSITY when compared to the density of the LIQUID. Three common cases: (1) OBJECT FLOATS ABOVE (2) OBJECT FLOATS UNDER (3) OBJECT SINKS VUNDER VTOTAL VUNDER VTOTAL VUNDER VTOTAL F B mg F B mg F B mg ρ OBJ ρ LIQ ρ OBJ ρ LIQ ρ OBJ ρ LIQ If object floats freely (no strings), there s a shortcut: EXAMPLE 1: A block of unknown material floats with 80% of its total volume under water. What is its density? EXAMPLE 2: An aluminum (2,700 kg/m 3 ) block with dimensions 1.0 x 1.0 x 1.0 m sits on a scale at the bottom of a 2.0 x 2.0 x 2.0 m tank that is filled with water to its top. How much does the scale read? Page 26
PRACTICE: BUOYANCY / DENSITY OF LIQUID WITH UNKNOWN BLOCK PRACTICE: A block floats with 40% of its volume above water. When you place it on an unknown liquid, it floats with 30% of its volume above. What is the density of the unknown liquid? Page 27
EXAMPLE: BUOYANCY / IS CROWN MADE OF GOLD? EXAMPLE: You want to verify if a 100-g crown is in fact made of pure gold (19.32 g/cm 3 ), so you lower it by a string into a deep bucket of water. When the crown is completely submerged, you measure the tension on the string to be 0.88 N. Is the crown made of pure gold? Page 28
PRACTICE: BUOYANCY / DENSITY OF SUBMERGED BLOCK ON STRING PRACTICE: An 8,000 cm 3 block of wood is fully immersed in a deep water tank, then tied to the bottom. When the block is released and reaches equilibrium, you measure the tension on the string to be 12 N. What is the density of the wood? Page 29
EXAMPLE: BUOYANCY / MAXIMUM LOAD ON A FLOATING BOARD EXAMPLE: A wooden board (700 kg/m 3 ) has 1m 2 in area, 10 cm in thickness, and floats on a salt-water lake. What is the maximum amount of mass you can place on top of the board such that this load does not get wet? Page 30
PRACTICE: BUOYANCY / HEIGHT OF SIDES ON A FLOATING CONTAINER PRACTICE: You want to build a large storage container, with outer walls and an open top, as shown, so that you can load things into it, while it floats on fresh water, without any water getting inside. If the bottom face of the container measures 3.0 m by 8.0 m, how high should the side walls be, such that the combined mass of container and inside load is 100,000 kg? Page 31
FLUIDS IN MOTION / IDEAL vs REAL FLUIDS The motion or FLOW of Fluids can get pretty complicated. We simplify things with a model called IDEAL Fluids. IDEAL FLUIDS REAL FLUIDS - Always: Incompressible (constant density) - Could be compressible - Always: Flow (steady) - Could have Flow (if too fast) - Always: Flow (no friction) Defining Characteristic: Flow - Viscosity = Friction - Most problems will be about Fluids. Assume IDEAL unless it says REAL or refers to Viscosity. - In most REAL Fluids problems you will have, but no or. Page 32
FLOW CONTINUITY There are TWO main terms that deal with how quickly a fluid FLOWS: 1. Fluid Speed (m/s), v = 2. Volume Flow Rate (m 3 /s), Q = EXAMPLE 1: A long, horizontal pipe has a 2 m 2 cross-sectional area. If it takes water 5 seconds to traverse an 80 m segment of the pipe, calculate: (a) the fluid speed; (b) the volume flow rate. Flow CONTINUITY: Because an IDEAL Fluid is incompressible, its ( ) never changes! = = = Constant - This is often called the Equation. - So if Area changes, changes: = - Pipe are usually cylindrical A = EXAMPLE 2: A garden hose with a radius of 2 cm has water flowing into it with 2 m/s. At the end of the hose is a nozzle with a radius of 1 cm. (a) Find the speed of water in the nozzle. (b) In how many minutes does it fill up a 350 L bath tub? Page 33
EXAMPLE: CONTINUITY / PROPORTIONAL REASONING EXAMPLE: Water flows in a horizontal cylindrical pipe. If the water has speed V at point A, and point B has double the diameter of A, what is the water speed at B? a) V / 4 b) V / 2 c) V d) 2 V e) 4 V Page 34
PRACTICE: CONTINUITY / SPEED OF OIL TO FILL UP TANK PRACTICE: A cylindrical pipe with inner diameter of 4 cm is used to fill up a 10,000 L tank with a 700 kg/m 3 oil. If it takes one hour to fill up the tank, calculate the speed, in m/s, with which the oil travels inside of the pipe. Page 35