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Chapter 11,12 Matter, Fluid Mechanics

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States of Matter Solid Solid Liquid Liquid Gas Gas Plasma Plasma

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Solids Has definite volume Has definite volume Has definite shape Has definite shape Molecules are held in specific locations Molecules are held in specific locations by electrical forcesby electrical forces vibrate about equilibrium positions vibrate about equilibrium positions Can be modeled as springs connecting molecules Can be modeled as springs connecting molecules

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More About Solids External forces can be applied to the solid and compress the material External forces can be applied to the solid and compress the material In the model, the springs would be compressedIn the model, the springs would be compressed When the force is removed, the solid returns to its original shape and size When the force is removed, the solid returns to its original shape and size This property is called elasticityThis property is called elasticity

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Crystalline Solid Atoms have an ordered structure Atoms have an ordered structure This example is salt This example is salt Gray spheres represent Na + ionsGray spheres represent Na + ions Green spheres represent Cl - ionsGreen spheres represent Cl - ions

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Amorphous Solid Atoms are arranged almost randomly Atoms are arranged almost randomly Examples include glass Examples include glass

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Liquid Has a definite volume Has a definite volume No definite shape No definite shape Exists at a higher temperature than solids Exists at a higher temperature than solids The molecules “wander” through the liquid in a random fashion The molecules “wander” through the liquid in a random fashion The intermolecular forces are not strong enough to keep the molecules in a fixed positionThe intermolecular forces are not strong enough to keep the molecules in a fixed position

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Gas Has no definite volume Has no definite volume Has no definite shape Has no definite shape Molecules are in constant random motion Molecules are in constant random motion The molecules exert only weak forces on each other The molecules exert only weak forces on each other Average distance between molecules is large compared to the size of the molecules Average distance between molecules is large compared to the size of the molecules

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Plasma Matter heated to a very high temperature Matter heated to a very high temperature Many of the electrons are freed from the nucleus Many of the electrons are freed from the nucleus Result is a collection of free, electrically charged ions Result is a collection of free, electrically charged ions Plasmas exist inside stars Plasmas exist inside stars

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Density The density of a substance of uniform composition is defined as its mass per unit volume: The density of a substance of uniform composition is defined as its mass per unit volume: Units are kg/m 3 (SI) Units are kg/m 3 (SI) Iron(steel) 7,800 kg/m 3 Water 1,000 kg/m 3 Air 1.3 kg/m 3

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Density, cont. The densities of most liquids and solids vary slightly with changes in temperature and pressure The densities of most liquids and solids vary slightly with changes in temperature and pressure Densities of gases vary greatly with changes in temperature and pressure Densities of gases vary greatly with changes in temperature and pressure

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Specific Gravity The specific gravity of a substance is the ratio of its density to the density of water at 4° C The specific gravity of a substance is the ratio of its density to the density of water at 4° C The density of water at 4° C is 1000 kg/m 3The density of water at 4° C is 1000 kg/m 3 Specific gravity is a unitless ratio Specific gravity is a unitless ratio Iron: 7.8 Water: 1.0 Air: 0.0013

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Fluids Liquids and gases do not maintain a fixed shape, have ability to flow Liquids and gases do not maintain a fixed shape, have ability to flow Liquids and gases are called fluids Liquids and gases are called fluids Fluids statics: study of fluids at rest Fluids statics: study of fluids at rest Fluids dynamics: study of fluids in motion Fluids dynamics: study of fluids in motion

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Pressure Pressure is force per unit area Pressure is force per unit area Ex: 60kg person standing on one Foot (10cm by 25cm). The force exerted by a fluid on a submerged object at any point if perpendicular to the surface of the object The force exerted by a fluid on a submerged object at any point if perpendicular to the surface of the object

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Measuring Pressure The spring is calibrated by a known force The spring is calibrated by a known force The force the fluid exerts on the piston is then measured The force the fluid exerts on the piston is then measured

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Variation of Pressure with Depth If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium All points at the same depth must be at the same pressure All points at the same depth must be at the same pressure Otherwise, the fluid would not be in equilibriumOtherwise, the fluid would not be in equilibrium The fluid would flow from the higher pressure region to the lower pressure regionThe fluid would flow from the higher pressure region to the lower pressure region

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Pressure and Depth Examine the area at the bottom of fluid Examine the area at the bottom of fluid It has a cross-sectional area AIt has a cross-sectional area A Extends to a depth h below the surfaceExtends to a depth h below the surface Force act on the region is the weight of fluid Force act on the region is the weight of fluid

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Pressure and Depth equation P atm is normal atmospheric pressure P atm is normal atmospheric pressure 1.013 x 10 5 Pa = 14.7 lb/in 21.013 x 10 5 Pa = 14.7 lb/in 2 The pressure does not depend upon the shape of the container The pressure does not depend upon the shape of the container

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Examples 1. Two levels in a fluid. 2. Pressure exerted by 10 m of water. 3. Pressure exerted on a diver 10 m under water.

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Pressure Measurements: Manometer One end of the U- shaped tube is open to the atmosphere One end of the U- shaped tube is open to the atmosphere The other end is connected to the pressure to be measured The other end is connected to the pressure to be measured Pressure at A is P=P o +ρgh Pressure at A is P=P o +ρgh

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Pressure Measurements: Barometer Invented by Torricelli (1608 – 1647) Invented by Torricelli (1608 – 1647) A long closed tube is filled with mercury and inverted in a dish of mercury A long closed tube is filled with mercury and inverted in a dish of mercury Measures atmospheric pressure as ρgh Measures atmospheric pressure as ρgh

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Pascal’s Principle A change in pressure applied to an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container. A change in pressure applied to an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container. First recognized by Blaise Pascal, a French scientist (1623 – 1662)First recognized by Blaise Pascal, a French scientist (1623 – 1662)

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Pascal’s Principle, cont The hydraulic press is an important application of Pascal’s Principle The hydraulic press is an important application of Pascal’s Principle Also used in hydraulic brakes, forklifts, car lifts, etc. Also used in hydraulic brakes, forklifts, car lifts, etc.

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Example Consider A 1 =5 A 2, F 2 =2000N. Find F 1.

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Archimedes 287 – 212 BC 287 – 212 BC Greek mathematician, physicist, and engineer Greek mathematician, physicist, and engineer Buoyant force Buoyant force Inventor Inventor

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Archimedes' Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object.

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Buoyant Force The upward force is called the buoyant force The upward force is called the buoyant force The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object The physical cause of the buoyant force is the pressure difference between the top and the bottom of the object

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Buoyant Force, cont. The magnitude of the buoyant force always equals the weight of the displaced fluid The magnitude of the buoyant force always equals the weight of the displaced fluid The buoyant force is the same for a totally submerged object of any size, shape, or density The buoyant force is the same for a totally submerged object of any size, shape, or density

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Buoyant Force, final The buoyant force is exerted by the fluid The buoyant force is exerted by the fluid Whether an object sinks or floats depends on the relationship between the buoyant force and the weight Whether an object sinks or floats depends on the relationship between the buoyant force and the weight

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Archimedes’ Principle: Totally Submerged Object The upward buoyant force is F B =ρ fluid gV obj The upward buoyant force is F B =ρ fluid gV obj The downward gravitational force is w=mg=ρ obj gV obj The downward gravitational force is w=mg=ρ obj gV obj The net force is F B -w=(ρ fluid -ρ obj )gV obj The net force is F B -w=(ρ fluid -ρ obj )gV obj ρ fluid >ρ obj floats ρ fluid >ρ obj floats ρ fluid <ρ obj sinks ρ fluid <ρ obj sinks

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Example A block of brass with mass 0.5 kg and specific gravity 8 is suspended from a string. Find the tension in the string if the block is in air, and if it is completely immersed in water.

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Totally Submerged Object The object is less dense than the fluid The object is less dense than the fluid The object experiences a net upward force The object experiences a net upward force

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Totally Submerged Object, 2 The object is more dense than the fluid The object is more dense than the fluid The net force is downward The net force is downward The object accelerates downward The object accelerates downward

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Fluids in Motion: ideal fluid laminar flow: path, velocity laminar flow: path, velocity Incompressible fluid Incompressible fluid No internal friction (no viscosity) No internal friction (no viscosity) Good approximation for liquids in general Good approximation for liquids in general Ok for gases when pressure difference is not too large Ok for gases when pressure difference is not too large

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Equation of Continuity A 1 v 1 = A 2 v 2 A 1 v 1 = A 2 v 2 The product of the cross-sectional area of a pipe and the fluid speed is a constant The product of the cross-sectional area of a pipe and the fluid speed is a constant Speed is high where the pipe is narrow and speed is low where the pipe has a large diameterSpeed is high where the pipe is narrow and speed is low where the pipe has a large diameter Av is called the flow rate Av is called the flow rate

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Equation of Continuity, cont The equation is a consequence of conservation of mass and a steady flow The equation is a consequence of conservation of mass and a steady flow A v = constant A v = constant This is equivalent to the fact that the volume of fluid that enters one end of the tube in a given time interval equals the volume of fluid leaving the tube in the same intervalThis is equivalent to the fact that the volume of fluid that enters one end of the tube in a given time interval equals the volume of fluid leaving the tube in the same interval Assumes the fluid is incompressible and there are no leaks Assumes the fluid is incompressible and there are no leaks

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Daniel Bernoulli 1700 – 1782 1700 – 1782 Swiss physicist and mathematician Swiss physicist and mathematician Wrote Hydrodynamica Wrote Hydrodynamica Also did work that was the beginning of the kinetic theory of gases Also did work that was the beginning of the kinetic theory of gases

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Bernoulli’s Equation Relates pressure to fluid speed and elevation Relates pressure to fluid speed and elevation Bernoulli’s equation is a consequence of Work Energy Relation applied to an ideal fluid Bernoulli’s equation is a consequence of Work Energy Relation applied to an ideal fluid Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steady-state manner Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steady-state manner

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Bernoulli’s Equation, cont. States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

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Applications of Bernoulli’s Principle: Venturi Tube Shows fluid flowing through a horizontal constricted pipe Shows fluid flowing through a horizontal constricted pipe Speed changes as diameter changes Speed changes as diameter changes Can be used to measure the speed of the fluid flow Can be used to measure the speed of the fluid flow Swiftly moving fluids exert less pressure than do slowly moving fluids Swiftly moving fluids exert less pressure than do slowly moving fluids

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An Object Moving Through a Fluid Many common phenomena can be explained by Bernoulli’s equation Many common phenomena can be explained by Bernoulli’s equation At least partiallyAt least partially In general, an object moving through a fluid is acted upon by a net upward force as the result of any effect that causes the fluid to change its direction as it flows past the object In general, an object moving through a fluid is acted upon by a net upward force as the result of any effect that causes the fluid to change its direction as it flows past the object

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Application – Golf Ball The dimples in the golf ball help move air along its surface The dimples in the golf ball help move air along its surface The ball pushes the air down The ball pushes the air down Newton’s Third Law tells us the air must push up on the ball Newton’s Third Law tells us the air must push up on the ball The spinning ball travels farther than if it were not spinning The spinning ball travels farther than if it were not spinning

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Application – Airplane Wing The air speed above the wing is greater than the speed below The air speed above the wing is greater than the speed below The air pressure above the wing is less than the air pressure below The air pressure above the wing is less than the air pressure below There is a net upward force There is a net upward force Called liftCalled lift Other factors are also involved Other factors are also involved

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Fluids Mechanics Carlos Silva December 2 nd 2009.

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