Chapter 11 Properties of Materials Properties of Materials We will try to understand how to classify different kinds of matter & some of their properties. States of Matter Solid Liquid Gas Plasma Solids Has definite volume Has definite shape Molecules are held in specific locations by electrical forces vibrate about equilibrium positions Can be modeled as springs connecting molecules More About Solids External forces can be applied to the solid and compress the material In the model, the springs would be compressed When the force is removed, the solid returns to its original shape and size This property is called elasticity Crystalline Solid Atoms have an ordered structure This example is salt Gray spheres represent Na+ ions Green spheres represent Cl- ions Amorphous Solid Atoms are arranged almost randomly Examples include glass Liquid Has a definite volume No definite shape Exists at a higher temperature than solids The molecules “wander” through the liquid in a random fashion The intermolecular forces are not strong enough to keep the molecules in a fixed position Gas Has no definite volume Has no definite shape Molecules are in constant random motion The molecules exert only weak forces on each other Average distance between molecules is large compared to the size of the molecules Plasma Matter heated to a very high temperature Many of the electrons are freed from the nucleus Result is a collection of free, electrically charged ions Plasmas exist inside stars Mechanical Properties (Hooke’s law) All objects are deformable It is possible to change the shape or size (or both) of an object through the application of external forces when the forces are removed, the object tends to its original shape This is a deformation that exhibits elastic behavior Elastic Properties Stress is the force per unit area causing the deformation Strain is a measure of the amount of deformation The elastic modulus is the constant of proportionality between stress and strain For sufficiently small stresses, the stress is directly proportional to the strain The constant of proportionality depends on the material being deformed and the nature of the deformation Stress Stress is the force per unit area causing the deformation Units: N/m²=Pascal (Pa), kPa, Gpa Strain Strain is a measure of the amount of deformation Stress is proportional to Strain Proportionality constant Modulus of elasticity Characterize material Elastic Modulus The elastic modulus can be thought of as the stiffness of the material A material with a large elastic modulus is very stiff and difficult to deform Young’s Modulus: Elasticity in Length Tensile stress is the ratio of the external force to the cross-sectional area Tensile is because the bar is under tension The elastic modulus is called Young’s modulus Young’s Modulus, cont. SI units of stress are Pascals, Pa 1 Pa = 1 N/m2 The tensile strain is the ratio of the change in length to the original length Strain is dimensionless Young’s Modulus, final Young’s modulus applies to a stress of either tension or compression It is possible to exceed the elastic limit of the material No longer directly proportional Ordinarily does not return to its original length Breaking If stress continues, it surpasses its ultimate strength The ultimate strength is the greatest stress the object can withstand without breaking The breaking point For a brittle material, the breaking point is just beyond its ultimate strength For a ductile material, after passing the ultimate strength the material thins and stretches at a lower stress level before breaking Shear Modulus: Elasticity of Shape Forces may be parallel to one of the object’s faces The stress is called a shear stress The shear strain is the ratio of the horizontal displacement and the height of the object The shear modulus is S Shear Modulus, final S is the shear modulus A material having a large shear modulus is difficult to bend Bulk Modulus: Volume Elasticity Bulk modulus characterizes the response of an object to uniform squeezing Suppose the forces are perpendicular to, and act on, all the surfaces Example: when an object is immersed in a fluid The object undergoes a change in volume without a change in shape Bulk Modulus, cont. Volume stress is the ratio of the force to the surface area This is also the Pressure The volume strain is equal to the ratio of the change in volume to the original volume Bulk Modulus, final A material with a large bulk modulus is difficult to compress The negative sign is included since an increase in pressure will produce a decrease in volume B is always positive The compressibility is the reciprocal of the bulk modulus Example How much will a 50-cm length of brass wire stretch when a 2-kg mass is hung from an end? The wire has a diameter of 0.10 cm. Example (Shear Stress) Motor is mounted on four foam rubber feet. The feet are in the form of cylinders 1.2 cm high and cross-sectional area 5.0 cm². How large a sideways pull will shift motor 0.10 cm? Density The density of a substance of uniform composition is defined as its mass per unit volume: Units are kg/m3 (SI) Iron(steel) 7,800 kg/m3 Water 1,000 kg/m3 Air 1.3 kg/m3 Density, cont. 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 Weight Density Weight per unit volume Specific Gravity 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/m3 Specific gravity is a unitless ratio Iron: 7.8 Water: 1.0 Air: 0.0013 Example A 50 cm3 beaker “weighs” 50 grams when empty and 97.2 grams when full of oil. What is the mass density of the oil? Weight density? Specific gravity? Fluids Liquids and gases do not maintain a fixed shape, have ability to flow Liquids and gases are called fluids Fluids statics: study of fluids at rest Fluids dynamics: study of fluids in motion Pressure 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 P=23520 Pa Measuring Pressure The spring is calibrated by a known force The force the fluid exerts on the piston is then measured Example Aluminum sphere 2 cm in radius is subjected to a pressure of 5x 109 Pa. What is the change in radius of the sphere? (50,000 atm, atmosphere pressure) Variation of Pressure with Depth 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 Otherwise, the fluid would not be in equilibrium The fluid would flow from the higher pressure region to the lower pressure region Pressure and Depth Examine the area at the bottom of fluid It has a cross-sectional area A Extends to a depth h below the surface Force act on the region is the weight of fluid Pressure and Depth equation Pa is normal atmospheric pressure 1.013 x 105 Pa = 14.7 lb/in2 (psi) The pressure does not depend upon the shape of the container Examples Two levels in a fluid. Pressure exerted by 10 m of water. Pressure exerted on a diver 10 m under water. Pressure Measurements: Manometer One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured Pressure at A is P=Po+ρgh Pressure Measurements: Barometer Invented by Torricelli (1608 – 1647) A long closed tube is filled with mercury and inverted in a dish of mercury Measures atmospheric pressure as ρgh 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. First recognized by Blaise Pascal, a French scientist (1623 – 1662) Pascal’s Principle, cont The hydraulic press is an important application of Pascal’s Principle Also used in hydraulic brakes, forklifts, car lifts, etc. Example Consider A1=5 A2, F2=2000N. Find F1. Archimedes 287 – 212 BC Greek mathematician, physicist, and engineer Buoyant force Inventor 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. 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 Buoyant Force, cont. 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 Buoyant Force, final 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 Archimedes’ Principle: Totally Submerged Object The upward buoyant force is FB=ρfluidgVobj The downward gravitational force is w=mg=ρobjgVobj The net force is FB-w=(ρfluid-ρobj)gVobj ρfluid>ρobj floats ρfluid<ρobj sinks Example Object weighs 5 N in air and has a volume of 200 cm3. How much will it appear to weigh when completely submerged in water? 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. Totally Submerged Object The object is less dense than the fluid The object experiences a net upward force Totally Submerged Object, 2 The object is more dense than the fluid The net force is downward The object accelerates downward Example What fraction of the volume of a piece of ice is submerged in water? (density of ice is 0.92 grams/cm3) Fluids in Motion: ideal fluid laminar flow: path, velocity Incompressible fluid No internal friction (no viscosity) Good approximation for liquids in general Ok for gases when pressure difference is not too large Equation of Continuity A1v1 = A2v2 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 diameter Av is called the flow rate Example Water flow Equation of Continuity, cont The equation is a consequence of conservation of mass and a steady flow 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 interval Assumes the fluid is incompressible and there are no leaks Daniel Bernoulli 1700 – 1782 Swiss physicist and mathematician Wrote Hydrodynamica Also did work that was the beginning of the kinetic theory of gases Bernoulli’s Equation Relates pressure to fluid speed and elevation 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 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 Applications of Bernoulli’s Principle: Venturi Tube Shows fluid flowing through a horizontal constricted pipe Speed changes as diameter changes Can be used to measure the speed of the fluid flow Swiftly moving fluids exert less pressure than do slowly moving fluids An Object Moving Through a Fluid Many common phenomena can be explained by Bernoulli’s equation At 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 Application – Golf Ball The dimples in the golf ball help move air along its surface The ball pushes the air down 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 Application – Airplane Wing 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 There is a net upward force Called lift Other factors are also involved Fluids in Motion Equation of Continuity A1v1 = A2v2 Bernoulli’s Equation Example Find force on the flat roof of a car with windows closed at 56 mph (25 m/s). The area of the roof is 2 m². Example Water flow in figure. Height at location 2 is 3 m higher than location 1. Pipe diameter is 4 cm at 1 and 3 cm at 2. Pressure is 200 kPa at 1 and 150 kPa at 2. What is the velocity at 2?