Archimedess Principle Archimedess Principle o o o Archimedess

หลกของอาคมดส (Archimedes’s Principle(

หลกของอาคมดส (Archimedes’s Principle) , ตอ o ขนาดของแรงลอยตวจะเทากบนำหนกของของเหลวทถ กแทนทโดยวตถ o นคอหลกของอาคมดส o Archimedes’s Principle does not refer to the makeup of the object experiencing the buoyant force n The object’s composition is not a factor since the buoyant force is exerted by the fluid

หลกของอาคมดส : วตถทจมอยในของเหลวทงกอน , ตอ o If the density of the object is less than the density of the fluid, the unsupported object accelerates upward o If the density of the object is more than the density of the fluid, the unsupported object sinks o The motion of an object in a fluid is determined by the densities of the fluid and the object

Archimedes’s Principle: Floating Object o The object is in static equilibrium o The upward buoyant force is balanced by the downward force of gravity o Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level

Archimedes’s Principle: Floating Object, cont o The fraction of the volume of a floating object that is below the fluid surface is equal to the ratio of the density of the object to that of the fluid

Archimedes’s Principle, Crown Example o Archimedes was (supposedly) asked, “Is the crown made of pure gold? ” o Crown’s weight in air = 7. 84 N o Weight in water (submerged) = 6. 84 N o Buoyant force will equal the apparent weight loss n Difference in scale readings will be the buoyant force

Archimedes’s Principle, Crown Example, cont. o o B = F g – T 2 (Weight in air – “weight” in water) o Archimedes’s principle says B = rg. V o Then to find the material of the crown, rcrown = mcrown in air / V

Archimedes’s Principle, Iceberg Example o What fraction of the iceberg is below water? o The iceberg is only partially submerged and so Vfluid / Vobject = applies o The fraction below the water will be the ratio of the volumes (Vwater / Vice)

Archimedes’s Principle, Iceberg Example, cont o Vice is the total volume of the iceberg o Vwater is the volume of the water displaced n This will be equal to the volume of the iceberg submerged o About 89% of the ice is below the water’s surface

Types of Fluid Flow – Laminar o Laminar flow n Steady flow n Each particle of the fluid follows a smooth path n The paths of the different particles never cross each other n The path taken by the particles is called a streamline

Types of Fluid Flow – Turbulent o An irregular flow characterized by small whirlpool like regions o Turbulent flow occurs when the particles go above some critical speed

Viscosity o Characterizes the degree of internal friction in the fluid o This internal friction, viscous force, is associated with the resistance that two adjacent layers of fluid have to moving relative to each other o It causes part of the kinetic energy of a fluid to be converted to internal energy

Ideal Fluid Flow o There are four simplifying assumptions made to the complex flow of fluids to make the analysis easier (1) The fluid is nonviscous – internal friction is neglected (2) The flow is steady – the velocity of each point remains constant

Ideal Fluid Flow, cont (3) The fluid is incompressible – the density remains constant (4) The flow is irrotational – the fluid has no angular momentum about any point

Streamlines o The path the particle takes in steady flow is a streamline o The velocity of the particle is tangent to the streamline o A set of streamlines is called a tube of flow

Equation of Continuity o Consider a fluid moving through a pipe of nonuniform size (diameter) o The particles move along streamlines in steady flow o The mass that crosses A 1 in some time interval is the same as the mass that crosses A 2 in that same time interval

Equation of Continuity, cont o m 1 = m 2 ® r A 1 v 1 = r A 2 v 2 o Since the fluid is incompressible, r is a constant o A 1 v 1 = A 2 v 2 n This is called the equation of continuity for fluids n The product of the area and the fluid speed at all points along a pipe is constant for an incompressible fluid

Equation of Continuity, Implications o The speed is high where the tube is constricted (small A) o The speed is low where the tube is wide (large A) o The product, Av, is called the volume flux or the flow rate o Av = constant is equivalent to saying the volume that enters one end of the tube in a given time interval equals the volume leaving the other end in the same time n If no leaks are present

Bernoulli’s Equation o As a fluid moves through a region where its speed and/or elevation above the Earth’s surface changes, the pressure in the fluid varies with these changes o The relationship between fluid speed, pressure and elevation was first derived by Daniel Bernoulli

Bernoulli’s Equation, 2 o Consider the two shaded segments o The volumes of both segments are equal o The net work done on the segment is W =(P 1 – P 2) V o Part of the work goes into changing the kinetic energy and some to changing the gravitational potential energy

Bernoulli’s Equation, 3 o The change in kinetic energy: n DK = ½ mv 22 - ½ mv 12 n There is no change in the kinetic energy of the unshaded portion since we are assuming streamline flow n The masses are the same since the volumes are the same

Bernoulli’s Equation, 4 o The change in gravitational potential energy: n DU = mgy 2 – mgy 1 o The work also equals the change in energy o Combining: W = (P 1 – P 2)V =½ mv 22 - ½ mv 12 + mgy 2 – mgy 1

Bernoulli’s Equation, 5 o Rearranging and expressing in terms of density: P 1 + ½ rv 12 + mgy 1 = P 2 + ½ rv 22 + mgy 2 o This is Bernoulli’s Equation and is often expressed as P + ½ rv 2 + rgy = constant o When the fluid is at rest, this becomes P 1 – P 2 = rgh which is consistent with the pressure variation with depth we found earlier

Bernoulli’s Equation, Final o The general behavior of pressure with speed is true even for gases n As the speed increases, the pressure decreases

Applications of Fluid Dynamics o Streamline flow around a moving airplane wing o Lift is the upward force on the wing from the air o Drag is the resistance o The lift depends on the speed of the airplane, the area of the wing, its curvature, and the angle between the wing and the horizontal

Lift – General o In general, an object moving through a fluid experiences lift as a result of any effect that causes the fluid to change its direction as it flows past the object o Some factors that influence lift are: n The shape of the object n The object’s orientation with respect to the fluid flow n Any spinning of the object n The texture of the object’s surface

Golf Ball o The ball is given a rapid backspin o The dimples increase friction n Increases lift o It travels farther than if it was not spinning

Atomizer o A stream of air passes over one end of an open tube o The other end is immersed in a liquid o The moving air reduces the pressure above the tube o The fluid rises into the air stream o The liquid is dispersed into a fine spray of droplets