Fluids Dr. Robert MacKay Clark College

Mass Density, Mass Density,

Weight Density, D

Volume

Some Densities

Specific Gravity

Pressure Pressure = Force Area P = F A F = P A

Quick Quiz 14.1 Suppose you are standing directly behind someone who steps back and accidentally stomps on your foot with the heel of one shoe. Would you be better off if that person were (a) a large professional basketball player wearing sneakers (b) a petite woman wearing spike-heeled shoes?

Answer: (a). Because the basketball players weight is distributed over the larger surface area of the shoe, the pressure (F / A) that he applies is relatively small. The womans lesser weight is distributed over the very small cross-sectional area of the spiked heel, so the pressure is high. Quick Quiz 14.1

In about 1657 Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres. Two teams of eight horses each could pull the hemispheres apart only on some trials, and then "with greatest difficulty," with the resulting sound likened to a cannon firing (Fig. P14.62). (a) Show that the force F required to pull the evacuated hemispheres apart is R 2 (P 0 – P), where R is the radius of the hemispheres and P is the pressure inside the hemispheres, which is much less than P 0. (b) Determine the force if P = 0.100P 0 and R = m. Figure P14.62

Pressure with increasing depth m= V & w=mg

Barometer P atms =1.013x10 5 N/m 2 =1.013x10 5 Pa =1.013 bar = 1 Atmosphere =1013 mb = 760 mm Hg = in Hg = 760 Torr =14.7 lb/in 2

manometer P abs =P atm +P gauge P gauge = gh

Quick Quiz 14.2 The pressure at the bottom of a filled glass of water (ρ = kg/m 3 ) is P. The water is poured out and the glass is filled with ethyl alcohol (ρ = 806 kg/m 3 ). The pressure at the bottom of the glass is (a) smaller than P (b) equal to P (c) larger than P (d) indeterminate

Answer: (a). Because both fluids have the same depth, the one with the smaller density (alcohol) will exert the smaller pressure. Quick Quiz 14.2

Archimedes Principle Active Figure 1409

Archimedes Principle The Buoyant force acting on an object submerged in a fluid equals the weight of fluid displaced by the object.

Buoyancy

Buoyancy 400 g 500g ? obj = ? V= m/ = 100 cm 3 f = 1.00 g/cm 3 m= 100 g

Buoyancy 400 g 500g ? obj = 5 g/cm 3 V= = m/ = 100 cm 3 f = 1.00 g/cm 3 m= 100 g

Buoyancy 400 g 500g ? obj =8.0 g /cm 3 V= ? f = ? m= 100 g

Buoyancy 400 g 500g ? obj =8.0 g /cm 3 V= 500g/8.0 g /cm 3 V=62.5 cm 3 f = 100g/62.5 cm 3 =1.6 g /cm 3 m= 100 g

Buoyancy A 20,000,000 tonn ship Floats in salt water. (1 tonn=1000kg) How much water does this ship Displace? Active Figure 1410

For a physics experiment, you drop three objects of equal mass into a swimming pool. One object is a piece of pine, the second object is a hunk of copper and the third object is a hunk of lead. The relationship between the magnitudes of the buoyant forces on these three objects will be a)F copper > F pine > F lead, b) F pine > F copper > F lead, c) F lead > F copper > F pine or d) F copper > F lead > F pine. (end of section 14.4) QUICK QUIZ 14.2

(b). From Archimedes principle, the magnitude of the buoyant force will be equal to the weight of the water displaced. From Table 14.1, lead and copper are more dense than water and will therefore sink, while pine is less dense than water and will therefore float. The buoyant force for the pine must equal the weight, mg, of the pine since these two forces balance. For completely submerged objects, the buoyant force will be equal to the weight of the water displaced, m w g, and will be less for the denser lead, because of its smaller volume, than for the copper. In addition, the mass of the water displaced will be less than the mass of the equal volume of metal displacing it, so that m w < m. Therefore, the buoyant force on each metal is less than the buoyant force on the pine. QUICK QUIZ 14.2 ANSWER

Buoyancy A cubical block of wood has 30.0 cm sides. If Its density is 600 kg/m 3 how much of it is submerged when floating in water?

Buoyancy A cubical block of wood has 30.0 cm sides. If Its density is 600 kg/m 3 how much mass must be placed on its top to just barely submerge it? M=?

Quick Quiz 14.6 A glass of water contains a single floating ice cube as in the figure below. When the ice melts, the water level (a) goes up (b) goes down (c) remains the same

Answer: (c). The ice cube displaces a volume of water that has a weight equal to that of the ice cube. When the ice cube melts, it becomes a parcel of water with the same weight and exactly the volume that was displaced by the ice cube before. Quick Quiz 14.6

A barge loaded with steel floats in a lock. If the steel is then thrown overboard. Does the water level in the lock a) go up b) go down c) stay at the same level?

Before H2O displaced =W(boat)+W(steel) After H2O displaced =W(boat)+little more

Quick Quiz 14.5 An apple is held completely submerged just below the surface of a container of water. The apple is then moved to a deeper point in the water. Compared to the force needed to hold the apple just below the surface, the force needed to hold it at a deeper point is (a) larger (b) the same (c) smaller (d) impossible to determine

Answer: (b). For a totally submerged object, the buoyant force does not depend on the depth in an incompressible fluid. Quick Quiz 14.5

Continuity

Equation of Continuity

Volume rate of flow Water flows out of a 1.0 inch hose at a speed of 2.0 ft/s. How long will it take this hose to fill a 50 gallon drum?

Quick Quiz 14.8 You tape two different sodas straws together end-to-end to make a longer straw with no leaks. The two straws have radii of 3 mm and 5 mm. You drink a soda through your combination straw. In which straw is the speed of the liquid the highest? (a) whichever one is nearest your mouth (b) the one of radius 3 mm (c) the one of radius 5 mm (d) Neither – the speed is the same in both straws.

Answer: (b). The liquid moves at the highest speed in the straw with the smaller cross sectional area. Quick Quiz 14.8

Example A 1.0 inch diameter hose is connected to a 0.25 in diameter nozzle. If the water shoots up in the air to a maximum height of 15.0 m, what is the flow rate (gallons/min) in the hose? (231 in 3 =1 gallon)

You would like to change the opening on the nozzle of a fire hose so that the water exiting the hose can reach a height that is four times the present maximum height the water can reach. To do this, you should decrease the cross sectional area of the opening by a factor of a) 16, b) 8, c) 4 d) 2. (end of section 14.5) QUICK QUIZ 14.5

(d). From the continuity equation, the velocity of the water exiting the hose is inversely proportional to the cross sectional area or v 1/A. However, the kinetic energy of the water that exits the hose will be equal to the potential energy of the water at its maximum height (when you point the hose straight up), or So to increase the height by a factor of four, you must decrease the area by a factor of 2. QUICK QUIZ 14.5 ANSWER

Bernoullis Principle Where the fluid speed is high the internal pressure is low. [based on the conservation of energy]

Bernoullis Principle

Bernoullis Principle

y1y1 y2y2 F 2 =P 2 A 2 F 1 =P 1 A 1 v1v1 A1A1 A2A2 v2v2 m2m2 x 2 =v 2 t x 1 =v 1 t m1m1 m 1 = m 2

Bernoullis Principle V 1 = 0.0 h 2 = 0.0, P 2 =P 1 h 1 = h 0.0

Bernoullis Principle V 1 = 0.0 h 2 = 0.0, P 2 =P 1 h 1 = h 0.0

Bernoullis Principle

(P 1 -P 2 )A=Lift

Surface Tension L = F Water, T N/m 0°C N/m 20°C N/m 100 °C

Capillary Action 2 r = r 2 h g r h

Viscosity

Viscosity Viscosity (Pa s) Fluid °C water °C water °C water °C blood

Poiseuilles Law

Stokes Law, terminal velocity of a sphere