# I N V E S T I C E D O R O Z V O J E V Z D Ě L Á V Á N Í

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1 MECHNICS O LUIDS luids are both liquids and gases. The common property of fluids is that the particles can be separated easily (liquids do not have their own shape etc.). Real fluids have something like internal friction = viscosity, so e.g. the stream in the middle of the tube in which a fluid moves is faster than near the walls. n ideal fluid would have the same speed in the whole cross-section. Ideal liquids are incompressible, which is valid in fact for real liquids as well. Ideal gases can be compressed to zero volume. : LUIDS T REST. Pressure in a liquid revision: pressure N m p = [ ] = Pa p (pascal) pressure is a scalar!!! = density m V kg = m = kg m 3 3 = [ ] a) pressure caused by outer force acting on a closed liquid Pascal s principle p = this pressure is the same within the whole liquid ccording to Pascal s principle, pressure applied to any part of an enclosed fluid is transmitted to all other parts. Use: hydraulic principle (brakes, jacks for lifting cars...) p p p = = TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY - - MECHNICS O LUIDS

2 b) pressure due to a liquid column (in the gravitational field) s the molecules of a liquid are attracted towards the Earth, this pressure will rise with increasing depth (measured from the open surface of the liquid!). mg Vg hg p h = = = = = hg h. Uses of liquid columns a) manometers (measurement of pressure) i) Toricelli s experiment used to state atmospheric pressure p a h 0.75 m mercury 3 5 h = hg = 0.75m 3600 kg m 9.8m s = 0 Pa p a p = ii) in labs extra pressure p h = extra pressure TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY - - MECHNICS O LUIDS

3 b) measurement of density i) immiscible liquids... known,... unknown p h = p h g h g h = h = h h h ii) miscible liquids should be separated by mercury derive the eqn. Hg Questions:. Pistons of a hydraulic press have cross-section areas 5 cm and 400 cm respectively. If we act by the force 00 N on the narrow piston, stipulate the pressure exerted in the liquid and also the force acting on the wider piston.. What hydrostatic force acts on the bottom of the water reservoir at a depth 5 m if the area of the bottom is 50 m? What is the pressure at this depth? 3. scuba diver descends to the bed of a lake to a depth m. What is the pressure there? 4. In one limb of the U-tube is water, in another is oil. The height of water above the common boundary is 7. cm, the height of oil is 8.0 cm. Stipulate the density of the oil. 5. What force acts on the area of. dm at atmospheric pressure 000 hpa? 6. tourist measures atmospheric pressure 05 hpa at the foot of a hill and pressure 950 hpa at the hilltop. What difference in altitude did he overcome? 7. glass stopper is weighed in air and afterwards it is immersed wholly in water and reweighed. The readings obtained are.4 N in the air and.0 N in the water. Given that the density of water is 000 kg m -3, calculate the density of the stopper. 8. What minimum force is needed to cover a hole from the inside on a ship that is immersed in water? The hole is at a depth of 4 m and has an area 5 cm. 9. t what depth is the pressure in water 0 times greater than atmospheric pressure 0 5 Pa? TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

4 3. rchimedes principle objects immersed in a liquid appear to lose weight - forces, have the same size (why?) and they cancel each other (why?) - = ph = hg force on the upper side - = ph = hg force on the bottom - upthrust (force of buoyancy, rchimedian force) U = h U = hg hg = g(h h ) = gh = V B L g h h V B... volume of the body immersed L... density of the liquid upthrust = weight of fluid displaced U depends only on...and... U does NOT depend on...and Behaviour of bodies immersed depends on the resultant force (components = force of gravity and upthrust) label in the figures : upthrust ( U ) force of gravity ( g ) volume of the body (V B ) density of the liquid ( ) L a) b) c) U g U = g U g VB Lg VBBg VB Lg = VBBg VB Lg VBBg L B L = B L B the body sinks the body floats the body rises until at NY depth partly immersed TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

5 5. Partly immersed bodies rise until partly immersed until the force of gravity balances the upthrust zero resultant = no motion / U = / g VB L g = mg / VB L = m... valid for PRTLY immersed bodies only (mass of the liquid displaced = mass of the whole object) V volume immersed (not the whole one) Use - hydrometer glass stem with scale inside (Lower values are up or down?) buoyancy bulb (What is its function?) lead shot (What is its function?) 6. Hydrostatic paradox The liquid in the vessels is the same and so is the height. The pressure on the bottom is the same though there is a different mass of the liquid in the vessels. ph = ph = ph3 = hg = K = force on the base 3 g g g3 K force of gravity 3 h Where is the force of gravity equal to the force on the base? Why is it in other examples bigger/smaller? TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

6 Questions: 0. n ice floe has a shape of a square board of an area m and width 35 cm. What is minimum mass of a weight which has to be put on the middle of the floe so that the floe is fully immersed in water?. 000 g weight of a) copper b) aluminium is immersed in water. Which upthrust is greater, on a) or on b)?. n iceberg floats in water. What part is immersed? (express as a percentage) 3. paper ship of mass 3 g floats on the surface of water. On the bottom of the ship is put a weight of mass 6 g. a) State the volume of the part immersed in water. b) State the change of volume when we take the weight out. L/ Discuss the motion of balloons: B: LUIDS IN MOTION Ideal fluid The following two equations will be valid for an ideal LIQUID only, because we assume constant density (incompressibility). We will assume a steady flow streamlines do not cross each other or turn back.. Equation of continuity = the law of conservation of MSS for a moving liquid mass of the liquid passing any cross-section must be the same Sketch streamlines and vectors of velocity v and v into the tube: m = m V = V v = v v = const m m TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

7 . Bernoulli s equation = the law of conservation of ENERGY for a moving liquid E = E K E E K P Vv + E = P = const mv = Vv = W = s = ps = p V + p V = const / V v v + p = const + p = v + p problem how to state p? p h p h v v Hydrodynamic paradox narrower part LOWER p!!! 3. pplications jets and nozzles paint spraying TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

8 ind in additional materials or other sources examples of use this effect and sketch pictures: Bunsen burner filter pump motion of a spinning ball aerofoil is the shape of a wing (seen in a cross-section area) producing a force perpendicular to the motion called LIT (similar to hydrofoil). 4. Speed of a liquid leaving the container E P mgh = hg = v v = E K mv hg / : m h v TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

9 Questions: l of water flow through a pipeline of = 400 cm in 0 min. Stipulate the speed of water. 5. Water flows at speed m s - through a hose of cross-section area 5 cm. Stipulate the speed of water flowing through a constricted place of cross-section area 0.6 cm? 6. garden hose has a cross-section area 5 cm and at a constriction it has an area cm. Water spouts horizontally from a nozzle at height 90 cm above the earth. The stream of water falls down at the distance m. What is the speed of water in the hose? 7. ind the speed of water flowing through a horizontal tube of a cross-section area 5 cm if in a constricted place of area 5 cm the pressure is lowered by 400 Pa? Neglect viscosity of water. 8. measuring cylinder contains 000 cm of water which fills it to a height 0 cm. The cylinder has a small hole near its base through which water leaks out. The diameter of the hole is 0.6 mm. a) What is the initial velocity of the escaping water? b) What is the initial rate of loss of water? L/ Real fluids ideal fluid zero viscosity real fluid viscous (η - coefficient of viscosity) speeds inside British system - motion of a SPHERE in any real fluid (η) Which forces act on a sphere moving vertically in a real liquid? U V g..., (mg) U..., (= L V B g ) V..., acts against the motion like the friction force V = 6πRηv Stoke s eqn radius and speed of the sphere g State the direction of motion of the sphere on the figure above and write the eqn for all the forces acting on the sphere. The viscous drag depends on the speed. So if the force of gravity is bigger than the upthrust (as it is on the figure), their resultant is initially pointed down and the sphere speeds up in this direction. s its speed rises, the viscous drag rises too which lowers the resultant (and therefore acceleration) until the forces are balanced. When there is zero resultant the sphere moves with a constant speed terminal velocity v t. when v v + + = 0 t when v = v = const. + + = 0 g t g V U R = V U R TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

10 Czech system motion of objects (having different shapes C) in the air, the coefficient of viscosity is in the eqn unfortunately replaced by density of the air V = Cv Newton s eqn C... coefficient of drag,33, 0,48 0,34 0,03 Problem: parachutist reaches terminal velocity after the jump. Then he opens a parachute and his terminal velocity decreases. State forces acting in both situations and discuss which forces changed when the parachute had opened. Questions: L/ nswers:. 00 kpa, 8 kn. 50 kpa, 500 kn kpa kg m N m kg m N m kg. 90 % 3. 9 cm 3, 6 cm m s m s m s m s - 8. m s - -, 0.56 cm 3 s TENTO PROJEKT JE SPOLUINNCOVÁN EVROPSKÝM SOCIÁLNÍM ONDEM STÁTNÍM ROZPOČTEM ČESKÉ REPUBLIKY MECHNICS O LUIDS

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