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class05lecnotes - Class 5, Friday, January 15,2010 Reading:...

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Class 5, Friday, January 15,2010 Reading: 15.5, 15.6, Applications of Bernoulli's Equation, and of the Continuity Equation. The figure below shows volume flow rates through different tubes connected to each other. Find out what the volume flow through the unmarked pipe is. •2. ~J . ~ .I i CM J,ls I --.J u I {rJ+ 2 -I+/-It -foX'=:O 10 ~ ~ : I~ 'fIM~ ((IIIIff1/4/J'D1f t/ tIt1af.1 ) (..1.ct . I I nl', * J X::-J 10 't I X ~~ It Is the situation illustrated in the picture below viable? When we discussed fluids in hydrostatic balance, we established that, if all three tubes open up into air, then the height of the column of liquid must be the same in all three. The situation illustrated below could not be a case of balance, because the pressure at level 1 would not be equal for all tubes. The pressure at 1 in the middle tube would be larger by an amount equal to pgh. ~A :: fo f,g;: 1'0 T f d It f,c. : Po Ii Can we ever make this possible? How? \ \ I I I I I I iJdAMl.t a/ /: "IJ;! fA t:c:: II; 1: 4 ::, fl t I; h I I .lfIMd "A-;' " c. ~ ttl> ::. /}; ::.) ~:: ~ t 'rA- ~)/(r.: ~-I)h ~JuJ. c") f/4 = fA. _/~ ~ ::: t". 5.1
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If we want the liquid to be in hydrostatic balance, we need to change the "atmospheric" pressure in each tube accordingly. Since we want the liquid to rise higher in the middle tube, we need to compensate in the other two tubes for this excess pressure. We can do that by artificially increasing the "atmospheric" pressure above the columns of liquid in tubes A and C, or likewise, by artificially decreasing the "atmospheric" pressure in tube B. If PS<Pft=PCbY the right amount, we are set. A: What is the right amount? lA-P~=pgh. Now the liquid can be in balance, even if in the middle tube the column of lIqUId is larger than in the other two tubes. A: How can we modify the "atmospheric" pressures in each individual tube? Last class we derived Bernoulli's equation from the simple principles of energy and mass conservation. S \r L f L COl-W1. P-t Z..... 'Tl.= It tells us that if the speed of a flow increases at a given point, then the pressure P must decrease in order for the sum to remain constant If we assume that the flow takes place without a change in height, the pgh-terms are the same on both sides" We call the pressure P the static pressure, and the term containing v 2 the dynamic pressure. So, we can decrease the static pressure by increasing the speed of the flow at that same point. A: How can we increase the speed of the flow? By forcing it to flow through a narrower tube (small A leads to large v!). If we close the tube-system with another device, that has a changing cross-section, as illustrated in the figure below, we can blow air through this closure-tube. The flow of air changes speed as it runs into the narrower horizontal tube that connects the two wider ones. In conclusion, we can decrease the pressure P 2
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This note was uploaded on 03/16/2010 for the course PHYS 6B taught by Professor Graham during the Winter '08 term at UCSC.

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class05lecnotes - Class 5, Friday, January 15,2010 Reading:...

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