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Midterm - Winter 2004 - The University of Michigan...

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Unformatted text preview: The University of Michigan Department of Mechanical Engineering ME 320 Profs. Akhavan and Hasselbrink Midterm Exam, March 8 2004 Open Book, One crib sheet (2 sides) allowed (10 points) Problem 1: A tank of density-stratified salt solution, having a depth h = 0.1m7 is prepared in the laboratory. The density of the liquid increases linearly with depth, as indicated in the figure, starting from the density of pure water pw at the top surface and increasing to a value of 1.6pw at the bottom of the tank. A rectangular block of wood of height h = 0.1771 and density p1, = 0.5pw is placed in the tank. Calculate the depth b below the surface of the liquid to which the bottom surface of the block sinks when it is floating in equilibrium. (25 points) Problem 2: A water tank of cross-sectional area A is placed on a cart and mounted on frictionless rollers as shown in the figure. Water in the tank discharges to the atmosphere through a short nozzle of area Aj << A. The water inside the tank is maintained at. a constant height hoabove the nozzle by an external supply which discharges in the tank at a volumetric flow rate Q. The cart is restrained from moving by a cable attached to an immovable Wall. The mass of the tank, cart, and water is initially Mo. In terms of the given parameters 13, Aj, Ma, and the water density p, (5 points) (a) Find the volumetric flow rate Q. (5 points) (b) Find the tensile force T in the cable. (10 points) (c) At time t = 0, the external supply of water is shut off and the cable holding the tank is cut. The cart begins to move to the left at a velocity Vc(t). Derive the differential equation governing Vc(t) for all times t Z 0. Assume the free surface of the fluid remains horizontal. (5 points) (d) How would your answer change if you considered the effect of acceleration on the free surface? Assume no sloshing; that is, assume the free surface results from quasi- hydrostatic equilibrium at any given moment. Q h A. 0 J ———-—>V Made m U S A ”alfanapmd 42-132 100 SHEETS f A www-mm 8 Y5 w lTwrg‘Eime 4%Naewwewzgmaummm Q fir. ML. E. P». (E mm ,: [Cavxam : C Q WHITE \ Ilov +70% {.mr HQ \P m; Nallana/"Brand 42'1” 10? SHEETS Made m U S /\ ”A’- rNfi ZN Who <33? NQQ? w h 30.3 HV W . flathsa N _flarr.a . To vswv he «mfloflmw up F . g ‘ ESH>N+m Q s «a ivggmurcsg kg E E: g @ \Oh ~93 nu Wu? .M ~.m\£ \0 Made m U S A ”aligns/am” 42-192 mo SHEETS ”A... A; 0‘ a; $15th 9 u - $33 :5 ¢ . n; 1 I Kl; (a h; f“? 930% Thain $928 Egg V® J1 - PM Me : My b <~ (avg m r. o .+~ ‘ »\ f ufirrw. <H+x «1. Lu. ‘ mm m \o % _\4H 0 m wm§~1§VHl®b+ fifl. 1.! <fi4r\. Awr?\ N m b D 4. .P _ P I! m “XYRWWAEMEJ, A? ,u. 3% 1a mi 1:? “Vim manna»? A» 35 a 1k 3% r WE wflTLF Tab. two a.<. . 0h M V fizphmzfifmése no @ \ Mariana/”Erma 42-18? 100 SHEETS Made m U S A r '3' _;,~ 7: ”M 3+» H5 «.fio‘rfi: “10». QCNDQP ~.m 30+ rolwfloiuk reg. 73 $2.»? 0.0 («$954 g I u + Dblr Fm. L4 m V 5% IS» 02m {umbilflm is. AW?» 9.5%.»? MB .tfi DrnsEzV \rec; <15 8 5 IRON 5e: lfiuémdmmgggghg NW3. 9% E g0 Wraix .235; :5 96‘??? ignf 0341 .86me .OCUMT‘TSMNL or: .311.» e 64f; gwqé QTSR was 9: E $3M. ...
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