Problem 1 (20 points)

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1 ME 309 Fall 01 Exam 1 Name: C Problem 1 0 points Short answer questions. Each question is worth 5 points. Don t spen too long writing lengthy answers to these questions. Don t use more space than is given. A. Explain the ifference between a pathline, streakline, an streamline. How woul you make each visible in a lab experiment? When are they the same an when are they ifferent? B. What o each of the terms of the Conservation of Linear Momentum equation represent? Page of 8

2 ME 309 Fall 01 Exam 1 Name: C Problem 1, cont C. Does the velocity fiel,,, = conserve mass? Assume incompressible flui. How many components an imensions oes the velocity fiel have? D. Why o ships mae of steel S.G. 7.8 float on water S.G. 1.0? Who first mae this observation? Draw a sketch showing the principles involve. Assuming a flat-bottome boat will make this easier. Page 3 of 8

3 COLM_BE_10 In the ME309 Jet Momentum Laboratory, you were aske to relate the volumetric flow rate of an upwarirecte water jet to the force require to hol a hemispherical cup in equilibrium, as shown in the figure. F hemispherical cup of iameter D an weight W H D g You may assume that H D, but you shoul not assume that the jet iameter remains constant. a. Determine the spee of the water relative to the groun at point shown in the figure in terms of a subset of the volumetric flow rate, the flui ensity ρ, the height H, the weight of the cup W, the iameter of the cup D, the iameter of the jet at the outlet, an the gravitational acceleration g. b. Determine the force F require to hol the cup in place in terms of a subset of the volumetric flow rate, the flui ensity ρ, the height H, the weight of the cup W, the iameter of the cup D, the iameter of the jet at the outlet, an the gravitational acceleration g. c. If the force F is suenly remove, what will be the acceleration of the cup relative to the groun in terms of a subset of the volumetric flow rate, the flui ensity ρ, the height H, the weight of the cup W, the iameter of the cup D, the iameter of the jet at the outlet, an the gravitational acceleration g? Page 1 of 5

4 COLM_BE_10 SOLUTION: F hemispherical cup of iameter D an weight W W H D g 1 z Apply Bernoulli s equation along a streamline from point 1 to point. " p!g + V g + z where p 1 = p atm p = p atm V 1 = /π /4 V =? z 1 = 0 z H Substitute an simplify. " p!g + V g + z V g + H = V 1 g V = "! 4 " p =!g + V g + z " p =!g + V g + z 1 1 gh 4 Note that applying conservation of mass to a control volume surrouning the jet from the jet outlet to the cup outlet will not provie the velocity V since the area A is not known. The conservation of mass statement assuming steay flow will simply give,!v 1 A 1 =!V A " V 1 A 1 = V A " = V A 5 We can fin the area A, however, after using Bernoulli s equation to etermine V Eq. 4, = A "! 4 gh A = "! 4 gh Page of 5

5 COLM_BE_10 Now apply the LME in the z irection to the control volume shown in the figure. Note that the coorinate system is fixe to the groun an, hence, is inertial. u z! V t " + " u z!u rel A = F B,z + F S,z 7 where t " u z! V = 0 the flow is steay 8 u z!u rel "A = V!V A =!m!v A " + V " = V!m 9 =!m The inlet an outlet to the are essentially at the same height, which means that the spees will be the same. The mass flow rate into the an out of the will also be the same from COM. F B,z =!W neglecting the weight of the liqui in the compare to the cup weight 10 F S,z =!F 11 Substitute an simplify.!v!m =!W! F 1 F =!W + V!m 13 F =!W + " 4! gh 14 where the mass flow rate is simply,!m =! 15 Determine by applying COM between the jet exit an the inlet of the shown. Note that using a control volume that surrouns the entire jet, as shown in the following figure, requires that the weight of the liqui in the be inclue in the boy force term it s the weight of the jet that causes the jet to ecelerate with height. This is much harer to work with because the weight of the jet must be etermine. F hemispherical cup of iameter D an weight W W H D g W j z where t u z! V t " + " u z!u rel A = F B,z + F S,z 16 " u z! V = 0 the flow is steay 17 Page 3 of 5

6 COLM_BE_10 u z!u rel "A = V 1!V 1 A 1 =!m!v A " + V " =!m V + V 1 =!m =! V 1 + V 18 F B,z =!W! W j 19 F S,z =!F 0 The weight of the jet can be foun by integrating the cross-sectional area of the jet over the height H. The cross-sectional area is foun using Bernoulli s equation an conservation of mass, ientical to the approach use to fin Eqs. 4 an 6, A z = "! 4 gz z=h z=h z + W j = gm j = g!v j = g! " A z z = g! " =! - z=0 z=0 4 * * gz, Substitute an simplify. =!F! W! ",!" V 1 + V!" 4 + 4!" 4 F =!W + " * 4!, 4 + -! gh / /. *! gh =!F! W! ", 4!, 4 + -! gh / /.. * gh 0 0 /! gh =!F! W 5 4! gh which is exactly the same as Eq. 14! 6 To evaluate the acceleration of the cup if the force F is remove, simply replace F with Ma in Eq. 14, where M is the mass of the cup = W/g an a is the cup s acceleration recall from Newton s n Law that F = Ma. Ma =!W + " 4! gh 7 a =!g + " g W 4! gh 8 Alternately, we coul apply the LME in the z-irection, using a coorinate system fixe to the now accelerating, to fin the acceleration. Use the shown in the first part of the solution. u z! V t " + " u z!u rel A = F B,z + F S,z " a z Z! V 9 where t " u z! V 0 the spee of material in the using the given coorinate system is ~zero 30 u z!u rel "A = V!V A =!m!v A " + V " = V!m 31 =!m F B,z =!W neglecting the weight of the liqui in the compare to the cup weight 3 Page 4 of 5

7 COLM_BE_10 F S,z = 0 There is no force applie now. 33 Substitute an simplify.!v!m =!W! am 34 a =!W + V!m =!W + V!m " =!g + V!m g M W g W 35 a =!g + " g W 4! gh which is exactly the same result as Eq. 8! 36 Page 5 of 5

8 A container is fille with an unknown flui. Its right sie is an incline gate θ respect to the horizon, which can rotate about the hinge A an stay at rest ue to an external force exerte at B. In orer to obtain the ensity of flui within the container, a manometer with mercury is applie to measure the pressure at C. The height of mercury is h an the ensity of mercury is ρ. The with of gate into the page w, the length of gate AB is L, the height of the free surface above C h 1, an the vertical istance from A to C h 3. The weight of the gate is negligible an both the back of the gate an the surfaces of flui an mercury are expose to atmosphere the atmospherical pressure is P. a. Determine the ensity of the flui ρ F. b. Calculate the gauge an absolute pressure respectively at point A within the container. c. Calculate the moment of the resultant force of flui about the hinge A.. Determine the minimum external force F neee to hol the gate at rest. Hg atm a. 6pts b. 5pts c. 0pts. 4pts

9 Solution: a 6pts Pressure of the static flui at the contact point is equal, ρ Hg gh = ρ gh F 1 The ensity of the flui, ρ F h = ρ h 1 Hg b 5pts The gauge pressure at point A, p = ρ g h + h gauge F 1 3 Then the absolute pressure, p = p + p = ρ g h + h + P abs gauge atm F 1 3 atm c 0pts We attach x- y coorinate to the gate such that the y irection is along the gate an the center is at A. We choose a small piece on the gate an calculate the force resulting from water, 5pts F = p gauge y wy Where pgauge y = ρg h1+ h3+ ysin θ is the gauge pressure, Such that the moment about A note the irection of F is perpenicular to the gate, Integral over the whole gate, M = yf = ρg h + h + y sin θ wyy 1 3

10 y= L M = ρg h + h + y sin θ wyy y= L L = ρgw h1+ h3 + ρgw sin θ 3 4pts When the moment ue to the external force is equal to M resulting from water, F reaches minimum, FL = M Thus, M L L F = = ρgw h1+ h3 + ρgw sin θ L 3

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