HW_13_Soln

HW_13_Soln - ME 309 Fall 2008 Section 2 (Merkle) Homework #...

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Unformatted text preview: ME 309 Fall 2008 Section 2 (Merkle) Homework # 13 Due Fri 26 Sep 2008 Problem 13-1 An initially stationary vane-cart assembly of mass M = 30 kg, rolls on a level track as shown in the figure. Friction and air resistance may be neglected. The vane receives a jet of water ( = 995 kg/m 3 ), which leaves a stationary nozzle horizontally at V = 20 m/s. Upon hitting the vane, the water is diverted entirely into the vertical direction (as seen in a frame moving with the cart). The nozzle exit area is A = 0.005 m 2 . a) Determine the velocity U(t) of the cart as a function of time using a frame of reference fixed on the ground. b) Determine the velocity U(t) of the cart as a function of time using a frame of reference fixed on the cart. Solution : Assumptions: No external forces Uniform 1-D flow across all inlet and outlet streams Neglect mass of water in contact with the vane Neglect rate of change of mass of water inside CV No body forces on incoming jet, and jet remains at constant velocity/area. Choose CV fixed on cart and enclosing cart plus vane and water adjacent to vane. Conservation of mass: ( ) = - CV CS dt d dA dVol n V This reduces to the trivial statement that the mass leaving the nozzle hits the vane, and all mass leaving the vane exits in the y direction but you must be careful in calculating the mass into the CV. ( ) c j out in u u A m m- = = & & Part a) Observer on Ground: Conservation of Momentum: M = 30 kg u c u j = 20 m/s y x A = 0.005 m 2 = 999 kg/m 3 Y X ( ) + = + - CV coor CV CS dVol a dt d dA dVol V F n V V For uniform flow across all areas of interest: Momentum simplifies to: ( ) ( ) dt du M u u A u u m c c j c j in =- =- 2 & The result is obtained by integration: ( ) ( ) --- = 2 c j c j u u u u d dt M A C t M A u u c j + =- 1 At time zero the cart is not moving: evaluate constant and solve: M t Au t...
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HW_13_Soln - ME 309 Fall 2008 Section 2 (Merkle) Homework #...

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