V Cart Y X Fy - Engineering.purdue.edu

COLM 07A jet of water is deflected by a vane mounted on a cart. The water jet has an area, A, everywhere and isturned an angle q with respect to the horizontal. The pressure everywhere within the jet is atmospheric.The incoming jet velocity with respect to the ground (axes XY) is Vjet. The cart has mass M. Determine:a. the force components, Fx and Fy, required to hold the cart stationary,b. the horizontal force component, Fx, if the cart moves to the right at the constant velocity, Vcart(Vcart Vjet)qAVjetVcartYFxXFyPage 1 of 5

COLM 07SOLUTION:Part (a):Apply conservation of mass and conservation of linear momentum to a control volume surrounding thecart. Use an inertial frame of reference fixed to the ground (XY).Vout (this velocity is currentlyan unknown quantity)qAVjetYFxXFyFirst apply conservation of mass to the control volume to determine Vout.dr dV r u rel dA 0dtòòCVwhereddt(1)CSò r dV 0(the mass within the control volume doesn’t change)CV u rel Aæ urel A ö éêç ˆˆˆˆr u rel dA ç r Vjet i - Ai ê r Vout cos q i sin q j A cos q ˆi sin q ˆjç êCSèø ë(ò) (left side(22 - rVjet A rVout A cos q sin q))ùúúúûright side 1 - rVjet A rVout A(Note that the jet area remains constant.)Substitute and re-arrange.- rVjet A rVout A 0Vout Vjet(2)Now apply conservation of linear momentum in the X-direction:du X r dV u X r u rel dA FB , X FS , XdtòòCVwhereddtòuCSXr dV 0 (the momentum within the control volume doesn’t change with time)CVPage 2 of 5(3)

COLM 07 u XòuX( r u rel dA ) (Vjet )CS u Xæ urel A öç r V ˆi - Aˆi V cos qjetç jet ç èø() u rel Aéꈈˆˆê r Vjet cos q i sin q j A cos q i sin q jêë(left side 2- rVjetA rVj2et A cos q( cos22q sin q)) ()ùúúúûright side 12 rVjetA ( cos q - 1)FB , X 0 (no body forces in the x-direction)FS , X - Fx (all of the pressure forces cancel out)Substitute and re-arrange.2rVjetA ( cos q - 1) - Fx2Fx rVjetA (1 - cos q )(4)Now look at the Y-direction:duY r dV uY r u rel dA FB ,Y FS ,YdtòòCVwhereddt(5)CSòuY r dV 0 (the momentum within the control volume doesn’t change with time)CV u rel AéꈈuY ( r u rel dA ) V jet sin q ê r Vjet cos q i sin q j A cos q ˆi sin q ˆjêCSë(ò uY()2rVjetA sin q( cos) ()ùúúúûright side2q sin 2 q) 12 rVjetA sin qFB ,Y - Mg (assume that the fluid weight in the CV is negligible compared to the cart weight)FS ,Y Fy(all of the pressure forces cancel out)Substitute and re-arrange.2rVjetA sin q - Mg Fy2Fy rV jetA sin q Mg(6)Page 3 of 5

COLM 07Part (b):Apply conservation of linear momentum to a control volume surrounding the cart. Use a frame ofreference fixed to the cart (xy). Note that this is an inertial frame of reference since the cart moves in astraight line at a constant speed. In addition, in this frame of reference, the cart appears stationary and thejet velocity at the left is equal to Vjet-Vcart.Vout (this velocity is currentlyan unknown quantity)qAVjet - VcartyxFxFyFirst apply conservation of mass to the control volume to determine Voutdr dV r u rel dA 0dtòòCVwhereddt(7)CSò r dV 0(the mass within the control volume doesn’t change)CV u rel u rel A A ù ééêêúr u rel dA ê r Vjet - Vcart ˆi - Aˆi ú ê r Vout cos q ˆi sin q ˆj A cos q ˆi sin q ˆjCSêëúû êë(ò()) (left side()right side( - r Vjet - Vcart A rVout A cos 2 q sin 2 q())ùúúúû) 1 - r Vjet - Vcart A rVout A(Note that the jet area remains constant.)Substitute and re-arrange.()- r Vjet - Vcart A rVout A 0Vout Vjet - Vcart(8)Now apply conservation of linear momentum in the x-direction:du x r dV u x r u rel dA FB , x FS , xdtòòCVwhereddtCSò u r dV 0x(the momentum within the control volume doesn’t change with time)CVPage 4 of 5(9)

COLM 07 u rel u rel u Xéé A A ùêêúˆˆˆˆu x ( r u rel dA ) V jet - Vcart ê r V jet - Vcart i - Ai ú V jet - Vcart cos q ê r V jet - Vcart cos q i sin q j A cos q ˆi sin q ˆjêêúCSêëúûêëò u x() ()())((left side( - r V jet - Vcart))right side(2) (ùúúúúûA r V jet - Vcart)2(A cos q cos 2 q sin 2 q) 1( r V jet - Vcart)2A ( cos q - 1)FB , x 0 (no body forces in the x-direction)FS , x - Fx (all of the pressure forces cancel out)Substitute and re-arrange.r (Vjet - Vcart ) A ( cos q - 1) - Fx2(Fx r V jet - Vcart2) A (1 - cosq )(10)Now solve the problem using an inertial frame of reference fixed to the ground (frame XY). From Eqn. (8)we know that using a frame of reference fixed to the cart, the jet velocity on the right hand side is:V V -Vcos q ˆi sin q ˆj(11)out,relative to cart(jetcart)()Hence, relative to the ground the jet velocity on the right hand side is:Vout, Vout, Vcart Vjet - Vcart cos q ˆi sin q ˆj Vcart ˆirelative toground)((relative tocart)(12)Now consider conservation of linear momentum in the X direction.du X r dV u X r u rel dA FB , X FS , XdtòòCVwhereddtòu(13)CSXr dV 0 (the momentum within the control volume doesn’t change with time)CV u X u rel u relé A A ùêêúˆˆéùu X ( r u rel dA ) V jet ê r V jet - Vcart i - Ai ú V jet - Vcart cos q Vcart ê r V jet - Vcart cos q ˆi sin q ˆj A cos q ˆi sin q ˆjëûêêúCSêëëêûú u X éò()()left side()(() ()right side)()()(2éù - rV jet V jet - Vcart A r ê V jet - Vcart cos q Vcart V jet - Vcart ú A cos 2 q sin 2 qëû) 1()2é 22 ù r ê -V jet V jetVcart V jet - Vcart cos q VcartV jet - Vcartúû Aë22ùé r ê V jet - Vcart cos q - V jet - Vcart ú Aëû(( r V jet - Vcart))2()( cos q - 1) AFB , X 0 (no body forces in the x-direction)FS , X - Fx (all of the pressure forces cancel out)Substitute and re-arrange.r (Vjet - Vcart ) A ( cos q - 1) - Fx2(Fx r V jet - Vcart2) A (1 - cosq )(Same answer as before!)(14)Note that using a frame of reference that is fixed to the control volume is easier than using one fixed to theground. This is often the case.Page 5 of 5ùúúúûú

VV V iji VV Vcos sinqqˆˆ ˆ- rel , , CV CS XX BXSX d u dV u d F F dt òòrr uA CV X 0 d u dV dt òr ()()() ()( )( ) rel rel jet jet cart jet cart cart jet cart CS left side ˆˆ ˆ ˆ ˆ ˆcos cos sin cos sin X X u u udVVV AVV V VV AXrr qr qqqq éù é ê