p0540 - 5.40. CHAPTER 5, PROBLEM 40 633 5.40 Chapter 5,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5.40. CHAPTER 5, PROBLEM 40 633 5.40 Chapter 5, Problem 40 Problem: A truck carries a tank of water that is open at the top with length , width w and depth h. If the truck is half full and = 5h, what is the maximum acceleration, a, that can be sustained without spilling any water? Assume constant acceleration, and note that the pressure is constant and equal to its atmospheric value at the free surface. . ..................... ...................... ... . ... . ................... .................. w..................................................................................................................................................................................................................................................................................................................... ´ ... ´ ´ . ................................................................................................................................................................ .....´................................................................................................................................................................................................ .. .................. .. .... ... .... ........................................................................................................................... .................. . .............. ........... .´...................................................................................................´................................. . .... ................................................................................................................................................................................................................. ....... ............................... ............................................................ ...... .... ..... . ...... ...... .................... ...................´............................................................................................................................................ .. .....´..................... .......... ......................... . .......... .................... ...................... ...... .......................... ......................................................... .... ..... ..... .. ´.............................. ...................... . ..................... . . . .... .................´................................................................................................................................. ................. . ............ ............................... ............... .................... ............. . .................................................................. ...................................................... ....................... ..... ..... ........ . .. .... .´...................................................................................................................´...................................................s .................................. . . ................. . ................ ............. . .............................. . . .......... ............... . ............................................................................................................................................................................c .................................... .... ................. .... ......... .... .............. . s. ........... .............................................................................................................................................................................. . .. . . ............... ..... . . c. . ... ..... ................................................ ... .......................................................................................................................................................................................................................................................................... . . . . ..... .. .. . . ........ .......... . . . .. . ........... . ......................... . . ...... .. ................................................................................................................................................................................................................... . .. .. . ...... ... ............................................... .. ... ............. ...... ......................................... . . ....... .. . ...................................................................................................................v...v......... . . . ... .... ..... ....... ........... . . .. .................................................................... .......... ................ . ...f...f... .......................................................................•...............................•.........•......... f v f v. • . .... .... ... . . . .. .. . . . . . . h . . . . . .. . .. . . . . . . . . . .. . . . ho . .. .. . . . . . . . Not moving x=a ¨ ´ ´ ´ aaa ´ ´ aa´ aa a a aa c s aa c s aa zy ff vv f v f v •• • • x Accelerating ......... ......................................................................................................... ......................................................................................................................................... . .. ....... ....... ..... .................................................................................................................................................................... ................ ............. ... ......... .... .. . ................................................. ...... ... .. ... ... ................. .. ........................ ...................... . ..... ........................................................................................................................................................................... ...................................................................................................................................................................................................... .. . .. . ................................................... .................................................................................................................................................................. ... . ......................................... . .... . .................... ......... .... ............................................................................................ ... ................................... ... . .................................................................................................................................................................... .................... . .......................... . .. . .. ..................... . ........... . ....................................... ................... ......... . ............................................................................................................................................................................................ ................. . ......................... . ................... .............. . ............ . ............................................. . .. ..... ............................................................................. ...................... ............. ....... .. . . ... ............................................................................................................................................................ . .............................................................. .. ............................... . ...................... .. . . ................ ............ . .......... ....... ..................................... ............ . ................. ......... ... ........................................................................................................................................................................................ . ......... . . . . . ................. . . . . .. . . . . ........................................................................... .... . .............................................................. . ........................ ...... ................................................ ............................................................................................................................................................................................................................................. ...... ........... . .... . . . . . . .. . . . .. . .. ... .. ................ . ............ . ..... ...................................................................................................................... ...................................................................................................................................... .. ...... ....................................................................................................................................... . .. . .. . . .. . . ... ................................................................................................................................... . .. . . . ....... ..... . . . .. .......................... ... ......... .. .. ............. ... . . ................................................................................................ .......... . ...... . . Solution: When the truck accelerates, the three components of the Euler equation are ∂p ∂x ∂p 0=− ∂y ∂p 0 = − − ρg ∂z ρa = − Clearly, since ∂ p/∂ y = 0, the pressure is at most a function of x and z . Integrating first over x, p(x, z ) = −ρax + f (z ) where f (z ) is a function of integration. Then, differentiating with respect to z and using the z component of Euler’s equation, we find ∂p = f (z ) = −ρg ∂z =⇒ f (z ) = C − ρgz where C is a constant. Thus, the pressure throughout the fluid in the truck is p(x, z ) = C − ρax − ρgz On the free surface, p = pa . So, the equation of the free surface is C − ρax − ρgz = pa 634 CHAPTER 5. MASS AND MOMENTUM PRINCIPLES Now, if we let x = 0 at the back of the truck, we will have z = h at x = 0 when the fluid is at the brink of spilling (see figure below). Hence, C = pa + ρgh and ax + gz = gh . . . . .. . . . . . . . . . . . . . . . . . h . . . . . . . . . . . . . . . . . .. .. . . ..................................................................................................................................................................................... .......................... ..... . .............................................................................................................................................................................................................................................................................................................................................. . .. .. ....... ....................... ..... ...... ..... ..... ....... .................................................................................................................................................................................................................................................................... .. .. . .. .. . . ............................... ........................................................................................................................................................ .. ... .... . .......... ......... .................. ... ................................................................................................................................................................................................................................................................................... .............................................................................................................................................................................................................................................................................................................................................................. . . . ... ... . ........................................................... ..... . .................................. .... . ... .......................... .. . . .... ................................................................................................................................................................................................................................................................................................................................................. ....... ....... ....................... ........ . ...... . ....................................................................................................................................................................................................................................................................................... ...................... .. ....... .................. . . .... ................................................................................................................................................................................................................................................................................................................. ... .... .... .... . ...... .. ........ . . .. ...................... .. ................................ .......................... ......... ...... . . .. ............................................... .. ............................................................................................................................................................................................................................................................................. ........ .. ............. . ...... .... ...... . .. .. . . . .......... .............. ................. ................................................................................................................................................................................................................................................................... ................. .................................................. .... ..... . ..... . ... .... ....... ...................... ..... .............. .................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................ .................................................... ... ............................... ................. ................................................................................................................................................................................................................................................................ ........................................................................................................................... . .. ... . . .................................................................................... ........................................................................................................................................................................................................................................................................................................................................................................ ..... .......................................................................................................................................................................................................................................................................................................................................................... ......................... ...................................................................................................................................................................................................................................................... .. ................. ...................................................................... . ... ...... . ....................... . ..... ........................................................................................................................................................................................................................................................................................................................... ............ ........ .. ... ............... . . . ...... ............ . .... ............ ...... .. .............................................................................................................................................................................................................................................................................................................................. ..... .................... ................................... ............. ...... .. .................................................................................................................................................................................................. x=0 . . .. .. . . . . . H . . . . . .. .. . . x= To evaluate H , note that =⇒ a + gH = gh H =h− a g To conserve mass, the volume of liquid must remain constant. Before any motion occurs, the volume is V = who During acceleration, the fluid in the tank has the shape of a rectangular parallelepiped for which the volume is 1 V = w(H + h) 2 Thus, we must have who = 1 w(H + h) 2 =⇒ 1 ho = (H + h) 2 So, substituting for H from above, ho = But, we are given ho = 1 h and 2 1 a a h− +h =h− 2 g 2g = 5h. Substituting these values gives 1 5ah h=h− 2 2g Thus, the acceleration is =⇒ 1 a= g 5 1 5a = 2 2g ...
View Full Document

This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

Ask a homework question - tutors are online