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FIGURE P179 foce Woter pressure lheupslrecm of o exerting on
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If M,,iszero andr : I l, calculate using(a) analyticalinM tcgration,(b) multiple-application trapezoidalrulc, and(c) multiplc-application Simpson's rulesFor (b) and (c) usel-m lncrements 1713 The total mass of a variable density rod is given bi71
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where u(l): width of the dam face (m) at elevation: (Fig Pl79b) The line of action can also be obtainedby evaluating IoDpgzu:(z)(D z)tJz, p,su(:)(D ;) t1:
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Use Simpson'srule to compute J, andcl 1710 The tbrce on a sailboatmastcan bc reprcscntcd the by following function:
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Jt) p(,r) : dcnsity, whererz : filoSS, A,(,r): cross-sectional area, -r : distance alongthe rod and L : the total lengthoftherod, The following datahasbeenmeasured a l0-m length for rod Determinethe massin grams to the best possibleaccurac),
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where: : the elevationabovethe deck and H : thc height of the mastThe total force F exertedon the mast can be dctermined by integratingthis function over the height of the mast:
x,m a 2 4 l0 3 6 B p,glcm34ao 395 389 3Bo 360 34t 33l A,, cm2 loo lo3 l06 I lo )2a I33 lsc
17,14 A transportation engineeringstudy requiresthatyou determinethe numbcr of cars that passthroughan intersection traveling during morning rush hour You standat the sideol the road and count the numberof carsthatpass every 4 minutesat severaltimes as tabulatedbelow Use thebest numericalmethodto determine(a) the total numberof can that passbetwecn7:30 and 9:15, and (b) the rateofcan going throughthe interscctionpcr minute (Hint: Be careful withunits) Time (hr) 73A Rote (cors per 4 min) I B 7 45 24 B:OO B:15 l4 24
O:J,,I k) dz
The linc of action can also be deterrnined integratron: by
r rH J0 :It:ta: -u
Use numerica data listed bel t, mln Q, m3/mi;r c, fng/Jn"
J,,'fr:\ 'l:
(a) Use the compositetrapezoidalrule to compute F rnd d for the casewhcre 11 : 30 (n : 6) (b) Repeat(a), but use the composite Simpson'sl/3 rule l7ll A wind fbrce distributed againstthe side of a skyscraperis measured as Height /, m Force, F(/), N/m Height l, m Force, F(/), N/m
1717 The c Putedas
B : 4 5 9 :i 5 21 9
A,: I /0
0 0 I50 3r00
30 340
60 l2aa
90 t2a t600 27aa
where R : t and,r : dist averagetlov
IB0 2)A 240 3200 3500 3800
1715 Determine average the valuefor thedatain FigPI 715 Perfbrm thc intcgral necdedtbr thc avcragcin theorder shown by the following equation:
L/", l Compute the net fbrce and thc line of action due to this diswind tributed 1716 Integrationprovides a meansto computehow much 1712 An I l-m beam is subjected a load, and the shear massentersor leavesa reactorover a specifiedtimeperiod to force follows the equation a si n V(-r):5+02512 where V is the shearforce, and,r is length in distancealong the beamWe know that V : dM ldx, and M is the bending moment Integrationyields the relationship
l- I ll J,
r, f r,-
/(\r),/rl,1r
w h e r eU : a numerica data:
)rfi I;1, ffi
M:M,+
I Vttt' l,t
Jt where I, and t2 - the initial and final times, respectively, This formula makesintuitive sense you rccall thc analogy if betweenintegrationand summationThus, the integral reprcscntsthe summationof the product of flow times concenqivo
et tlr
U, mls -
PROBTEMS
10 12
F I G U R EP I 7 I 5
numerical Use inte_uration evaluate to this equation the fbr data listed bclow: l, min Q, m3/min c, mg/m3
l7lll The average concentfation ofa substance -(g/rn3)in a lake u,hercthe arcaA1m:) varieswith depth l(m) can be computcd integration: by
0 4 t0
t0 2a 30 4B 52 50 3_s -55 52
35 4A 45 46 43 43 4A 37 32
50 50 34
,t'(,+,da,
wherc Z : the total depth (ni) Determinethe averageconcenlrationbasedon thc followins data:
17,17 The cross-sectional area of a channelcan bc computed as
A,= I H(r')r1r
O 4 z,n B 12 )6 A , l a 6 m 2 9 8 1 7 5 5 t 0 5 t 1 9 6 3 5A 3 5 2 7 O O O O O c,g/m3 )a'2 85 74 52 4l
wherc : the total channelwidth (rn), H : rhedepth(m), B and : distance r' liom the bank (m ) In a similarfashion, thc l1ou,' (mr/s) can be contputccl average as 0 0 : rl] I U{rlrrlr,r)z/r Jo
1719 As wasdonein Section179 cletermine work perthe fbrmed ifa constant force of I N appliedat an anglegresults in the iollowing displacernents the curnLrcipzfunction Use g 1odetcrmine cunrulativc the work andplot the resultversus r,h 0,rad 0 0 I 30 27 60 38 90 37 l2O 3 150 14 lB0
(m/s)Use these where : watervelocity' U relationships and methodto determineA,and B fbr the fbllorving a nurnerical data:
n)/1
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