Mass Linear Momentum Moment of Momentum Energy Putting it all together!

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1 inie Cnrl lue nalsis vin fr a Sse a inie Cnrl lue a Linear enu en f enu Ener Puin i all eer! D Cnservain f a B = Tal aun f a in e e b = a er uni a = DB ˆ b b n ˆ n ˆ equain Bu D / =! Cninui Equain a leavin - a enerin = - rae f increase f a in ˆ Cnservain f a n ˆ n ˆ ˆ ˆ If a in is cnsan ˆn [/T] Uni vec ˆn is nal surface an ine u f We aue unif n e cnrl surface is e saiall averae l i l

2 Cninui Equain f Cnsan Densi an Unif elci n ˆ n ˆ [L /T] Densi is cnsan acr Densi is e sae a an Sile versin f e cninui equain f cniins f cnsan ensi. I is uners a e velciies are eier unif saiall. averae Eale: Cnservain f a? Te flw u f a ervir is L/s. Te ervir surface is 5 5. Hw fas is e ervir surface rin? ˆ n ˆ Cnsan ensi u in elci f e ervir surface u Eale: Cnservain f a Te flw ru e ifice is a funcin f e e f waer in e ervir C in e ie f e ervir level r fr c 5 c. Te ervir surface is 5 c 5 c. Te ifice is in iaeer an is c ff e b f e ervir. Te ifice cefficien is.6. C wi cnsan canin a. Draw C, label CS, slve usin variables sarin wi ˆ Eale Cnservain f a Cnsan lue ˆ n ˆ n ˆ C

3 Eale Cnservain f a Canin lue ˆ C C Eale Cnservain f a / / / s 4..8 / / C 59 s Linear enu Equain Linear enu Equain DB B D ˆ b b n enu b ˆ n D ˆ R.T.T. enu/uni a Sea sae D D D ecs!!! ˆ n ˆ n ˆ uins Unif ensi Unif velci Sea flui velci

4 Sea Cnrl lue f Newn s Secn Law D Wa are e fces acin n e flui in e cnrl vlue? Gravi Sear fces a e walls Psure fces a e walls Psure fces n e ens W W wall wall Linear enu Equain W W W W Te enu vecs ave e sae irecin as e velci vecs Reucin elbw in verical lane wi waer flw f L/s. Te vlue f waer in e elbw is L. Ener l is neliible. P=5 kpa Calculae e fce f e elbw n e flui. W = -96 N secin secin W D 5 c c /s 4. /s 5 kpa? -459 N 69 N Direcin f vecs 9,4 N? Eale: Reucin Elbw Eale: Wa is? z z z z.5 /s 4. /s 5 Pa 98 N/ 9.8 /s 9.8 /s P = kpa = -94 N

5 Eale: Reucin Elbw Hiznal ces Eale: Reucin Elbw erical ces W W W W W W W W N - N W W 459N,96N 9,4N 7.9 k N 8 kn acin wnwar n flui W.7kN Pie wans ve e lef ce f ie n flui Pie wans ve u Eale: ire nzzle ire nzzle sall fire nzzle is use creae a werful je reac far in a blaze. Esiae e fce a e waer eers n e fire nzzle. Te sure a secin is kpa (ae. Ine fricinal les in e nzzle. 8 c.5 c Ienif wa u nee knw Cun ur unknwns Deerine wa equains u will use 8 c kpa.5 c

6 in e elciies ire nzzle: Sluin z z D D 4 D D D D 4 4 D D secin secin D P Pa /s 57 N N -4 N. L/s W W 8 c kpa Wic irecin es e nzzle wan? Is is e fce a e firefiers nee brace aains? NO!.5 c fce alie b nzzle n waer Eale: enu wi Cle Geer in, an fce n e wee. L/s /s k / 5 Unknwn:,,,, Cninui 5 Unknwns: Nee 5 Equains Ienif e 5 equains! Bernulli ( z z enu (in an W W Unknwns:,,,,

7 Slve f an Slve f an W W sin sin sin sin Cnen f velci in irecin a cnservain Neliible les al Bernulli aseric sure sin sin sin sin sin sin sin sin sin a f af af sina 5f sin sin sin sin sin sin 6. L / s.867 L / s Slve f ec sluin cs cs cs cs cs cs 6 N L c k / a / sf N ce f wee n flui c af O c a f P. / s cs c.6 / s cs a f.867 / s cs 5 N.66N 77.4N L / s 6. L / s.867 L / s

8 ec iin Eale T=5 C Were is e line f acin f? Given: iure in: Hiznal fce require l lae in siin Sluin: z B z B B B B *75/ 999. / s =75 kpa =.4 /s i u u C CS u CS ( 999*.4*. 4.9kN B Eale Eale Given: iure in: Eernal reacins in an irecins neee l fie vane. Sluin: u u C CS u CS ( ( cs ( ( cs ( ( cs.9**.(8 7cs 9.5kN ( e lef =8 /s =. /s =7 /s v v C CS v CS ( sin ( sin ( ( sin.9**.(7sin.4kn (wn =8 /s =. /s =7 /s

9 Eale Eale Given: iure in: ce alie flanes l ie in lace Sluin: Cninui equain P= kpa, ae =.6 /s /.6/( *. / / s enu u u C CS ( ( (,( *. / 4 *8.49**.6 4,5 N v v C CS Wb W f 5.*98 48N D= c l=. W=5 N W b +W f Given: Waer je, 6 c iaeer, wi velci /s is vane vin a 7 /s. in: in fce n vane b waer. Sluin: Selec C vin wi e vane a cnsan velci. Te aniue f e velci aln e vane is cnsan v ( v u u C CS ( v [ ( v ] ( v cs45 [( v ] ( v ( cs45 ( 7 (( *.6 / 4( cs N 7.9 N v v C CS ( v sin 45 [( v ] ( v sin 45 ( 7 (( *.6 / 4(sin 45 Sluice Gae Eercise: 5.4, 5.9, 5., 5.5, 5.5, 5.56 in: ce ue sure n face f ae Sluin: ue: v an v are unif (s sure is rsaic u u C CS v[ v ] v[ v ] G ( v v ( ( b b G ( v v b G ( v v (

10 Cninui Equain Renls Trans Tee (eensive B b B B (inensive b b C CS C CS CS Unsea Case Sea Case Sea flw CS Incsible flui -D lw in a Cnui Cninui Eq. C CS enu Equain Renls Trans Tee B b b C CS b = velci; B = e enu C CS ec equain -- cnens, e.., u u C CS B ui vj wk