LINEAR MOMENTUM. Derivation of the Linear Momentum Equation. Newton s second law of motion for a system is. Momentum is mass times velocity.

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1 LNEAR MOMENTUM Deriatin f the Linear Mmentum Equatin Newtn s secnd law f mtin fr a sstem is Time rate f change f the linear mmentum f the sstem Sum f the eternal frces acting n the sstem Mmentum is mass times elcit. Then, the Newtn s secnd law becmes ; D Dt VdV ss ss urthermre, fr a sstem and the cntents f a cincident cntrl lume that is fied and nn-defrming cntrl lume, the Renlds transprt therem (with belcit) allws us t cnclude that ; Time rate f change f the linear mmentum f the sstem Time rate f change f the linear mmentum f the cntents f the cntrl lume + Net rate f flw f linear mmentum thrugh the cntrl surface 1

2 t can be written as ; D Dt ss VdV t c VdV + cs VV nda ˆ r a cntrl lume that is fied and nn-defrming, the apprpriate mathematical statement f Newtn s secnd law f mtin is ; VdV + V V nda t c cs ˆ cntents f the cntrl lume We call abe equatin is the linear mmentum equatin. 2

3 rces Due t luids in Mtin Newtn s secnd law f mtin is ma n fluid flw prblems, we use mass flw rate (kg/s) t determine mass that inle in the mtin. V m ma m t t Mass flwrate can be written as ; m& Q r fluid, Newtn s secnd law f mtin is ; m ma m& Q t 3

4 Linear mmentum idea is usuall used fr water jet and ane prblems. Because elcities has magnitude and directin, frces act n ane can be determine as ; 2 2 R Q Q + 4

5 rce n -directin ) ( Q Q Q R rce n -directin ) ( Q Q Q R 5

6 QUESTON 1 TUTORAL OR LNEAR MOMENTUM igure 1 Water flws thrugh the 20º reducing bend shwn in igure 1 at rate f 0.025m 3 /s. The flw is frictinless, graitatinal effects are negligible, and the pressure at sectin (1) is 150kPa. Determine the and cmpnents f frce required t hld the bend in place. 1

7 QUESTON 2 igure 2 Determine the magnitude and directin f the anchring frce needed t hld the hrizntal elbw and nzzle cmbinatin shwn in igure 2 in place. Atmspheric pressure is 100kPa. The gage pressure at sectin (1) is 100kPa. At sectin (2), the water eits t the atmsphere. 2

8 QUESTON 3 igure 3 Water flws as tw free jets frm the tee attached t the pipe shwn in igure 3. The eit speed is 15m/s. f iscus effects and grait are negligible, determine the and cmpnents f the frce that the pipe eerts n the tee. 3

9 QUESTON 4 igure 5 A free jet fluid strikes a wedge as shwn in igure 5. Of the ttal flw, a prtin is deflected 30º, the reminder is nt deflected. The hrizntal and ertical cmpnents f frce needed t hld the wedge statinar are H and V, respectiel. Grait is negligible, and the fluid speed remains cnstant. Determine the frce rati, H / V. 4

10 QUESTON 5 igure 5 A cnerging elbw as shwn in igure 5 turns water thrugh an angle f 135º in a ertical plane. The flw crss sectin diameter is 400mm at the elbw inlet, sectin (1), and 200mm at the elbw utlet, sectin (2). The elbw flw passage lume is 0.2m 3 between sectins (1) and (2). The water lume flwrate is 0.4m 3 /s and the elbw inlet and utlet pressures are 150kPa and 90kPa. The elbw mass is 12kg. Calculate the hrizntal (-directin) and ertical (z-directin) anchring frces required t hld the elbw in place. 5

11 QUESTON 6 igure 6 Water flws frm a large tank int a dish as shwn in igure 6. a) f at the instant shwn the tank and the water in it weigh, W 1 in kg, what is the tensin, T 1, in the cable supprting the tank? b) f at the instant shwn the dish and the water in it weigh W 2 in kg, what is the frce, 2, needed t supprt the dish? 6

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