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Homework3 - EXAMPLE 2.13 The coffee cup in Example 2.12 is...

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Unformatted text preview: EXAMPLE 2.13 The coffee cup in Example 2.12 is removed from the drag ra tated about its central axis until a rigid-body mode occurs. F will cause the coffee to just reach the lip of the cup and (b) th condition. MAE 101A: Introductory Fluid Solution Mechanics Homework Part (a) The cup contains 7 cm of coffee. The remaining distance of 3 distance h Due Friday February 5, 5:00 PM/2 in Fig. 2.23. Thus 2 22 h 2 0.03 m R 4g (0.03 4(9.81 Solving, we obtain Problem 1 The coffee cup shown in the figure below is placed on a turntable, and rotated 2 about its central axis until a rigid-body mode occurs. Find (a) the angular velocity which 1308 or 36.2 rad/s will cause the coffee to just reach the lip of the cup and (b) the gage pressure at it is convenient to put the origin of point A for Part (b) To compute the pressure, this condition. of the free-surface depression, as shown in Fig. E2.13. The z point A is at (r, z) (3 cm, pA 4 cm). Equation (2.63) can the 0 (1010 kg/m3)(9.81 m/s2 1 2 (1010 kg/m3)(0.03 m)2(1 594 N/m2 3 cm 396 N/m2 0 r 99 This is about 43 percent greater than the still-water pressure 7 cm Ω A 3 cm 3 cm Here, as in the linear-acceleration case, it should be pressure distribution (2.63) sets up in any fluid under of the shape or size of the container. The container may fluid. It is only necessary that the fluid be continuously container. The following example will illustrate a pecu sualize an imaginary free surface extending outside the E2.13 Figure 1: Problem 1 EXAMPLE 2.14 Problem 2 A spherical balloon (specific gravity of skin diameter D, and skin thickness t = 0.013 mm is filled with −10o C . The balloon lifted a payload of mass M = 230 kg to an altitude of 49 km. Calculate the diameter of the balloon and its mass, provided that at h = 49000 m, air is at pressure p = 0.95 mbar and temperature T = −20o C . A U-tube with a radius of 10 in and containing mercury to a center at 180 r/min until a rigid-body mode is achieved. The material SGballoon =2116 lbf/ftofFind the pressure a ble. Atmospheric pressure is 1.28) 2. See Fig. at temperature THe = heliumE2.14. Problem 3 A block of mass 30 kg and volume 0.024 m3 is allowed to sink in water as shown in figure. A circular rod 5 m long and 25 cm2 in cross section is attached to the weight and also to the wall. If the rod mass is 1.25 kg , what will be the angle θ, for equilibrium? Problem 4 A U-tube accelerometer is a U-tube filled with a fluid of density ρ being accelerated with a horizontal acceleration a towards the left, as it’s shown in the figure below. The fluid height difference is h, the separation between the two vertical tubes is L and b is the level of fluid on the left hand side of the U-tube. Obtain an expression for the acceleration a in terms of h and L. 1 Problem *3.97 [4] Figure 2: Problem 2 Problem 3.99 [3] ENT NEEDED m2 k and rod rium Figure 3: Problem 3 (L + c)/2 L/2 c Problem *3.113 #0 FB # ρ$ g$ V (Buoyancy) !" FBR FBB WR a [2] hown. FBB and FBR are the buoyancy of the is the (unknown) exposed length of the rod L WB nge ( θ) % FBR$ ( L ( c) L $ cos ( θ) ( WR$ $ cos ( θ) # 0 2 2 FBR # ρ$ g$ ( L % c) $ A ( L ( c) L ( MR$ # 0 2 2 Figure 4: Problem 4 WR # MR$ g FBB # ρ$ g$ VB % ρ$ VB $ L % ρ$ A$ ( L % c) $ L %c 2 2 ' 2 ' # 2$ )MB % ρ$ VB ( 1 $ MR, $ L * + 2 . Problem 5 The U-tube shown is filled with water at T = 68 o F . It is sealed at A and open to the atmosphere at D. The tube is rotated about vertical axis AB. For the dimensions Problem *3.115 compute the maximum angulas speed if there !"# to be no cavitation. shown, is /3)1,)4'/)/'3 //3)948)'4):1;&,1,&4' Figure 5: Problem 5 Problem 6 Fluid with 65 lbm/f t3 density is flowing steadily through the rectangular " box shown. Given A1 = 0.5 f t2 , A2 = 0.1 f t2 , A3 = 0.6 f t2 , V1 = 10i f t/s, and V2 = ? " ! ! % $ ρ# 18 $ "ρ# $ " ρ# ( [1] Problem 4.21 j$f"t/s,determine%velocity V3 . 20 ρ# ω # 8 8 !8 !@ @ % $ " ρ# ( % $ " ρ# ( " 1'3)04 %B " %C $ "ρ# (# F %J " %< $ "ρ# (# F % & '% %C < L GHI 1'3)04 G"I @ % $ ρ# ω # 8 8x "L C " %< $ ρ # ω # " " 1'3)04 % " H 3% $ & ρ# ω # 8 38 'K Figure 6: Problem 6 GMI Problem C $ %1,7 ( ρ# (# F essibleC " ρ3) Uniform flow < $ % flow # ω # " " 7 You are filling your car with gasoline at a rate 5.3 gals/min. Although you can’t see it, the gasoline is rising in the tank at a rate of 4.3 in. per minute. What is the " " L horizontal cross-sectional area of your L " gas tank? Is this a realistic answer? 04 %< $ %1,7 ( ρ# (# F " ρ# ω # " " Problem V1" A1 ( V2" A2 ( V3" A3 & 0 8 A film of oil of thickness h is flowing steadily "L 04 $ %< " ρ# (# F %J $ %1,7 " ρ# ω # J θ) as shown in the figure, with a"velocity profile such as: down an inclined plane (at angle n at location 1 and out at location 2; we assume outflow at location 3 ρg sin θ &',)J)G'4,)<I u= (hy A2 µ ft ft 0.5 ft 0.1 V2" V3 & 5 00-8/)49)A1,/8)&0 %;V$ KQMMU"# %0& 3 & 10 s ) 0.6 ' 20" s ) 0.6 A3 s 0" deg) "# % 1,7 " %; ω$ " ρ# L ( 60" deg) ft 0 +,'2.5"HTX# 813 ω$ 1 − y2 ) 2 (1) ) * ft " V2 & 4.33 M V x H #5 / , H"# &'3 + 02-(9, 6 3"# G HVQW "s KQMMUI # 2.9 + 9, $ + +0 " HQUV# 02-( " . H# 9, 1 " 3 &' G M# &'I 0 # 2.9 6 4 & '2.5 ft 7 Vy s ω $ HWSV# 8%7 Calculate the mass flow rate per unit width at which the film of oil is flowing. H Problem 4.29 [2] Figure 7: Problem 8 Problem 9 Water flows steadily through a pipe of length L and radius R = 75 mm. Calculate the uniform inlet velocity, U , if the velocity distribution across the outlet is given by: r2 u = umax 1 − 2 . (2) R Problemand u 4.31 max = 3 m/s. [2] Figure 8: Problem 9 w at inlet and outlet of pipe !0 Problem 10 Water is flowing in the two-dimensional square channel of width h = 75.5 mm shown in the figure. Knowing that at the outlet the velocity profile is linear, at which vmax = 2vmin , and that at the inlet the flow is uniform of value U = 7.5 m/s, find the value of the minimum velocity at the outlet vmin . 2) Incompressible flow 2& ρ" U" π" R ) ' ρ" u ( r ) " 2" π" r dr ! 0 (0 2 * 2 1 2max" + R % 2 " R . ! R " U , / R & 0 * r - 23 ' 2 umax" 11 % + . 4 " 2" r dr ! R " U ' R/ 5 2, (0 U! 1 "u 2 max m s R ! 1 m 6 3" 2 s U ! 1.5" 4 Problem 4.35 [2] Figure 9: Problem 10 5 ...
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