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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 rigidbody 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 rigidbody 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 freesurface 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 stillwater pressure 7 cm Ω A 3 cm 3 cm Here, as in the linearacceleration 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 Utube with a radius of 10 in and containing mercury to a center at 180 r/min until a rigidbody 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 Utube accelerometer is a Utube 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 Utube. 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 Utube 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 crosssectional 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 & 5 008/)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 twodimensional 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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 Spring '08
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