# Problem 1. 12ft. Find: Velocity of truck for both drag situations. Equations: Drag F Weight. For force balance analysis: Lift and Drag: Solution:

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1 Problem 1 Given: Truck traveling down 7% grade Width 10ft m 5 tons 50,000 lb Rolling resistance on concrete 1.% weight C 0.96 without air deflector C 0.70 with air deflector V ft Find: Velocity truck for both drag situations θ For force balance analysis: F ma F ma x x y y Fz maz rag F Weight ift and rag: drag C F lift C F 1. Use force balance to find equation for motion truck.. Calculate drag due to rolling resistance by 0.01*weight.. Calculate aerodynamic drag using both given C (separately) and eqn. for drag. 4. Plug information into original force balance eqn. to find truck velocity. Numbers:

2 Problem Given: Box kite blowing in wind U 0ft/s V wind 0 ft/s W 0.9 lb Tension in string.0 lb ngle to ground 0 o Frontal area 6.0 ft θ Find: a) etermine the lift and drag coefficients b) If V wind 0 ft/s and C and C are as found, will the kite rise or fall? ssumptions: y For force balance analysis: F ma F ma x x y y Fz maz x T W ift and rag: drag C F lift C F a) 1. Find the density air from Table.9.. Use force balance in both x and y to find equations for drag and lift respectively.. Using equations for lift and drag calculate C and C. b) 1. Plug new velocity into equations for lift and drag and recalculate θ.

3 Numbers:

4 Problem Given: Old airplane with wires used to strengthen design, with: U 70mph, wing 148 ft, C wing 0.00, wire 160 ft, wire 0.05 in Find: Ratio drag from wire bracing to that from the wings ssumptions: (1) Wires modeled as smooth cylinders ift and rag: F drag C 1 ρ V F lift C 1 ρ V 1. Calculate the Reynolds number the wire braces.. ook up C for wires in Fig Calculate wire / wing. Numbers:

5 Problem 4 Given: Old airplane with wires used to strengthen design, with: U 150 km/h, wire 60 m, wire 6 mm Find: Power required to move the guy wires ssumptions: (1) Wires modeled as smooth cylinders ift and rag: F drag C 1 ρ V F lift C 1 ρ V 1. ook up the kinematic viscosity and density air at 15 o C in Table ν 1.45x 10 m / s, ρ 1.kg / m. Calculate the Reynolds number the wire braces. km 1000m 1hr 1m 150 6mm U Re hr 1km 600s 1000mm ν 1.45x10 m / s 5. ook up C (Re) for wires in Fig 9.1. C ~ 1. 17,41 4. Calculate the force drag using the equation for the coefficient drag. F C 1 ρv F 1. (0.5)(1.kg / m )(150km / hr *1000m / km *1hr / 600s) (0.006m)(60m) F 461.5N 5. Now calculate power necessary, knowing P FV (461.5 N)(41.7m/s) 19. kw.

6 Problem 5 Given: Optimally faired wires the same length and other properties as in problem 4 Find: Percent power saved by fairing guy wires ift and rag: F drag C 1 ρ V F lift C 1 ρ V 1. ook up the kinematic viscosity and density air at 15 o C in Table ν 1.45x 10 m / s, ρ 1.kg / m. Calculate the Reynolds number the wire braces. km 1000m 1hr 1m 150 6mm U Re hr 1km 600s 1000mm ν 1.45x10 m / s 5 17,41. ook up C (Re) for streamlined strut in Fig C ~ Calculate the force drag using the equation for the coefficient drag. F C 1 ρv F 0.06 (0.5)(1.kg / m )(150km / hr *1000m / km *1hr / 600s) (0.006m)(60m) F N 5. Now calculate power necessary, knowing P FV ( N)(41.7m/s) 960 W. 6. Comparing this to the power in problem 4: P P streamline P 19,00W 960W x100 % x100% 95% less power necessary!!! 19,00W

7 Problem 6 Given: Spherical hailstones, 10mm, falling through air at STP Find: Terminal velocity (1) Coefficien t drag C F 5 1. ook up properties for air in Table.10: ν 1.45x 10 m / s, ρ 1.kg / m. ook up the specific gravity ice in Table.1 and calculate the density ice. SG ρ SG ρ (0.917)(1000kg / m ) 917kg m ice ice ice H 0 /. Since the sum the forces in y is zero when terminal velocity is reached, sum forces: Fy 0 mg F 4. Rearrange equation (1) for the drag force and plug it into the force balance equation just found to find V t. mg C (0.5) ρv t V t mg C ρ π (0.01m) 4 5. Calculate the mass the hail: m ρ V 917kg / m 4.8x10 kg 6 6. Calculate the area: π 4 π x10 5 m 7. Now calculate V t (C ): V t 4 (4.8x10 kg)(9.81m / s ) 5 C (1.kg / m )(7.85x10 m ) 9.87m / s C 8. ssume C is in the flat region Figure 9.11 so that C ~ m / s 9. Now calculate: V t 14.4m / s 0.47 V (14.4 / )(0.01 ) 10. For this velocity calculate: Re t m s m ν 1.45x10 m / s 11. Now go back to Figure 9.11 and look up C for this Reynolds number: C ~ m / s 1. Recalculate: V t 15.4m / s 0.41 * lthough hailstones are not perfectly spherical this is probably a good estimate

8 Problem 7 Given: Bicycle with frontal.9 ft, and C 0.88, and peddling at 15 mph in still air Find: Extra power necessary to peddle against a 0 mph head wind (1) Coefficien t drag C F 1. ook up density air in Table.10: ρ 0.008slug / ft. Change units bike velocity: V bike 15 mph (88ft/s)/(60mph) ft/s. Wind speed relative to bike: V rel (15+0)mph(88ft/s)/(60mph)51. ft/s 4. Rearrange equation (1) into an equation for drag: F.5C ρv (0.5)(0.88)(0.008slug / ft )(09 ft ) V 4.08x rel Vrel 5. Get equation for power necessary to peddle bike: P V F bike ( ft / s)(4.08x10 ) Vrel Vrel 6. Calculate power for peddling in still air: P Vrel (0.0898)( ft / s) 4.5 ft lb / s 7. Calculate power for peddling in 0mph headwind: Prel Vrel (0.0898)(51. ft / s) 6 ft lb / s 8. dditional power necessary: P rel P (6-4.5)(ft-lb/s)(1hp/550ft/lb/s) 0.5 hp

9 Problem 8 Given: Power to overcome aerodynamic drag: P ~ U n Find: n (1) Power UF rag () drag C F 1. Combining equations (1) and () for power: P 0.5UC ρu. For many vehicles the drag coefficient is essentially independent Reynolds number, thus C is not a function U so that P 0.5C ρu. So n.

10 Problem 9 Given: Cork ball in water with: d 0.m, SG 0.1, angle 0 o Find: Current speed, U θ U (1) For force balance for stationary object: F F 0 F 0 x 0 y z F Buoyancy force F B () drag C F W mg Tension T 1. o some geometry and use equations (1) to find in the x-direction: F Tcos0 o, and in the y-direction: F B W + Tsin0 o.. Calculate the buoyancy force and weight the ball: F B ρgvol (9.80 kn/m )(4π/)(0./ m) kn, and W SG F B 0.1 (0.185 kn) kn. Now balancing forces in the y-direction and using equation (): kn kn + tan0 o, Or kn, where C ½ U C U (1/) (999 kg/m ) (π(0.m) /4) 5.C U N, where U ~ m/s. HENCE: (eq. ) 5. C U 189 (in N) or C U 5.5 (eq. B) Re U/ν 0.m U/ 1.1x10-6 m /s.68 x 10 5 U 4. Now the strategy is to use trial and error to determine U. ssume C, then calculate U from Eq. () and Re from Eq. (B); check C from Fig. 9.11, and iterate until C s converge. See below. ssume C 0.5 then from Eq. () U.7 m/s and Re 8.76 x 10 5, and from Fig 9.11 C 0.15 which does not equal 0.5 so try again ssume C 0.15 then from Eq. () U 5.97 m/s and Re 1.6 x 10 6, and from Fig C 0.0 which does not equal 0.15 so try again ssume C 0.19 then from Eq. () U 5.1 m/s and Re 1.4 x 10 6, and from Fig C 0.19 so U 5.1 m/s

11 Problem 10 Given: Flow around UN building where: w 87.5m, h 154m, C 1., U 0 m/s Find: rag ssumptions: (1) ir is at STP (1) drag C F 1. ook up density air at STP in Table.10. ρ 1.kg / m. Rearrange equation (1) to find: F 0.5C ρv. Plug in the given conditions and calculate the drag: 6 F ( 0.5)(1.)(1.kg / m )(0m / s) (87.5m)(154m) 4.1x10 N

12 Problem 11 Given: Flow around UN building where: w 87.5m, h 154m, C 1., U 0 m/s at y h/ 77m Find: rag ssumptions: (1) ir is at STP (1) Velocity prile given: u Cy 0.4 () drag C F () total drag F df 1. ook up density air at STP in Table.10. ρ 1.kg / m. Rearrange equation (1) to find C, knowing u(h/): C 0/( ).5. Rearrange equation () to find: F 0.5C ρv 4. Plug this into equation () to find: F 0.5C ρ u dxdy 5. Now plug in equation (1) and known values and integrate from the ground to the top the building. F F 0.5C ρw u dy 0.5C ρwc y 1.8 (0.5)(1.)(1.kg / m )(87.5m)(.5) * rag is just less than with a uniform flow prile x10 6 N

13 Problem 1 Given: Nuclear submarine cruising submerged V 7 knots δ y x 107m w pi* 4.6m Find: a) Estimate percentage hull length with laminar B b) Calculate drag due to skin friction c) Estimate power consumed ssumptions: (1) Can treat hull as a flat plate with same wetted area () Neglect laminar boundary layer (if small percent hull length) Boundary ayer: Re 500,000 transition * displacement thickness δ momentum thickness θ 0 u U u U u U dy dy δ 0 δ 0 1 u U u 1 U dy u U dy Flow over flat plate parallel to the flow: friction drag F τ d plate surface Turbulent Boundary ayer: 5 x 10 5 < Re < 10 7 : C Re 1 5 Re < 10 9 : w

14 0.455 C (log Re ).58 ift and rag: drag C F lift C F a) 1. ook up value viscosity seawater in Table... Calculate Re.. Calculate the ratio Re xt / Re which will give us the ratio x t /. 4. Multiply this ratio by 100% to get the percent hull length with B (it is small). b) 1. Find C using equation for turbulent boundary layer. 4. Plug C into equation to find F. c) 1. Power F V, so calculate the power.

15 Fox

16 Problem 1 Given: Paddle wheel is immersed in a river current. Find: a) Expression for force produced by the wheel b) Expression for torque produced by the wheel c) Expression for power produced by the wheel d) Find the optimum angular speed, ω ssumptions: (1) Neglect air resistance since ρ air << ρ water () Use the velocity relative to the paddle () ssume the equivalent one paddle is constantly immersed ift and rag: drag C Other useful equations: F lift C F Torque F R Power Tω a) 1. Calculate the relative velocity, V rel V U V - Rω.. Plug into equation for drag force and find the expression. b) 1. Now plug this F into the equation for torque and find the expression. c) 1. Now plug this T into the equation for power and find the expression. d) 1. ifferentiate P with respect to U, where U Rω.. Set this equal to zero to find the minima and maxima.. Plug back in Rω U and find the relationship ω f (V,R).

17 Fox

18 Problem 14

19 Problem 15

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