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1. Outline • Kinetics (external) • Forces in human motion • Impulse-momentum • Mechanical work, power, & energy • Locomotion Energetics

2. Outline • Kinetics • Forces in human motion • Gravity • Ground reaction • Inertial (F = ma) • Centripetal • Friction • Fluid Resistance • Multi force Free body diagrams • Dynamic and Static Analysis with Newton’s Laws

3. Reading • Newton’s Laws • Ch 2: pages 41-44; 46-61 • Friction • Ch 2: pages 61-62 • Static/Dynamic Analyses & FBDs • Ch 3: pages 107-124 • Fluid Resistance • Ch 2: pages 63-68 • Linear Impulse/Momentum • Ch 2: pages 68-72 • Mechanical Energy/Work/Power • Ch 2: pages 81-90 • Applications (Locomotion, Jumping) • Ch 4: pages 145-159

4. Human movements where fluid resistance is important?

5. Factors affecting fluid resistance • Density • mass per unit volume • resistance to motion through a fluid increases with density • Viscosity • a measure of the fluid’s resistance to flow

6. Figure 2.20 Components of Fluid Resistance Drag Force: Opposes motion Lift Force: perpendicular to motion

7. Components of drag force • Surface drag: friction of fluid rubbing on surface • Pressure drag: front-back pressure differential • Wave drag: waves at interface of two fluids.

8. Streamlines Drag force is effected by: 1) different velocities of the streamlines 2) the extent to which the relative motion of the streamlines is disturbed

9. Laminar flow Uniform layers of different speed Slowest layer closest to the surface of the object

10. Laminar flow: Surface drag dominates Velocity of air Air direction relative to ball

11. Surface drag • also called skin friction • Depends on • velocity of fluid relative to surface • roughness of surface • surface area of object • properties of fluid

12. Reducing surface drag • Speed skater: wearing a smooth spandex suit • 10% less surface drag than wool clothes • Cyclist: wearing Lycra long sleeved shirt, tights, and shoe covers • Swimmer: Shaving body hair

13. Surface drag • Surface drag: Friction within boundary layer • human movement in air: surface drag (3-5%) • small compared to pressure drag (95-97%)

14. Pressure drag: dominant form of drag in human movement • Turbulent flow: Non-uniform flow of fluid around an object • Pressure differential causes a “pressure drag force”. Higher Pressure Lower Pressure

15. Streamlining reduces turbulence and pressure drag • Flow remains laminar for longer -- less turbulence • less pressure drag Enoka, Figure 2.3A

16. Pressure drag vs. surface drag • Pressure drag: dominates for large objects moving in low density & viscosity fluids • e.g., human running, cycling in air • Surface drag: dominates when small objects moving in high viscosity fluids, e.g. sperm swimming

17. Pressure drag force • Fd = (0.5 CD)Av2 •  = fluid density • air: 1.2 kg/m3 • water: 1000 kg/m3 • CD = coefficient of drag • A = projected area (m2, frontal area as object moves through the fluid) • v = velocity of the fluid relative to the object (m/s)

18. Coefficient of drag (CD): combines shape & aspect ratio index • Unitless • Magnitude depends on • shape of object • orientation of object relative to fluid flow • Independent of size • Streamlining reduces CD

19. Coefficient of drag examples • Mackerel: 0.0053 • Rainbow trout: 0.15 • Pigeon or vulture: 0.4 • Sphere: 0.47 • Human swimmer: 0.66 • Cyclist and bike: 0.9 • Runner: 0.9 • Flat plate: 1.0

20. How can we measure frontal area?

21. vobject Velocity (v) of fluid relative to the object v = vobject - vair • Example: vcyclist = 7 m/s • Still air: vair = 0 • Headwind: vair = 7 m/s • Tailwind: vair = 7 m/s

22. vair vobject Velocity (v) of fluid relative to the object v = vobject - vair • Example: vcyclist = 7 m/s • Still air: vair = 0 v = 7 m/s • Headwind: vair = -7 m/s v = 14 m/s • Tailwind: vair = 7 m/s v = 0 m/s

23. Components of drag force • Surface drag: friction of fluid rubbing on surface • Pressure drag: front-back pressure differential • Wave drag: waves at interface of two fluids.

24. Figure 2.20 Components of Fluid Resistance Drag Force: Opposes motion Lift Force: perpendicular to motion

25. Lift Force Asymmetric objects Spinning object Bernoulli’s Principle: Pressure is inversely proportional to the velocity of the fluid

26. High Velocity Low Pressure Low Velocity High Pressure

27. Figure 2.22

28. Drag acts in horizontal (x) direction, opposite to the direction of locomotion Drag

29. Drag in locomotion (Fd = 0.5 CDAv2) • Walking or running in air (CD = 0.9,  = 1.2 kg/m3) • 0.5 CD = 0.55 kg/m3 • Fd = 0.55Av2 • Frontal area (A) = 0.4 m2 • Fd (Newtons) = 0.22 * v2

30. Role of Fd in locomotion • Person in still air • Walk (1.25 m/s): Fd ~ 0.001 Fg,x • Run (4 m/s): Fd ~ 0.01 Fg,x • Run (8 m/s): Fd ~ 0.025 Fg,x • Person in headwind of 17 m/s (~ 35 mph) • Run (8 m/s): Fd ~ 0.25 Fg,x

31. Role of Fd in locomotion

32. Drag in cycling (Fd = 0.5 CDAv2) • For cyclist in air (CD = 0.9,  = 1.2 kg/m3) • 0.5 CD = 0.55 kg/m3 • Fd = 0.55Av2 • Frontal area (A) of cyclist & bike • Touring position (upright): 0.5 m2 • Racing position: 0.3 m2 • Recumbent position: 0.2 m2

33. Touring Cycling Racer Recumbent

34. Swimming • Water density >> air density • greater pressure drag • Fd = 0.5 CDAv2 •  = 1000 kg/m3 • CD = 0.66 • A = 0.073 m2 • Fd (swimming) = 24* v2 • Comparison: Fd (walk, run) = 0.22 * v2

35. Drag: walking vs. swimming • Drag force comparison at a given speed • Fd (swimming) ~ 100 x > Fd (walk, run in air) • Reasons • Water density >> air density • frontal area less • Cd less for swimming position

36. Total force: walking vs. swimming • Swimming • Drag: largest force • 2 m/s ---> Fd ~ 0.14 * body weight • Walking • Ground reaction force: largest force • 2 m/s ---> Fg ~ 1.5 * body weight

37. Figure 2.23

38. Figure 2.24

39. Fn mg q Problem: Friction force on slope Find maximum friction force in terms of mg, q, & µs.

40. Fn mg q Friction force on slope Fs,max = Fn• µs Fn= mg cosq Fs,max = µs • mg cosq Fparallel (force pulling downhill parallel to slope) = mg sin q

41. Friction vs. Gravity force parallel • m=70kg • µs = 0.5 • theta = 30 degrees • Solve for static friction force and the component of gravitational force pulling parallel to the slope.

42. Recitation • a skier starts at the top of a 30 degreeincline,init. vel. = 0 • considering gravity, air resist. & friction, draw a FBD.

43. Recitation • a skier starts at the top of a 30 degreeincline,init. vel. = 0 • considering gravity, air resist. & friction, draw a FBD • If m = 0.050 and mass is 70.0kg, what is max. frictional force? add that number to FBD

44. Recitation • If frontal area is 0.600 m^2, air density is 1.200 kg/m^3, Cd is 0.9, what is air resist force when velocity = 10 m/sec • add this # value to your FBD

45. Neglect or Do Not Neglect? • if we include air resistance, kinematic problems get more difficult. • In the bike lab we will take aero force into account and use an iterative computer approach.

46. Recitation • What is the fastest velocity that can be reached by the skier. i.e. what is terminal velocity?