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

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

Reading Newton’s Laws – Ch 2: pages 41-44; Friction – Ch 2: pages Static/Dynamic Analyses & FBDs – Ch 3: pages Fluid Resistance – Ch 2: pages Linear Impulse/Momentum – Ch 2: pages Mechanical Energy/Work/Power – Ch 2: pages Applications (Locomotion, Jumping) – Ch 4: pages

Human movements where fluid resistance is important?

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

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

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.

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

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

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

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

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

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

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

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

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

Pressure drag force F d = (0.5  C D )Av 2  = fluid density – air: 1.2 kg/m 3 – water: 1000 kg/m 3 C D = coefficient of drag A = projected area (m 2, frontal area as object moves through the fluid) v = velocity of the fluid relative to the object (m/s)

Coefficient of drag (C D ): combines shape & aspect ratio index Unitless Magnitude depends on – shape of object – orientation of object relative to fluid flow Independent of size Streamlining reduces C D

Coefficient of drag examples Mackerel: 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

How can we measure frontal area?

Velocity (v) of fluid relative to the object Example: v cyclist = 7 m/s Still air: v air = 0 Headwind: v air = 7 m/s Tailwind: v air = 7 m/s v object v = v object - v air

Velocity (v) of fluid relative to the object Example: v cyclist = 7 m/s Still air: v air = 0 v = 7 m/s Headwind: v air = -7 m/s v = 14 m/s Tailwind: v air = 7 m/s v = 0 m/s v object v air v = v object - v air

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.

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

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

Low Velocity High Pressure High Velocity Low Pressure

Figure 2.22

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

Drag in locomotion ( F d = 0.5  C D Av 2 ) Walking or running in air (C D = 0.9,  = 1.2 kg/m 3 ) – 0.5  C D = 0.55 kg/m 3 – F d = 0.55Av 2 Frontal area (A) = 0.4 m 2 – F d (Newtons) = 0.22 * v 2

Role of F d in locomotion Person in still air – Walk (1.25 m/s): F d ~ F g,x – Run (4 m/s): F d ~ 0.01 F g,x – Run (8 m/s): F d ~ F g,x Person in headwind of 17 m/s (~ 35 mph) – Run (8 m/s): F d ~ 0.25 F g,x

Role of F d in locomotion

Drag in cycling ( F d = 0.5  C D Av 2 ) For cyclist in air (C D = 0.9,  = 1.2 kg/m 3 ) – 0.5  C D = 0.55 kg/m 3 – F d = 0.55Av 2 Frontal area (A) of cyclist & bike – Touring position (upright): 0.5 m 2 – Racing position: 0.3 m 2 – Recumbent position: 0.2 m 2

Touring Racer Recumbent Cycling

Swimming Water density >> air density – greater pressure drag F d = 0.5  C D Av 2 –  = 1000 kg/m 3 – C D = 0.66 – A = m 2 F d (swimming) = 24* v 2 – Comparison: F d (walk, run) = 0.22 * v 2

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

Total force: walking vs. swimming Swimming – Drag: largest force – 2 m/s ---> F d ~ 0.14 * body weight Walking – Ground reaction force: largest force – 2 m/s ---> F g ~ 1.5 * body weight

Figure 2.23

Figure 2.24

Problem: Friction force on slope  FnFn mg Find maximum friction force in terms of mg, , & µ s.

Friction force on slope F s,max = F n µ s F n = mg cos  F s,max = µ s mg cos  F parallel (force pulling downhill parallel to slope) = mg sin   FnFn mg

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

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

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

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

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. Neglect or Do Not Neglect?

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