Chapter 12Linear Kinetics of Human

MovementBasic Biomechanics, 6th edition

Susan J. Hall

Presentation Created byTK Koesterer, Ph.D., ATCHumboldt State University

Objectives• Identify Newton’s laws of motion and gravitation and

describe practical illustrations of the laws• Explain what factors affect friction and discuss the role

of friction in daily activities and sports• Define impulse and momentum and explain the

relationship between them• Explain what factors govern the outcome of a collision

between two bodies• Discuss the interrelationship among mechanical work,

power, and energy• Solve quantitative problems related to kinetic concepts

Newton’s Laws

• Isaac Newton developed the theories of gravitation in 1666, when he was 23 years old.

• In 1686 he presented his three laws of motion.

Newton’s LawsLaw of Inertia

• A body will maintain a state of rest or constant velocity unless acted on by an external force that changes its state.

• Inertia is the property of a body that causes it to remain at rest if it is at rest or continue moving with a constant velocity unless a force acts on it.

Newton’s First Law of Motion• Law of Inertia

Every body will remain in a state of rest or constant motion (velocity) in a straight line unless acted on by an external force that changes that state

• A body cannot be made to change its speed or direction unless acted upon by a force(s)

• Difficult to prove on earth due to the presence of friction and air resistance

Newton’s LawsLaw of Acceleration

• A force (F) applied to a body causes an acceleration (a) of that body of a magnitude proportional to the force, in the direction of the force, and inversely proportional to the body’s mass (m).

• Forms the link between force and motion:

• Force = mass x acceleration or acceleration = Force

mass

• Assuming mass remains constant, the greater the force the greater the acceleration

• Acceleration is inversely proportional to mass– if force remains the same

and mass is halved, then acceleration is doubled

– if force remains the same and mass is doubled, then acceleration is halved

Applications of Newton’s 2nd Law

F = m × a

a = 333 m·s-2

Fa = = m

5001.5

F = 500 N a = ?

m = 1.5 kg

Newton’s LawsLaw of Reaction

• For every action, there is an equal and opposite reaction

• When one body exerts a force on a second, the second body exerts a reaction force that is equal in magnitude and opposite in direction of the first body

Newton’s LawsLaw of Reaction

• During gait, every contact of a foot with the ground generates an upward reaction force by the ground called (GRF).

• GRF has both horizontal and vertical components.

• What is magnitude of vertical GRF running?

Newton’s LawsLaw of Gravitation

• All bodies are attracted to one another with a force proportional to the product of the masses and inversely proportional to the square of the distance between them

• Fg = G(m1m2 / d2)– Fg = Force of attraction– Earth mass = 5.972 x 1024 kg– Earth radius = 6.378 x 103 km– Greater mass, greater attraction– Greater distance, less attraction

Implications of Newton’s Law of Gravitation

• MassMass– Greater mass =

greater gravitational force– Smaller mass =

lower gravitational force

• DistanceDistance– Greater distance =

smaller gravitational force– Smaller distance =

greater gravitational force

• Most bodies in sport have relatively small mass– Attractive force between

them can be considered negligible

Newton’s LawsLaw of Gravitation

Latitude Acceleration due to Gravity

Sample Locations

0 9.780 Ecuador, Kenya

15 9.784 Philippines, Guatemala

30 9.793 Texas, Israel

45 9.806 Oregon, France

60 9.819 Alaska, Sweden

75 9.829 Greenland

90 9.832 North Pole

Acceleration Because of Gravity at Sea Level by Latitude

Gravitational variations largely accounted for by the equatorial bulge of the earth rather than altitude above sea level.

rrpolespoles

Weight• Weight (W) is the attractive force

between the earth and any body in contact with it or close to its surface

• Product of the mass (m) of the body and the acceleration caused by the attractive force between it and the earth(g = 9.81 m·s-2)

i.e. W = m × g

• Gravity is based on:– Mass of bodies– Distance between bodies

rrequatorequator

r=radius of earthrequator > rpoles

Gequator < Gpoles

Wequator < Wpoles

Mechanical Behavior of Bodies in ContactFriction

• Friction is a force that acts parallel to the surfaces in contact and opposite to the direction of motion.

• If there is no motion, friction acts in opposite direction to any force that “tends to produce motion”.

• Friction is a necessity for and a hindrance to motion.

12-5

Magnitude of friction forces determine relative ease or difficulty of motion for two objects in contact.

Mechanical Behavior of Bodies in ContactFriction

• Starting friction is greater than moving friction.• It takes more force to start moving an object

than it does to keep it moving.Maximum static friction (Fm): • As magnitude of applied force becomes greater

and greater, magnitude of opposing friction force increases to a critical point.

Kinetic (sliding) friction (Fk): • Magnitude of kinetic friction remains constant.

Mechanical Behavior of Bodies in ContactFriction

For static bodies, friction is equal to the applied force.

For bodies in motion, friction is

constant and less than maximum

static friction.

StaticFm = sR

DynamicFk = kR

Applied external force

Fric

tion

Mechanical Behavior of Bodies in ContactFriction

Ff = RFf = frictional force; = coefficient of friction; R = normal (┴)

reaction force

Coefficient of friction: number that serves as index • Coefficient of static friction (s):

• Coefficient of kinetic (sliding) friction (k) :

Normal reaction force (┴): force acting . . .

Rolling friction: influence by weight, radius, deformability of rolling object, plus

Mechanical Behavior of Bodies in ContactFrictionFf = R

Amount of friction changed by altering μ.– Gloves in racquetball, golf, batting– Wax on surfboard or cross country skis– Rosin on dance floor

• Provides large s

• Provides significantly smaller k

Amount of friction changed by altering R.– Press surfaces together (+ or – weight)

Mechanical Behavior of Bodies in ContactFriction

Material Starting Friction Sliding Friction

Hardwood on hardwood Steel on steelSteel on steel (oiled)Rubber on dry concreteRubber on wet concrete

0.400.580.132.01.5

0.250.200.131.00.97

Synovial fluid present in many joints reduces friction between articulating bones.

Mechanical Behavior of Bodies in ContactFriction

• Artificial turf and cleats coefficient of friction cause more injuries?

Shoe traction dry• μ = .90 to 1.50Shoe traction wet• μ = 1.10 to 1.50FIFA recommendation• μ = 0.35 to 0.75

Mechanical Behavior of Bodies in Contact

Momentum is quantity of motion a body possesses.Linear Momentum: M or p• M = m • v• Units: kg • m/s (or slug • ft/s)• Downhill skier example: 55 kg • 30 m/s = 1650 kg • m/s

Newton’s laws can be expressed in terms of momentum.

Mechanical Behavior of Bodies in Contact

• Newton’s first law (inertia) states that in the absence of external forces the momentum of an object remains constant.

M = constantPrinciple of Conservation of Momentum

• Newton’s second law (acceleration) states that the rate of change of momentum equals the net external force acting on it.

Mechanical Behavior of Bodies in Contact

Newton’s third law (action-reaction) may be stated in momentum terms as whenever two bodies exert forces on one another, the resulting changes of momentum are equal and opposite.

Principle of conservation of momentum:In the absence of external forces, the total

momentum of a given system remains constant. M1 = M2

Mechanical Behavior of Bodies in Contact

Principle of conservation of momentum:

In the absence of external forces, the total

momentum of a given system remains constant. M1 = M2 or (mv)1 = (mv)2

• Initial momentum (M1) of objects before collision

• Final momentum (M2) of objects after collision

Mechanical Behavior of Bodies in Contact

Golf Ball and Club example:Golf Ball mass = .045 kg, Golf Club mass = .200 kgGBv1 = 0, GCv1= 270 m/s; GBv2=360 m/s, GCv2= ?• How much does Golf Club slow down? Momentum before Momentum after impact.045 · 0 + .2 · 270 m/s = .045 · 360 m/s + .2 · GCv2

0 + 54 kg · m/s = 16.2 kg · m/s + .2 kg· GCv2

37.8 kg · m/s .2 kg = 189 m/s = GCv2

Golf Club Velocity Decreases (189 – 270) = - 81 m/s

Mechanical Behavior of Bodies in ContactImpulse

Changes in momentum depend on force and length of time during which force acts.

Impulse: product of force and time interval the force acts• Impulse (J) = F tDerived from Newton’s Second law:• F = ma; (a = v2 - v1 / t)• F = m ([v2 - v1] / t) = (mv2 – mv1)/ t• Ft = (mv2) - (mv1)• Ft = M2 – M1 = MThis is the impulse-momentum relationship.

Mechanical Behavior of Bodies in Contact Impulse

• Bunch start results in clearing blocks sooner but with less velocity.

• Highest proportion of best runs are from 16-in stance.

• Elongated stance of 26- in results in greater velocity leaving blocks but lost within 10 yds

Block Spacing

11 in. 16 in. 21 in. 26 in.

Time on Blocks, i.e. time from gun to foot leaving

0.345 s 0.374 s 0.397 s 0.426 s

Block Velocity, i.e. horizontal velocity as leave blocks

6.63 ft/s 7.41 ft/s 7.50 ft/s 7.62 ft/s

Time to 10 yd (9.1 m)

2.070 s 2.054 s 2.041 s 2.049 s

Time to 50 yd (45.7 m)

6.561 s 6.479 s 6.497 s 6.540 s

Adapted from Henry, F. M. (1952) Research Quarterly, 23:306.

12-10

Since Impulse = F x t, i.e. the amount of force applied during a period of time,

impulse is the area under the force curve.

Which jump generated greater change in momentum (vertical velocity)?

Impulse

ImpulseHorizontal (sagittal) Plane• If initial negative impulse <

push off positive impulse, horizontal velocity increased.

• If initial negative & push off impulse equal, no change in horizontal velocity.

• If initial negative impulse > push off positive impulse, horizontal velocity decreased.

Mechanical Behavior of Bodies in ContactImpact

Impact: collision of two bodies over small time Elasticity: an object’s ability to return to its

original size and shape when outside forces are removed.

Perfectly elastic impact: relative velocities samePerfectly plastic impact: at least one body loses

velocity, bodies don’t separateMost impacts are neither perfectly elastic nor

perfectly plastic.

Mechanical Behavior of Bodies in Contact

Ball velocities before impact

Ball velocities after impact

u1 u2

v1 v2

v1 - v2 = -e ( u1 - u2)

The differences in two balls’ velocities before impact is proportional

to the difference in their velocities after impact. The factor of proportionality is the

coefficient of restitution.

Mechanical Behavior of Bodies in ContactImpact

Impact (cont.)Coefficient of restitution:

When two bodies undergo a direct collision, the difference in their velocities immediately after impact is proportional to the difference in their velocities immediately before impact

-e = relative velocity after impact = v1 - v2

relative velocity before impact u1 - u2

Elasticity

• In case of impact between moving body and stationary one, e= hb/hd

• Coefficient of restitution describes interaction between two bodies, not a single object or surface.

Elasticity

Factors Affecting C of R1.Velocities2.Temperature3.Material4.Spin

C of R Concrete Wood

Basketball 0.57 0.80

Golf ball 0.89 0.66

Racquetball 0.86 0.84

Baseball 0.57 0.45

Elasticity - Spin

• Magnitude of horizontal forces exerted on ball are influenced by amount of spin.

• Horizontal velocity of points on ball sum of 2 component velocities – translational component & rotational component.

Elasticity - Spin

• The angle of incidence (approach) equals the angle of reflection in perfectly elastic impact with no spin imparted.

Elasticity - Spin• When topspin applied,

translational component of part of ball that contacts the floor is offset by rotational component; evoked frictional force is less as is the decrease in ball’s forward velocity.

Elasticity - Spin• When backspin applied,

the translational & rotational components complement each other, the evoked frictional reaction is increased, and post-impact velocity is less.

• Backspin causes ball to bounce more slowly & at higher angle.

Summary

• Linear kinetics is the study of the forces associated with linear motion

• Friction is a force generated at the interface of two surfaces in contact

• Magnitudes of maximum static friction and kinetic friction are determined by the coefficient of friction and normal reaction force pressing the two surfaces together.

• Linear momentum is the product of an object’s mass and its velocity

Summary

• Total momentum in a given system remains constant barring the action of external forces

• Changes in momentum result from impulses, external forces acting over a time interval

• The elasticity of an impact governs the amount of velocity in the system following impact

• The relative elasticity of is represented by the coefficient of restitution