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Announcements

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• TTU Football Away
• Saturday vs. OSU

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Newtons Laws

• ESS 5306-001
• Lecture 4

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Part 1

• Newtons Laws governing Linear Motion

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First some terminology
5
Linear Motion

• Motion of an object in linear or curvilinear path
• Key is the path of the object does not follow
  along a circular path
• Examples

6
Forces Acting on Links

• Force Classifications
• External Forces
• Ground Reaction Forces (GRF)
• Internal Forces
• Muscle and Ligament Forces
• Another classification can be
• Contact
• Non-Contact

7
Non-Contact Forces

• Newtons Law of Gravitational Force
• The force of gravity is inversely proportional to
  the square of the distance between attracting
  objects and directly proportional to the product
  of their masses.
• G universal gravitational constant (6.67
  10-11 Nm2/kg2)
• m1 mass of one object
• m2 mass of other object
• r distance between the mass centres of the
  objects

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Gravitational Acceleration

• Fg -9.81 m/s2
• Fg (G mobject Mearth) / r2
• Attractive force of the earth on an object
• F ma mg
• Weight mg (for stationary objects on Earth)
• g is acceleration due to gravity (same as Fg
  above)
• Bodyweight is mass times gravitational
  acceleration

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Weight and Mass

• F m a
• (F is force, m is mass, a is acceleration)
• Weight m a
• SoOn Earth
• Weight m g
• (where g is Earths constant gravity -9.81 m/s2)

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Contact Forces

• Forces resulting from the interaction of two
  objects
• Types of Contact Forces
• Ground Reaction Force (GRF)
• Joint Reaction Force
• Friction
• Fluid Resistance
• Inertial Force
• Muscle Force
• Elastic Force

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Ground Reaction ForceGRF

• Force equal in magnitude and opposite in
  direction to that which is applied
• Force Platform

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Properties of Force Curves VGRF Force Traces
Landing from a drop
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Walking versus Running
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Soft and Stiff Landings
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Representation of Forces on System
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COG versus COM

• Center of Gravity (COG)
• Center of Mass (COM)
• Gravity is dependent upon the square of the
  distance from the center of the earth
• What affects gravitational acceleration (g)?

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Inertial ForceInertia

• Force resulting from an objects motion (or lack
  thereof, where a is simply g).
• F ma
• Segments often exert a force on a more distal
  segment due to inertial force rather than
  muscular force.
• A body in motion
• A body at rest

The toughest part of the day is overcoming
inertia.
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Normalized VGRF

• Vertical Ground Reaction Forces (VGRF)
• Normalize to body weight in Newtons
• Allows relative comparison between subjects
• VGRFnorm VGRF / BW
• Units are BW (body weights)
• Ex Body weight is 600 N, and VGRF 3000 N
• VGRFnorm 5 BW

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Joint Reaction ForceJRF

• Resultant force acting across the joint
• Forces from above
• Forces below
• Force of distal bony surface on proximal bony
  surface (adjacent segments)
• Usually (truly) unknown
• Need appropriate kinematic, kinetic and
  anthropometric (body dimensions) data

We make stick people
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Bone-on-Bone Force (FB)

• The sum of the muscle force (Fm) and other
  external forces.
• Composite of all active forces acting across the
  joint
• Fm JRF mg
• Not to be confused with the net joint reaction
  force

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Friction Force

• Friction
• Force acting parallel between two contacting
  surfaces that tend to rub or slide past each
  other
• Friction can exist without movement
• Example?

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Coefficient of Friction

• µ coefficient of friction
• Ff friction force
• N normal force
• FsMAX maximum static friction force
• µs static coefficient of friction
• Fk kinetic force ( FsMAX)
• µk kinetic coefficient of friction

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Friction Force
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Muscle Force
Remember Muscles can only pull
Fm
When will muscle strength capability be maximal?
25
Elastic Force

• k constant of proportionality (stiffness)
• ?s change in length
• In springboard diving
• Body weight deflects the springboard
• The springboard stores an elastic force (elastic
  energy) which is returned to fling the diver
  upward

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Fluid Resistance

• Fluids
• Gas (e.g., air)
• Liquid (e.g., water)
• Fluids resist the movement of objects through
  them
• Determining the magnitude and direction of fluid
  resistance is very complex
• Fluid properties which influence resistance
• Density
• Mass per unit volume
• Increase density, increase resistance
• Air density is affected by humidity, temperature,
  and pressure
• Viscosity
• Fluids resistance to flow
• Air viscosity increases with air temperature

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Fluid Resistance

• Object disturbs fluid
• Disturbance is dependent upon density and
  viscosity of fluid
• Increased disturbance, increased energy passing
  from the object to the fluid
• Transfer of energy is termed fluid resistance
• 2 components of fluid resistance are drag and lift

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Now for Newtons Laws Governing Linear Motion
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Isaac Newton?

• Isaac Newton (1642 1727)
• Laws relating observed motion with the causative
  forces.
• Principia Mathematica (1686)
• Philosophiae naturalis principia mathematica
• Fluxiones
• Physics and Quantum Physics
• Other Contributions to Science
• Calculus
• Big battle with Liebniz regarding who discovered
  it first
• Theories on Prisms, Light and Colour
• Opticks, 1704
• Law of Gravitational Acceleration

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Newtons 1st Law of Linear Motion

• Law of Inertia
• If no external forces act on a body then the
  velocity of that body remains constant
• A body at rest tends to remain at rest
• A body in motion tends to remain in motion

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Law of Inertia

• Inertia
• Describes an objects resistance to motion
• Mass
• Measure of the amount of matter constituting an
  object (SI unit kg)
• Inertial Force ma mg
• Force which must be overcome for an object to
  move from rest or change direction

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Newtons 2nd Law

• Law of Acceleration
• The rate of change of momentum of a body is
  proportional to the applied force and takes place
  in the direction which the force acts.
• F ma
• Force equals mass times acceleration
• SI unit of force is the Newton (N kg.m/s2)
• Example
• If I try to accelerate (move) someone who has a
  mass of 50 kg, then I will need twice the force
  to similarly accelerate someone of 100 kg.

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Newtons 3rd Law

• Action Reaction
• To every action there is an equal and opposite
  reaction
• Bodies exert equal and opposite forces on each
  other
• Action and reaction are immediate. There is no
  delay between the action and reaction.
• Example
• What would be the magnitude of the vertical
  ground reaction force if I were to stand still on
  one foot?

34
Newtons Laws

• Understanding Forces
• Direct Extensions and Applications

35
Differentiation and Integration

• Differentiation (Derivative, d/dt)
• Slope of the tangent line of a curve at any point
• The change in a variable across time
• The derivative of displacement is velocity
• ?d / ?t
• The derivative of velocity is acceleration
• ?v / ?t
• Integration (Integral, ?v dt)
• Area under the curve
• Opposite function of differentiation
• The integral of acceleration is velocity
• The integral of velocity is displacement

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Differentiation and Integration
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Momentum

• Momentum (?)
• The product of the mass of the object and the
  change in velocity
• ? m(vf vi)
• ? m(?v)
• The proof
• Since F ma
• then F m(?v/ ?t)
• soF (m?v)/?t
• and soF ??/?t

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Law of Conservation of Momentum

• Unless an external force acts on a system then
  the total momentum (?) will be conserved
• i.e., ? will be constant from start to finish
• So if you are told linear momentum is conserved,
  you can assume initial and final momentum are the
  same (e.g. m?vi m?vf)
• Thus no energy is lost in the system
• This is an application of Newtons Laws to the
  linear motion of objects.
• Example (2-ball collision)

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Part 1 - Summary

• Types of forces
• Noncontact
• Contact (grf, jrf, friction, fluid resistance,
  inertia, muscle force, elastic force)
• Coefficients of friction, elasticity, drag,
  viscosity
• Lift, drag (relation with air/fluid pressure)
• Representation of forces acting on a system
• Drawing diagrams
• Newtons 3 Laws Governing Linear Motion
• Differentiation and Integration
• Conservation of Linear Momentum

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Part 2

• Newtons Laws governing Angular Motion

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First some terminology
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Key Terms

• Mass - Measure of the amount of matter comprising
  an object kg
• Weight Measure of the amount of gravitational
  attraction between an object and the earth (e.g.
  mass on moon is same as mass on Earth) N
• Segment Mass Mass of a segment kg
• Center of Mass Location for which mass of a
  body is evenly distributed m
• Sometimes called the balance point
• Normally lies within the body, but does not have
  to (e.g. a ring)
• Whole Body Center of Mass Center of mass of
  system of articulating rigid bodies

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Center of Mass

• Center of Mass
• The point about which the mass is evenly
  distributed
• It is the point about which the sum of torques is
  equal to zero
• The point about which objects rotate when in
  flight

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Segment and Total Body COM
45
Segment and Total COM Relation
? 55 BH ? 56-57 BH
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Moment of Inertia

• Moment of Inertia
• The resistance of a body to rotation about a
  given axis
• I S mi ri2
• I moment of inertia about a given axis
• np number of particles making up rigid body
• mi mass of particle
• ri distance between particle and axis
• Thus, the moment of inertia is a function of the
  mass and distribution of that mass.

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Moment of Inertia

• Units
• Consider the equation
• I S mr2
• Basic Units for each component
• kg m2
• Units for moment of inertia are kgm2

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Mass versus Mass Distribution

• Which component mass (m) or mass distribution
  (r) has the greater influence on moment of
  inertia (I)?
• Consider the equation
• I S m r2
• So, which measurement precision is more
  important, mass or center of mass position?

49
Moment of Inertia of the Human Body

• Influence of body position on moment of inertia
• Influence of moment of inertia on rotation

50
Parallel Axis Theorem

• If I (moment of inertia) is known about a given
  axis, then a new I can be found about a second
  axis provided we know how far apart the two axes
  are.
• I1 Icm mass d2
• I1 is the moment of inertia about first axis
• Icm is the moment of inertia about center of mass
• d is the distance between the two axes

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Example Calculation

• IProx ICOM mass d2
• Given
• mass 10kg
• d 0.25m
• ICOM 10 kgm2
• IProx ? (i.e., find IProx)

Answer 10.63 kgm2
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Parallel Axis Theorem

• Example Calculation
• The moment (torque) about the center of mass of a
  20kg segment is 5.4 Nm2. If the moment of
  inertia about the segments proximal end is 42
  Nm2, what is the distance between the proximal
  end and the center of mass?

53
Angular Momentum

• The product of angular velocity and the moment of
  inertia
• where, H angular momentum
• I moment of inertia
• ? angular velocity

H I ?
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Angular Momentum

• Angular Momentum (H)
• Product of moment of inertia and angular velocity
• The rotational equivalent of linear momentum (mv)
• Moment is equal to the rate of change of angular
  momentum

55
Now for Newtons LawsGoverning Angular Motion
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Newtons 1st Law

• Law 1
• If no external moments act on a body, the angular
  momentum remains constant (H I?)
• A moment is required to start, stop, or alter
  angular motion
• Applications to flight or motion on frictionless
  surface
• No external moments act, so
• Decrease moment of inertia Increase angular
  velocity
• Increase moment of inertia Decrease angular
  velocity
• Examples Ice skater, twisting gymnast, rotating
  discus thrower

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Angular Momentum

• Angular Momentum (H)
• Product of moment of inertia and angular velocity
• Units are kgm2/s
• The rotational equivalent of linear momentum
  (mv)
• Moment is equal to the rate of change of angular
  momentum
• Moment (M)
• Torque (T)
• Moment of force

58
Conservation of Angular Momentum

• From Newtons first Lawif no external moment
  acts on a body, the angular momentum remains
  constant H I ?
• Thus, moment of inertia and angular velocity are
  inversely related
• In other words
• Decrease moment of inertia (I)
• Increase angular velocity (?)
• Increase moment of inertia (I)
• Decrease angular velocity (?)

59
Conservation of Angular Momentum
When gravity is the only external force acting on
an object, the angular momentum (H) remains
constant
? I then ? ? ? I then ? ?
Derived from Newtons first law for angular motion
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Conservation of Angular Momentum

• Example Calculation
• The moment of inertia (I) of a diver in the
  layout position is 24 kgm2, and 16 kgm2 in the
  tuck position. During a somersault routine, her
  initial angular momentum is 57 kgm2/s with an
  angular velocity of 13 rad/s in the layout
  position. When she changes to a tuck position,
  what will be her new angular velocity be in
  radians/second and in degrees/second if angular
  momentum is conserved?
• Now, lets do an experiment!

61
Newtons 2nd Law

• Law 2
• The angular acceleration of a body is
  proportional to the moment causing it.
• The moment can be computed in 2 ways

• M F d-
• M is moment (torque)
• F is force causing moment
• d is moment arm of force

• M I a
• M is moment (torque)
• I is moment of inertia
• a is angular acceleration

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M F d-

• More applications
• F Muscle force
• d- Moment arm of muscle
• M Resulting moment from force application

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Moment of ForceTorque

• Units
• M F d-
• ? units (Newtons) (meters)
• Units of moment are Nm (Newton meters)

64
Moment Arm

• In order to calculate the torque or moment of
  force (generally called moment)
• The moment arm is defined as
• Perpendicular distance from axis to line of force
  application
• It is not just the distance along the segment or
  lever from the axis to the point of force
  application
• Draw 3 examples

65
Newtons 3rd Law

• Law 3
• Bodies exert equal and opposite moments on each
  other.
• Moments about a joint (e.g. knee) will be equal
  and opposite
• Injury often occurs if the moment caused by
  external forces exceeds that of the internal
  muscle and ligament forces

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Applications of these laws

• Somersault and Twist

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Body Axes of Rotation
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Somersaulting

• Body position will influence moment of inertia
  and thus number of revolutions during flight
• Example If a gymnast takes off with 60 kgm2/s of
  angular momentum about the M-L axishow many
  somersaults can they do?
• Assume tucked position has I of 4 kgm2, and
  layout position has I of 12 kgm2
• Remember H I ?
• Work this out on your ownrecall that ? is in
  rad/s
• Hint Figure how many revolutions in the radian
  velocity answer
• Solutions
• 2.4 revolutions for tucked
• 0.8 revolutions for layout

69
Somersaulting

• Middle of Skill
• Position Tucked or Piked (straight when they
  took off)
• Mechanics
• Reduced moment of inertia
• Increased angular velocity
• Impression
• Gymnast has initiated somersault in the air
• End of Skill
• Position Straightens from shape
• Mechanics
• Increase moment of inertia
• Reduces angular velocity
• Impression
• Stopping rotation to drop and land on feet

70
Generating Twist

• 3 ways to generate twist for aerials
• Contact Twist
• Cat-Twist
• Tilt-Twist
• Many gymnasts use some combination of these.

71
Contact Twist

• While in contact with the ground, the gymnast
  generates a moment about their longitudinal axis
  (e.g. turn shoulders just prior to take-off)
• Once in flight
• Increase rate of twist by pulling the arms in
  closer to the body (thus reduce the moment of
  inertia and increase spin)
• Vice versa if slowing spin is desired (abduct or
  flex arms to increase moment of inertia and
  reduce spin)

72
Cat Twist

• Twist is initiated after take-off
• Rabbits and cats
• When dropped upside down with no angular
  momentum
• Create twist by rotating the upper body one way
  while holding the lower body fixed
• This creates twist about longitudinal axis
• Why does this occur?
• Angular momentum is constant
• Thus rotation in one direction is countered by
  rotation in opposite direction, and net momentum
  remains 0.
• How much twist can be generated this way?

73
Cat Twist

• Does the cat
• generate angular
• momentum
• during the descent?

74
Tilt Twist

• When somersaulting
• If the body is tilted away from the plane of
  rotation, then twist will be initiated.
• This twist results so the cumulative (net)
  angular momentum will remain constant about all
  the axes.
• So how can the body be tilted?
• Changing arm position (raising or lowering)
• Turning shoulders when body is in piked position
• Extending from pick asymmetrically
• Realizewhen jumping, if one arm is raised the
  longitudinal axis of the body is tilted from
  vertical.

75
Tilt Twist

• Points to remember
• Tilt twist only works when somersaulting
• More tilt results in faster twist
• Twist is stopped by the reverse process
• Movements to produce twist are often subtle
• These movements can be combined with another
  twist generating method to enhance twist

76
Long Jumping

• During the flight phase of a long jump, jumpers
  can have 15 kgm/s of angular momentum about their
  M-L axis. Ideally they want to land with their
  feet forward not rotated part way through a
  somersault
• So, how do they control their body rotation to
  ensure their feet land first and their body
  continues forward past their feet?

77
Summary Questions

• What is the principal of conservation of angular
  momentum
• How can a gymnast generate twist?
• A free-style skiing aerialist performs a triple
  somersaulting skill with one and a half twists in
  the last somersault. How was this twist
  produced? Explain.
• How does a long jumper control the forward
  rotation from take-off?

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Part 2 - Summary

• Center of Mass
• Moment of Inertia (I S mr2)
• Parallel Axis Theorem (IProx Icm mass d2)
• Angular Momentum (H I ?)
• Newtons Laws
• Angular momentum
• Angular acceleration
• Action-Reaction
• Conservation of Angular Momentum
• Applications for aerial movements

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For next time

• Applications of Newtons Laws
• Projectile Motion

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