Chapter 12:
The Conditions of
Linear Motion
KINESIOLOGY
Scientific Basis of Human Motion, 11th edition
Hamilton, Weimar & Luttgens
Presentation Created by
TK Koesterer, Ph.D., ATC
Humboldt State University
Revised by Hamilton & Weimar
© 2008 McGraw-Hill Higher Education. All Rights Reserved.
Objectives
1. Name, define, and use the terms of linear motion.
2. Define magnitude, direction, and point of application
of force and use terms properly.
3. Explain changes magnitude, direction, and point of
application of force on the motion state of a body.
4. Define and give examples of linear forces, concurrent
forces, and parallel forces.
5. Determine magnitude, direction, and point of
application of muscles forces.
6. State Newton’s laws as they apply to linear motion.
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Objectives
7. Explain cause and effect relationship between forces of
linear motion and objects experiencing the motion.
8. Name & define basic external forces that modify
motion.
9. Draw and analyze a 2D free-body diagram.
10. Explain the work-energy relationship applied to a body
experiencing linear motion.
11. Define and use properly the terms work, power, kinetic
energy, and potential energy.
12. Perform a mechanical analysis of a motor skill.
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THE NATURE OF FORCE
• Force is that which pushes or pulls through
direct mechanical contact or through the force
of gravity to alter the motion of an object.
• Internal forces are muscle forces that act on
various structures of the body.
• External forces are those outside the body:
– Weight, gravity, air or water resistance,
friction, or forces of other objects acting on
the body.
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Aspects of Force
• Force is a vector quantity:
– Magnitude and direction
– Also has a point of application
• All three characteristics must be identified.
– For a weight lifter to lift a 250 N barbell:
• Lifter must apply a force greater than
250 N, in an upward direction, through
the center of gravity of the barbell.
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Magnitude
• Amount of force being applied.
– Force exerted by the barbell had a
magnitude of 250 N.
– This force was the result of gravity acting
on the mass of the barbell.
• In this case, the force is referred to as weight.
• Weight is mass times acceleration due to
gravity:
w = mg
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Magnitude of Muscular Force
• In direct proportion to the number & size of
fibers contracting in a muscle.
• Muscles normally act in groups whose force or
strength is measured collectively.
• Maximum muscular strength is measured by a
dynamometer.
• Measures force applied by a group of muscle
through an anatomical lever.
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Point of Application
• Point at which force is applied to an object.
• Where gravity is concerned this point is
always through the center of gravity.
• For muscular force, this point is assumed to
be the muscle’s attachment to a bony lever.
• Technically, it is the point of intersection of the
line of force and the mechanical axis of the
bone.
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Direction
• Direction of a force is along its action line.
– Gravity is a downward-directed vector
starting at the center of gravity of the
object.
– Direction of a muscular force vector is the
direction of line of pull of the muscle.
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Direction of Muscular
Force Vector
• Muscle angle of pull: the angle between the
line of pull and the mechanical axis of the
bone.
Fig 12.1
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Resolution of
Forces
• Magnitude is line A.
• Point of application is at
point B.
• Direction is represented
by the arrowhead and the
angle .
Fig 12.2
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Angle of Pull
• Force may be resolved into vertical and a
horizontal components.
• Size of each depends on angle of pull.
• Since a muscle’s angle of pull changes with
every degree of joint motion, so do the
horizontal & vertical components .
• The larger the angle (0° - 90°), the greater the
vertical and less the horizontal component.
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Angle of Pull
• The vertical component is
perpendicular to the lever,
called rotary component.
• The horizontal component is
parallel to the lever and is the
nonrotary component.
• Most resting muscles have
an angle of pull < 90°.
Rotary
component
Nonrotary
component
Fig 12.1a
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Rotary vs. Nonrotary Components
Angle of pull < 900
• Nonrotary force is
directed toward fulcrum.
• Helps maintain integrity
of the joint (stabilizes).
Rotary
component
Nonrotary
component
Fig 12.1a
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Rotary vs. Nonrotary Components
Angle of pull > 900
• Dislocating force is
directed away fulcrum.
• Does not occur often.
• Muscle is at limit of
shortening range and not
exerting much force.
Fig 12.1c
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Rotary vs. Nonrotary Components
Angle of pull = 90°
• Force is all rotary.
Angle of pull = 45°
• Rotary & nonrotary
components are equal.
Muscular force functions:
• Movement
• Stabilization
Fig 12.1b
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Anatomical Pulley
• Changes the angle of
pull of the muscle
providing the force.
• This increase in angle
of pull increases the
rotary component.
– e.g. Patella for the
quadriceps.
Fig 12.4
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Resolution of External Forces
• Accomplished in the
same manner as
muscular forces applied
at an oblique angle.
• Only horizontal force will
move the table.
• Vertical force serves to
increase friction.
Fig 12.7
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Composite Effects of
Two or More Forces
• Two or more forces can be applied to objects.
– A punted ball’s path is the result of force of
the kick, force of gravity, and force of wind.
– Muscles work in groups, e.g. the 3
hamstrings.
• Composite forces on the body may be
classified according to their direction and
application as linear, concurrent, or parallel.
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Linear Forces
• For forces applied in the same direction, the
resultant is the sum of the forces:
a+b=c
a
+
b
c
=
• For forces applied in the opposite directions,
the resultant is the sum of the forces:
a + (-b) = c
a
+
b
=
c
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Concurrent Forces
• Act at the same point of
application at different
angles.
• Resultant of two or more
concurrent forces
depends on both the
magnitude of each force
and the angle of
application.
Fig 12.8
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Parallel Forces
• Forces not in the same
action line, but parallel to
each other.
• Three parallel forces:
– two upward
– one downward
Fig 12.9
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Parallel Forces
• 10 N weight at 90°.
• Gravity acts at points B &
C.
• A is the force of biceps.
• Effect of parallel forces
on an object depends on
magnitude, direction &
application point of each
force.
Fig 12.10
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NEWTONS’ LAWS OF MOTION
Law of Inertia
A body continues in its state of rest or of uniform
motion unless an unbalanced force acts on it.
– An object at rest remains at rest.
– An object in motion remains in same motion
– Unless acted upon by an outside force.
• Friction & air resistance effect objects in motion.
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Law of Inertia
• A body continues in its
state of rest or of
uniform motion unless
an outside, unbalanced
force acts on it.
Vx
Vy
Gravity
Fig 12.11
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Law of Acceleration
F = ma
The acceleration of an object is directly
proportional to the force causing it and
inversely proportional to the mass of the
object.
What is the force needed to produce a given
linear acceleration?
• Since m = w/g, F = (w/g) x a
• Force to accelerate a 300 N object 2 m/sec2
• F = (300 N / 9.8m/s2) x 2 m/s2 = 61 N
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Impulse
Ft = m(vf – vi)
The product of force and
the time it is applied.
F = ma
• Substitute (vf – vi) / t for a:
F= m (vf – vi) / t
• Multiply both sides by
time:
Ft = m (vf – vi)
Fig 12.12
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Momentum
Ft = mvf - mvi
The product of mass and velocity
• 20 N force applied for 5 sec has equal
momentum to a 100 N force falling for 1 sec.
Why?
• Any change in momentum is equal to the
impulse that produces it.
• Force applied in direction of motion will
increase momentum.
• Force applied opposite to direction of motion
will decrease momentum.
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Law of Reaction
For every action there is an equal and opposite
reaction.
Fig 12.13 & 12.14
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Conservation of Momentum
In any system where forces act
on each other the momentum
is constant.
Fig 12.15
• An equal and opposite
momentum change must
occur to object producing
reactive force.
• Therefore:
m1vf1 – m1vi1 = m2vf2 – m2vi2
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Summation of Forces
Force generated by muscle may be summated from one
segment to another.
Typical throwing pattern
• Force from legs is transferred to the trunk.
• Further muscular force increases momentum and is
transferred to upper arm.
– Mainly as an increase velocity because mass is
smaller.
• Sequential transfer of momentum continues with mass
decreasing and velocity increasing.
• Finally momentum is transferred to thrown ball.
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FORCES THAT MODIFY MOTION
Weight
• The force of gravity is
measured as the weight of
the body applied through
the center of gravity of the
body and directed toward
the earth’s axis.
W = mg
Weight
Fig 12.16
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Contact Forces:
Normal Reaction
• For every action there
is an equal and
opposite reaction.
– The jumper pushes
off the ground and
the ground pushes
back.
Fig 12.17
Reaction
Action
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Contact Forces:
Friction
• Friction is the force that opposes efforts to
slide or roll one body over another.
– In some cases we try to increase friction
for a more effective performance.
– In other cases we try to decrease friction
for a more effective performance.
• The amount of friction depends on the nature
of the surfaces and the forces pressing them
together.
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Friction
Friction is proportional to
the force pressing two
surfaces together.
• Force of friction acts
parallel to the surfaces
and opposite to the
direction of motion.
Fig 12.18
W = weight
T = reactive force of table
P = force needed to move
F = force resisting motion
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Coefficient of friction, 
• The ratio of force needed to overcome the
friction, P, to the force holding the surface
together, W:
 = P / W or
 = Fmax/FN
– Large coefficient surfaces cling together.
– Small coefficient surfaces slide easily.
– Coefficient of 0.0 = frictionless surface.
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Coefficient of Friction
• May be found by:
– Placing one object on a second and tilt the
second until first begins to slide.
– The tangent of the angle with horizontal is
the coefficient of friction.
Fig 12.19
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Elasticity and Rebound
• Objects rebound in a predictable manner.
• The nature of rebound is governed by
elasticity, mass, and velocity of rebounding
surface, friction between surfaces, and angle
of contact.
• Elasticity is the ability to resist distorting
influences and to return to the original size
and shape.
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Elasticity and Rebound
• Stress is the force that
acts to distort.
• Strain is the distortion
that occurs.
• Stress may take the
form of tension,
compression, bending,
or torsion.
Fig 12.21b
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Coefficient of Elasticity
• Is defined as the stress divided by the strain.
• Most commonly determined in the
compression of balls by comparing drop
height with the bounce height.
e=
bounce height
drop height
• The closer to 1.0 the more perfect the
elasticity.
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Coefficient of Elasticity
• Also may be found using the Law of
Conservation of Momentum:
– Using the change in velocity of the two
objects, assuming masses remain
constant:
e = (vf1 – vf2) / (vi1 – vi2)
Where vf2 and vf1 are velocities after impact,
and vi1 and vi2 are velocities before impact.
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Angle of Rebound
• For a perfectly elastic object, the angle of
incidence (striking) is equal to the angle of
reflection (rebound).
• As coefficient of elasticity varies variations will
occur.
Fig 12.22
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Effects of Spin on Bounce
• A ball with topspin will rebound from horizontal surface
lower and with more horizontal velocity.
• A ball with backspin will rebound higher and with less
horizontal velocity.
• A ball with no spin will develop topspin.
• A ball with topspin will gain more topspin.
• A ball with backspin may be stopped or reversed.
• Spinning balls hitting vertical surfaces will react in the
same manner as with horizontal surfaces, but in
relation to the vertical surface.
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Fluid Forces
• Water and air are both fluids and as such are
subject to many of the same laws and
principles.
• The fluid forces of buoyancy, drag, and lift
apply in both mediums and have
considerable effect on the movements of the
human body.
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Buoyancy
• Archimedes’ Principle states: a body
immersed in a liquid is buoyed up by a force
equal to the weight of the liquid displaced.
• This explains why some things float and
some things sink.
• Density is a ratio of the weight of an object to
its volume.
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Specific gravity
• Ratio of the density of an object to the density
of water.
• An object the same weight for volume as
water has a specific gravity of 1.0.
• An object with specific gravity > 1.0 will sink.
• An object with specific gravity < 1.0 will float.
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Lift and Drag
Drag is the resistance to
forward motion through a
fluid.
Result of :
• fluid pressure on the
leading edge of the
object.
• amount of backward pull
produced by turbulence
on the trailing edge.
Fig 12.24 b
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Lift and Drag
Laminar flow is a smooth, unbroken flow of fluid
around an object.
• A smooth surface will have better laminar flow
than a rough surface, resulting in less drag.
Fig 12.24 a
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Lift and Drag
Lift is the result of changes in fluid pressure as the
result of difference in air flow velocities.
Bernoulli’s Principle states: the pressure in a
moving fluid decreases as the speed increases.
V P
Lift
Drag
Fig 12.24 c
V P
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Ball Spin (Magnus Effect)
• Bernoulli’s Principle
applies here also.
• A ball will move in the
direction of least air
pressure.
• A ball spinning drags a
boundary layer of air
with it, causing air to
move faster & reducing
pressure on one side.
Fig 12.25
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FREE BODY DIAGRAMS
• In analyzing any technique, one should
consider all external forces, by accounting for
effect of each one of the body.
• The isolated body is considered a separate
mechanical system.
• Easier to identify forces & represent as
vectors.
• Can help determine the application and
direction of forces acting on the body.
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Direction & Point of
Application of External Forces
Force
Direction of Force
Point of Application
Weight (W)
Downward
Center of Gravity
Normal (R)
Perpendicular
Point of contact
Friction (F)
Along surface
Point of Contact
Buoyancy (B) Upward
Center of buoyancy
Drag (D)
Opposite flow
Center of Gravity
Lift (L)
Perpendicular to drag Center of Gravity
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Free Body Diagram
• Magnitude
– arrow length
• Direction
– arrow head
• Point of application
– arrow tail
• Weight (W)
• Reactive force (R)
• Friction (F)
Fig 12.26
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Free Body Diagram
•
•
•
•
Weight (W)
Buoyancy (B)
Lift (L)
Drag (D)
State of motion or rest of
the body depends on
the vector sum of all
these forces.
Fig 12.27
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Free Body
Diagram
Fig 12.28
• Also used to show forces
on a body segment.
• Thigh is isolated:
– Weight of thigh (W)
– Muscle force Hip (MH)
– Reactive Forces
• Hip (Hx & Hy)
• Knee (Kx & Ky))
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WORK, POWER, AND ENERGY
Work
• Work is the product of force expended and
the distance over which force is applied.
W = Fs
– Work (W), Force (F), Distance (s)
• Units are any combination of force &
distance:
– foot/pounds,
– joule = 107 x 1 gram / 1 centimeter
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Work
• A 20 N suitcase is place on a shelf 2 m above
the ground:
– Work done against gravity= 40 Nm
• Same suitcase lifted along a 4 m incline is still
40 Nm of work against gravity.
– Horizontal distance not included.
4m
2m
30o
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Positive & Negative Work
• Positive work – force acts in the same
direction to that of the objects motion.
• Negative work – force acts in the opposite
direction to that of the objects motion.
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Mechanical Muscular Work
Example: a rectangular muscle 10 cm x 3 cm,
that exerts 240N of force.
• Average muscle fiber shortens 1/2 its length.
W = Fs
W = 240N x 5cm
W = 1200N•cm or 120 Nm
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Force per Muscle Cross Section
• If force of the muscle is not known, it is
computed form the muscle’s cross section.
Example: Assume same muscle is 1cm thick:
Cross section = width x thickness
3 cm X 1 cm = 3 sq cm
Average force = 360 N per sq cm
F = 360 x 3 = 1080N
W = Fs
W = 1080N x 5cm = 5400 N cm or 540 Nm
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Muscular Work
• If the internal structure of the muscle is
rectangular, a simple geometric crosssectional measure can be used.
• For penniform & bipenniform muscle,
physiological cross section must be
determined.
• “s” represents 1/2 the length of the average
fiber.
• Force per square inch depends on whose
research the student accepts.
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Muscle Work by
Physiological Cross Section (PCS)
W = Average force x PCS (sq cm) x .5 length of
fibers (cm)
Divide by 100 to convert N-cm to Nm
W (Nm) = 360 x PCS (sq cm) x .5 fiber length (cm)
100
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Power
• The rate at which work is done.
P = Fs / t or P = W / t or
P = Power
W = work
P = Fv
t = time
v = velocity = s / t
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Energy
• The capacity to do work.
Law of Conservation of Energy:
The total amount of energy possessed
by a body or an isolated system
remains constant.
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Potential Energy
• Potential energy: energy based on position.
• Potential energy is the product of the weight
of an object and the distance over which it
can act:
PE = mgh
m = mass, g = gravity, h = height
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Kinetic Energy
• Energy based on motion:
KE = 1/2 mv2
m = mass, v = velocity
• Work done is equal to the kinetic energy
acquired, or
Fs = 1/2 mv2
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ANALYSIS OF LINEAR MOTION
• First identify the nature of the forces
involved in the motion of interest:
– Weight
– Propulsive forces
– Ground Reaction Force
– Friction
– Buoyancy, Drag, & Lift
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ANALYSIS OF LINEAR MOTION
• The principles that govern the mechanical
aspects of a movement can be summarized
by examining some of the basic concepts
involved in the kinetics of linear motion:
– Inertia
– Impulse
– Work & Power
– Potential & Kinetic Energy
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Chapter 12:
The Conditions of
Linear Motion
© 2008 McGraw-Hill Higher Education. All Rights Reserved.