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Chapter 10:
Terminology and Measurement
in Biomechanics
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. Define mechanics & biomechanics.
2. Define kinematics, kinetics, statics, & dynamics, and
state how each relates to biomechanics.
3. Convert units of measure; metric & U.S. system.
4. Describe scalar & vector quantities, and identify.
5. Demonstrate graphic method of the combination &
resolution of two-dimensional (2D) vectors.
6. Demonstrate use of trigonometric method for
combination & resolution of 2D vectors.
7. Identify scalar & vector quantities represented in
motor skill & describe using vector diagrams.
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Mechanics
• Area of scientific study that answers the
questions, in reference to forces and motion
– What is happening?
– Why is it happening?
– To what extent is it happening?
• Deals with force, matter, space & time.
• All motion is subject to laws and principles of
force and motion.
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Biomechanics
• The study of mechanics limited to living
things, especially the human body.
• An interdisciplinary science based on the
fundamentals of physical and life sciences.
• Concerned with basic laws governing the
effect that forces have on the state of rest or
motion of humans.
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Statics and Dynamics
• Biomechanics includes statics & dynamics.
Statics: all forces acting on a body are balanced
F = 0 - The body is in equilibrium.
Dynamics: deals with unbalanced forces
F  0 - Causes object to change speed or
direction.
• Excess force in one direction.
• A turning force.
• Principles of work, energy, & acceleration are
included in the study of dynamics.
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Kinematics and Kinetics
Kinematics: geometry of motion
• Describes time, displacement, velocity, &
acceleration.
• Motion may be straight line or rotating.
Kinetics: forces that produce or change motion.
• Linear – causes of linear motion.
• Angular – causes of angular motion.
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QUANTITIES IN BIOMECHANICS
The Language of Science
• Careful measurement & use of mathematics
are essential for
– Classification of facts.
– Systematizing of knowledge.
• Enables us to express relationships
quantitatively rather than merely descriptively.
• Mathematics is needed for quantitative
treatment of mechanics.
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Units of Measurement
• Expressed in terms of space, time, and mass.
U.S. system: current system in the U.S.
• Inches, feet, pounds, gallons, second
Metric system: currently used in research.
• Meter, kilogram, newton, liter, second
Table 10.1 present some common conversions
used in biomechanics
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Units of Measurement
Length:
• Metric; all units differ by a multiple of 10.
• US; based on the foot, inches, yards, & miles.
Area or Volume:
• Metric: Area; square centimeters of meters
– Volume; cubic centimeter, liter, or meters
• US: Area; square inches or feet
– Volume; cubic inches or feet, quarts or
gallons
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Units of Measurement
Mass: quantity of matter a body contains.
Weight: product of mass & gravity.
Force: a measure of mass and acceleration.
– Metric: newton (N) is the unit of force
– US: pound (lb) is the basic unit of force
Time: basic unit in both systems in the second.
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Scalar & Vector Quantities
Scalar: single quantities
– Described by magnitude (size or amount)
• Ex. Speed of 8 km/hr
Vector: double quantities
– Described by magnitude and direction
• Ex. Velocity of 8 km/hr heading
northwest
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VECTOR ANALYSIS
Vector Representation
• Vector is represented by an arrow
• Length is proportional to magnitude
Fig 10.1
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Vector Quantities
• Equal if magnitude & direction are equal.
• Which of these vectors are equal?
A.
B.
C.
D.
E.
F.
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Combination of Vectors
• Vectors may be combined be addition, subtraction,
or multiplication.
• New vector called the resultant (R ).
Fig 10.2
Vector R can be achieved by different combinations.
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Combination of Vectors
Fig 10.3
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Resolution of Vectors
• Any vector may be
broken down into two
component vectors
acting at a right angles
to each other.
• The arrow in this figure
may represent the
velocity the shot was
put.
Fig 10.1c
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Resolution of
Vectors
• What is the vertical
velocity (A)?
• What is the horizontal
velocity (B)?
• A & B are
components of
resultant (R)
Fig 10.4
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Location of Vectors in Space
• Position of a point (P) can be located using
– Rectangular coordinates
y
– Polar coordinates
• Horizontal line is the x axis.
• Vertical line is the y axis.
x
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Location of Vectors in Space
• Rectangular coordinates for point P are
represented by two numbers (13,5).
– 1st - number of x units
y
– 2nd - number of y units
P=(13,5)
5
13
x
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Location of Vectors in Space
• Polar coordinates for point P describes the
line R and the angle it makes with the x axis.
It is given as: (r,)
– Distance (r) of point P from origin y
– Angle ()
P
13.93
21o
x
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Location of Vectors in Space
Fig 10.5
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Location of Vectors in Space
• Degrees are measured in a counterclockwise
direction.
Fig 10.6
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Graphic Resolution
• Quantities may be handled graphically:
– Consider a jumper who takes off with a
velocity of 9.6 m/s at an angle of 18°.
– Since take-off velocity has both magnitude &
direction, the vector may be resolved into x & y
components.
– Select a linear unit of measurement to
represent velocity, i.e. 1 cm = 1 m/s.
– Draw a line of appropriate length at an angle
of 18º to the x axis.
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Graphic Resolution
• Step 1- Use a protractor to measure
18o up from the positive x-axis.
• Step 2- Set a reference length. For
example: let 1 cm = 1 m/s.
• Step 3- Draw a line 9.6 cm long, at
an angle of 18° from the positive xaxis.
• Step 4- Drop a line from the tip of
the line you just drew, to the x-axis.
• Step 5- Measure the distance from
the origin, along the x-axis to the
vertical line you just drew in step 4.
This is the x component (9.13cm =
9.13 m/s).
•Step 6- Measure the length of the
vertical line. This is the ycomponent (2.97cm = 2.97m/s).
y
9.6 m/s
18°
2.97 cm =
2.97 m/s
9.13 cm = 9.13 m/s
x
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Combination of Vectors
• Resultant vectors may be obtained
graphically.
• Construct a parallelogram.
• Sides are linear representation of two
vectors.
• Mark a point P.
• Draw two vector lines to scale, with the same
angle that exists between the vectors.
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Combination of Vectors
• Construct a parallelogram for the two line
drawn.
• Diagonal is drawn from point P.
• Represents magnitude & direction of resultant
vector R.
Fig 10.8
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Combination of Vectors
with Three or More Vectors
• Find the R1 of 2 vectors.
• Repeat with R1 as one of the vectors.
Fig 10.9
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Another Graphic Method of
Combining Vectors
• Vectors added head
to tail.
• Muscles J & K pull on
bone E-F.
• Muscle J pulls 1000 N
at 10°.
• Muscle K pulls 800 N
at 40°.
Fig 10.10a
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Another Graphic Method of
Combining Vectors
• 1 cm = 400 N
• Place vectors in
reference to x,y.
• Tail of force vector of
Muscle K is added to
head of force vector
of Muscles J.
• Draw line R.
Fig 10.10b
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Another Graphic Method of
Combining Vectors
• Measure line R &
convert to N:
• 4.4 cm = 1760 N
• Use a protractor to
measure angle:
• Angle = 23.5°
• Not limited to two
vectors.
Fig 10.10b
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Graphical Combination of
Vectors
• Method has value for portraying the situation.
• Serious drawbacks when calculating results:
– Accuracy is difficult to control.
– Dependent on drawing and measuring.
• Procedure is slow and unwieldy.
• A more accurate & efficient approach uses
trigonometry for combining & resolving
vectors.
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Trigonometric Resolution
of Vectors y
• Any vector may be
resolved if
trigonometric
relationships of a right
triangle are employed.
• Same jumper example
as used earlier.
• A jumper leaves the
ground with an initial
velocity of 9.6 m/s at an
angle of 18°.
9.6m/s
18o
x
Find:
• Horizontal velocity
(Vx)
• Vertical velocity (Vy)
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Trigonometric Resolution
of Vectors
Given: R = 9.6 m/s
 = 18°
To find Value Vy:
opp Vy
sin  

hyp R
Vy = sin 18° x 9.6m/s
= .3090 x 9.6m/s
= 2.97 m/s
Fig 10.11
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Trigonometric Resolution
of Vectors
Given: R = 9.6 m/s
 = 18°
To find Value Vx:
adj Vx
cos  

hyp R
Vx = cos 18° x 9.6m/s
= .9511 x 9.6m/s
= 9.13 m/s
Fig 10.11
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Trigonometric Combination
of Vectors
• If two vectors are applied at a right angle to
each other, the solution process is also
straight-forward.
– If a baseball is thrown with a vertical
velocity of 15 m/s and a horizontal velocity
of 26 m/s.
– What is the velocity of throw & angle of
release?
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Trigonometric Combination
of Vectors
Given:
Vy = 15 m/s
Vx = 26 m/s
Find: R and 
Solution:
R2 = V2y + V2x
R2 = (15 m/s)2 + (26 m/s)2 = 901 m2/s2
R = √ 901 m2/s2
R = 30 m/s
Fig 10.12
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Trigonometric Combination
of Vectors
Solution:
  arctan
Vy
Vx
15 m s
  arctan m
26 s
  30o
Velocity = 30 m/s
Angle = 30°
Fig 10.12
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Trigonometric Combination
of Vectors
• If more than two vectors are involved.
• If they are not at right angles to each other.
• Resultant may be obtained by determining
the x and y components for each vector and
then summing component to obtain x and y
components for the resultant.
• Consider the example with Muscle J of 1000
N at 10°, and Muscle K of 800 N at 40°.
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Muscle J
R = (1000N, 10°)
y = R sin 
y = 1000N x .1736
y = 173.6 N (vertical)
x = R cos 
x = 1000N x .9848
x = 984.8 N (horizontal)
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Muscle K
R = (800N, 40°)
y = R sin 
y = 800N x .6428
y = 514.2 N (vertical)
x = R cos 
x = 800N x .7660
x = 612.8 N (horizontal)
Sum the x and y components
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Trigonometric Combination
of Vectors
Given:
Fy = 687.8 N
Fx = 1597.6N
Find:
 and r
Fig 10.13
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Trigonometric Combination
of Vectors
Solution:
  arctan
  arctan
 Fy
 Fx
687.8 N
1597.6 N
  23.3o
R 2   Fy2   Fx2
R 2  (687.8 N ) 2  (1597.6 N ) 2
R 2  3025395 N 2
R  1739 N
Fig 10.13
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Value of Vector Analysis
• The ability to understand and manipulate the
variables of motion (both vector and scalar
quantities) will improve one’s understanding
of motion and the forces causing it.
• The effect that a muscle’s angle of pull has on
the force available for moving a limb is better
understood when it is subjected to vector
resolution.
• The same principles may be applied to any
motion such as projectiles.
© 2008 McGraw-Hill Higher Education. All Rights Reserved.
Chapter 10:
Terminology and Measurement in
Biomechanics
© 2008 McGraw-Hill Higher Education. All Rights Reserved.