Introduction to Biomechanics
and Vector Resolution
Applied Kinesiology
420:151
Agenda
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Introduction to biomechanics
Units of measurement
Scalar and vector analysis
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Combination and resolution
Graphic and trigonometric methods
Introduction to Biomechanics
Biomechanics
The study of biological motion
Statics
The study of forces on the body in equilibrium
Kinetics and Kinematics
Dynamics
The study of forces on the body subject to unbalance
Kinetics and Kinematics
Kinetics: The study of the effect of forces on the body
Kinematics: The geometry of motion in reference to time and displacement
Linear vs. Angular
Linear vs. Angular
Linear: A point moving along a line
Angular: A line moving around a point
Agenda
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Introduction to biomechanics
Units of measurement
Scalar and vector analysis
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Combination and resolution
Graphic and trigonometric methods
Units of Measurement
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Systeme Internationale (SI)
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Base units
Derived units
Others
SI Base Units
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Length: SI unit  meter (m)
Time: SI unit  second (s)
Mass: SI unit  kilogram (kg)
Distinction: Mass (kg) vs. weight (lbs.)
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Mass: Quantity of matter
Weight: Effect of gravity on matter
Mass and weight on earth vs. moon?
SI Derived Units
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Displacement: A change in position
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Velocity: The rate of displacement
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SI unit  m
Displacement vs. distance?
SI unit  m/s
Velocity vs. speed?
Acceleration: The rate of change in velocity
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SI unit  m/s/s or m/s2
SI Derived Units
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Force: The product of mass and acceleration
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SI Unit  Newton (N)  The force that is able to
accelerate 1 kg by 1 m/s2
How many N of force does a 100 kg person exert
while standing?
Moment: The rotary action of a force
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Moment = Fd
SI Unit  N*m  When 1 N of force is applied at
a distance of 1 m away from the axis of rotation
SI Derived Units
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Work: The product of force and distance
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SI Unit  Joule (J)  When 1 N of force moves
through 1 m
Note: 1 J = 1 N*m
Energy: The capacity to do work
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Deadlift
Example
SI Unit  J
Note: 1 J = ~ 4 kcal
Power: The rate of doing work (work/time)
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SI Unit  Watt (W)  When 1 J (or N*m) is
performed in 1 s
Note: Also calculated as F*V
Other Units
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Area: The superficial contents or surface
within any given lines
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2D in nature
SI Unit  m2
Volume: The amount of space occupied
by a 3D structure
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SI Unit  m3 or liter (l)
Note: 1 l = 1 m3
Agenda
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Introduction to biomechanics
Units of measurement
Scalar and vector analysis
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Combination and resolution
Graphic and trigonometric methods
Scalar and Vector Analysis
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Scalar defined: Single quantities of magnitude
 no description of direction
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A speed of 10 m/s
A mass of 10 kg
A distance of 10 m
Vector defined: Double quantities of
magnitude and direction
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A velocity of 10 m/s in forward direction
A vertical force of 10 N
A displacement of 10 m in easterly direction
Scalar and Vector Representation
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Scalars are represented as values that
represent the magnitude of the quantity
Vectors are represented as arrows that
represent:
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The direction of the vector quantity (where
is the arrow pointing?)
The magnitude of the vector (how long is
the arrow?)
Figure 10.1, Hamilton
Combination of Vectors
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Vectors can be combined which results
in a new vector called the resultant.
We can combine vectors three ways:
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Addition
Subtraction
Multiplication
Vector Combination: Addition
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Tip to tail method
The resultant vector is represented by
the distance between the tail of first
vector and the tip of the second
+
Vector 1
=
Vector 2
Resultant
Vector Combination: Subtraction
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Tip to tail method
Resultant = Vector 1 – Vector 2 or . . .
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Resultant = Vector 1 + (- Vector 2)
Flip direction of negative vector
+
Vector 1
=
Vector 2
Resultant
Vector Combination: Multiplication
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Tip to tail method
Only affects magnitude
Same as adding vectors with same
direction
X 3 =
Vector Resolution
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Resolution: The breakdown of vectors
into two sub-vectors acting at right
angles to one another
Resultant velocity of shot at take off is
a function of the horizontal velocity (B)
and the vertical velocity (A)
Location of Vectors in Space
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Frame of reference:
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Reality = 3D  2D for simplicity
Two types:
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Rectangular coordinate system
Polar coordinate system
Rectangular Coordinate System
Y
(-,+)
(+,+)
X
(-,-)
The vector starts at (0,0) and
ends at (x,y)
(+,-)
Example: Vector (4,3)
Figure 10.5, Hamilton
Polar Coordinate System
Coordinates are (r,q) where r
= length of resultant and q=
angle
Figure 10.6, Hamilton
Graphic Resolution of Vectors
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Tools: Graph paper, pencil, protractor
Step 1: Select a linear conversion factor
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Example: 1 cm = 1 m/s, 1 N or 1 m etc.
Step 2: Draw in force vector based on frame
of reference
Step 3: Resolve vector by drawing in vertical
and horizontal subcomponents
Step 4: Carefully measure and convert length
of vectors to quantity
Combination? Tip to tail method!
Conversion factor:
1 cm = 1 m
With the protractor and ruler,
measure measure a vector that is
5.5 cm long with a take-off angle
of 18 degrees at (0,0)
Horizontal velocity = 5.2 m/s
Vertical velocity = 1.7 m/s
5.5 cm
1.7 cm
18 deg
5.2 cm
Assume a person performs a long jump with a take-off
velocity of 5.5 m/s and a take-off angle of 18 degrees.
What are the horizontal and vertical velocities at take-off?
Trigonometric Resolution of Vectors
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Advantages:
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Does not require precise drawing
Time efficiency and accuracy
Trigonometry Terminology
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Trigonometry: Measure of triangles
Right triangle: A triangle that contains
an internal angle of 90 degrees (sum =
180 degrees)
Acute angle: An angle < 90 deg
Obtuse angle: An angle > 90 deg
Trigonometry Terminology
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Hypotenuse: The side of the triangle opposite
of the right angle (longest side)
Opposite leg: The side not connected to
angle in question
Adjacent leg: The side connected to angle in
question (but not hypotenuse)
H
O
Angle in Q
A
Trigonometry Functions
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Sine: Sine of an angle = O/H
Cosine: Cosine of an angle = A/H
Tangent: Tangent of an angle = O/A
Soh Cah Toa
Online Scientific Calculator 
http://www.creativearts.com/scientificcalculator
Trigonometric Resolution of
Vectors
Figure 10.11, Hamilton
Trigonometric Resolution of
Vectors
Pythagorean Theorum
Figure 10.12, Hamilton
Trigonometric Combination of
Vectors
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Step 1: Resolve all vertical and
horizontal components of all vectors
Step 2: Sum the vertical components
together for a new vertical component
Step 3: Sum the horizontal components
for a new horizontal component
Step 4: Generate new vector based on
new vertical and horizontal components
Figure 10.13, Hamilton
Figure 10.13, Hamilton
Trigonometric Combination of
Several Vectors
Figure 10.14, Hamilton