1. Introduction to Biomechanics and Vector Resolution
Applied Kinesiology
420:151

2. Agenda Introduction to biomechanics Units of measurement
Scalar and vector analysis
Combination and resolution
Graphic and trigonometric methods

3. Introduction to Biomechanics
The study of biological motion
Statics
Dynamics
The study of forces on the body in equilibrium
The study of forces on the body subject to unbalance
Kinetics and Kinematics
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

4. Agenda Introduction to biomechanics Units of measurement
Scalar and vector analysis
Combination and resolution
Graphic and trigonometric methods

5. Units of Measurement Systeme Internationale (SI) Base units
Derived units
Others

6. SI Base Units Length: SI unit  meter (m) Time: SI unit  second (s)
Mass: SI unit  kilogram (kg)
Distinction: Mass (kg) vs. weight (lbs.)
Mass: Quantity of matter
Weight: Effect of gravity on matter
Mass and weight on earth vs. moon?

7. SI Derived Units Displacement: A change in position
SI unit  m
Displacement vs. distance?
Velocity: The rate of displacement
SI unit  m/s
Velocity vs. speed?
Acceleration: The rate of change in velocity
SI unit  m/s/s or m/s2

8. SI Derived Units Force: The product of mass and acceleration
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
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

9. SI Derived Units Work: The product of force and distance
Deadlift Example
Work: The product of force and distance
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
SI Unit  J
Note: 1 J = ~ 4 kcal
Power: The rate of doing work (work/time)
SI Unit  Watt (W)  When 1 J (or N*m) is performed in 1 s
Note: Also calculated as F*V

10. Other Units
Area: The superficial contents or surface within any given lines
2D in nature
SI Unit  m2
Volume: The amount of space occupied by a 3D structure
SI Unit  m3 or liter (l)
Note: 1 l = 1 m3

11. Agenda Introduction to biomechanics Units of measurement
Scalar and vector analysis
Combination and resolution
Graphic and trigonometric methods

12. Scalar and Vector Analysis
Scalar defined: Single quantities of magnitude  no description of direction
A speed of 10 m/s
A mass of 10 kg
A distance of 10 m
Vector defined: Double quantities of magnitude and direction
A velocity of 10 m/s in forward direction
A vertical force of 10 N
A displacement of 10 m in easterly direction

13. Scalar and Vector Representation
Scalars are represented as values that represent the magnitude of the quantity
Vectors are represented as arrows that represent:
The direction of the vector quantity (where is the arrow pointing?)
The magnitude of the vector (how long is the arrow?)

14. Figure 10.1, Hamilton

15. Combination of Vectors
Vectors can be combined which results in a new vector called the resultant.
We can combine vectors three ways:
Addition
Subtraction
Multiplication

16. Vector Combination: Addition
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

17. Vector Combination: Subtraction
Tip to tail method
Resultant = Vector 1 – Vector 2 or . . .
Resultant = Vector 1 + (- Vector 2)
Flip direction of negative vector + =
Vector 1
Vector 2
Resultant

18. Vector Combination: Multiplication
Tip to tail method
Only affects magnitude
Same as adding vectors with same direction
X 3 =

19. Vector Resolution
Resolution: The breakdown of vectors into two sub-vectors acting at right angles to one another

20. Resultant velocity of shot at take off is a function of the horizontal velocity (B) and the vertical velocity (A)

21. Location of Vectors in Space
Frame of reference:
Reality = 3D  2D for simplicity
Two types:
Rectangular coordinate system
Polar coordinate system

22. Rectangular Coordinate System
(-,+)
(+,+) X
(-,-)
(+,-)
The vector starts at (0,0) and ends at (x,y)
Example: Vector (4,3)

23. Polar Coordinate System
Figure 10.5, Hamilton
Coordinates are (r,q) where r = length of resultant and q= angle

24. Figure 10.6, Hamilton

25. Graphic Resolution of Vectors
Tools: Graph paper, pencil, protractor
Step 1: Select a linear conversion factor
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

26. Combination? Tip to tail method!
Conversion factor:
1 cm = 1 m
Combination? Tip to tail method!
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?

27. Trigonometric Resolution of Vectors
Advantages:
Does not require precise drawing
Time efficiency and accuracy

28. Trigonometry Terminology
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

29. Trigonometry Terminology
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

30. Trigonometry Functions
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 

31. Trigonometric Resolution of Vectors
Figure 10.11, Hamilton

32. Trigonometric Resolution of Vectors
Pythagorean Theorum
Figure 10.12, Hamilton

33. Trigonometric Combination of Vectors
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

34. Figure 10.13, Hamilton

35. Figure 10.13, Hamilton

36. Trigonometric Combination of Several Vectors
Figure 10.14, Hamilton