Lecture outline
Centre of Gravity
• definition of centre of gravity (COG)
• calculating COG of multisegment bodies
• measuring COG with the Reaction Board
• COG considerations in vertical jumping
• COG considerations in balance
Ozkaya and Nordin
Chapter 4, p. 73-77
Stephen Robinovitch, Ph.D.
KIN 201
2007-1
Lecture 5: COG
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More on COG of Limb Segments
Centre of Gravity of Limb Segments
• we have assumed that the gravitational force
acting on a body can be represented by a
single force W, applied at the centre of
gravity (COG) of the body
• W represents the sum effect of a large
number of small gravitational forces
distributed over the body (one acting on each
particle forming the body)
2
xCG
W
• the COG represents a balance point where
application of a single force (-W) prevents
the body from translating and rotating under
the action of gravity
• if gravity can be considered uniform, the terms
centre of gravity (COG) and centre of mass
(COM) refer to the same thing
• COG data for individuals body segments have
been determined from cadaver measures,
mathematical modelling, and imaging studies
(radiosotope scanning); this general area of
biomechanics is termed anthropometry (study
of body size)
• regression equations exist for predicting
segment COG values from height and weight
• COG can be located outside the material of
the body (e.g., consider a hoop)
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• values for the 50th percentile male will be
posted to the website
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Calculation of COG
Measuring the COG with the Reaction Board
(Kin 142 lab)
The mathematical definition of COG is:
n
n
! mi xi
x CG =
i=1
Mtotal
n
! mi yi
yCG =
! mi zi
i=1
zCG =
Mtotal
R1
i=1
Mtotal
mass of board (kg) = mb
length of board (m) = 2
scale reading (N) = R1
+ !MA= 0:
-2R1 + bmbg = 0
(equation 1)
x
A
mbg
R1
2m
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mass of human (kg) = mh
scale reading (N) = R2
centre of gravity of human (m) = xCG
+ !MA= 0:
R2
-2R2 + bmbg + xCGmhg = 0
b
(equation 2)
x
R2
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-2R2 + bmbg + xCGmhg = 0 (equation 1)
-2R1 + bmbg = 0
(equation 2)
eliminate the bmbg term:
2(R2 - R1) = xCGmhg
We’ve measured
2( R2 ! R1 )
xCG =
xCG, the location of
mh g
the centre of mass
of the whole body!!
x
A
R2
mhg
mbg
2m
b
mhg
R2
xCG
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mbg
2m
b
A
xCG
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Example: calculating the COG of mulisegment bodies.
The mass and (x,y) coordinates of the hand, forearm, and
arm segments are located as shown in the figure. Using
the equations shown below, calculate the x and y
coordinates of the COG of the entire upper extremity.
Whole-body COG and
vertical jumping
!
n
! mi xi
Y
0.4 kg
(0.2, 0.5)
xCG = i=1
Mtotal
1.2 kg
(0.3, 0.4)
n
2.1 kg
(0.4, 0.2)
0
!
! mi yi
yCG = i=1
Mtotal
case (a) shows how raising
only one hand (versus both
hands) overhead results in a
“higher” jump
case (b) shows how the
whole-body COG may
never pass over the bar
during pole vaulting!
(a)
(b)
X
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Whole-body COG and balance
a state of balance exists only
when the whole-body COG is
located within the base of
support provided (typically)
by the feet
! base of support = area where
the centre of pressure (COP)
can be located
! static equilibrium truly exists
only when the COG is exactly
in line with the COP
! this rarely exists; even quiet
stance involves continuous
movement of the COG and
COP, or “sway”
Review Questions
!
what is the physical meaning of the COG?
! how can the COG of a multisegmented system
be calculated?
! how can a reaction board be used to measure
whole-body COG?
! can a pole-vaulter clear a jump without their
whole-body COG passing over the bar?
! what is the “base of support”?
!
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