Center of Gravity
Pages 129 - 143
The earth pulls down on each particle of
an object with a gravitational force that
we call weight. Although individual
particles throughout an object all
contribute to weight in this way, the net
effect is as if the total weight of the
object were concentrated in a single
point - the object's center of gravity.
4/15/2015
Dr. Sasho MacKenzie HK 376
1
Why is Jordan reaching down with
his left hand?
4/15/2015
Dr. Sasho MacKenzie HK 376
2
Working Definitions
1. Balance point of the body.
2. Point about which the body rotates while
free in the air.
3. Point of the body where we can consider
all of the weight of the body to act.
Near the surface of the earth, the force of gravity
is the same on all parts of the body. This means
that the center of gravity and center of mass of
an object are the same location.
4/15/2015
Dr. Sasho MacKenzie HK 376
3
CM: Simple Example in 2D (7-iron)
 There are only two parts
1. Shaft + Grip
2. Clubhead
 Butt of the grip is the origin (0 cm, 0 cm)
Y
X
Dr. Sasho MacKenzie
Where is the CM of the 7-iron?
Y
X
Grip + Shaft
Xshaft = 33 cm
Yshaft = 0 cm
Mshaft = 156 g
XCM =
Xshaft*Mshaft + Xhead*Mhead
Mshaft + Mhead
Dr. Sasho MacKenzie
Clubhead
Xhead = 94 cm
Yhead = 4 cm
Mhead = 260 g
YCM =
Yshaft*Mshaft + Yhead*Mhead
Mshaft + Mhead
Where is the CM of the 7-iron?
Y
X
Grip + Shaft
Xshaft = 33 cm
Yshaft = 0 cm
Mshaft = 156 g
71.125 cm =
33*156 + 94*260
156 + 260
Dr. Sasho MacKenzie
Clubhead
Xhead = 94 cm
Yhead = 4 cm
Mhead = 260 g
2.5 cm =
0*156 + 4*260
156 + 260
Formal/Mathematical Definition
N
CG   mi gri  0
i 1
•
•
•
•
•
i represents a particular particle in the body
N represents the number of particles
m is the mass of the particle
g = 9.81 m/s2
r is the moment arm distance from the CG to the
particle
N
CM   mi ri  0
i 1
4/15/2015
The equation for center of mass
is conceptually the same. The
difference is the lack of g.
Dr. Sasho MacKenzie HK 376
7
Example: A uniform stick
Center of gravity
There are numerous particles in the stick that each
have a mass. Gravity acts on all of these masses
producing forces which act at a distance from the
center of gravity. Because these forces act at a
distance, they produce a torque. The point where all
these torques balance out is the center of gravity.
4/15/2015
Dr. Sasho MacKenzie HK 376
8
Center of Mass and Torque
• A common working definition for the CG is:
– The balance point of an object.
• The point where we can place a single finger
and keep the object from rotating.
• A net torque produces rotation
• Therefore, the CG is the point about which
the gravitational forces produce no net
torque.
Center of Mass and Torque
 Torque = Force x Moment Arm

Moment Arm: Perpendicular distance from the force to
the point of rotation
The CoG is not the
156 g
point that equally
divides the weight on
either side of an
object.
Dr. Sasho MacKenzie
38.125 cm
Torque:
5947.5
22.875
cm
Torque:
5947.5
260 g
The center of gravity is not the point that equally
divides the weight on either side of a body or object.
Consider the extreme example of a sledge hammer?
C of G
There is much more weight to the right, but the moment
arm is also much smaller on the right, therefore the
torques will balance out.
This is why you can safely remove weights from the bar on
a squat rack.
4/15/2015
Dr. Sasho MacKenzie HK 376
11
cg
1
cg
2
Center of gravity moves
in the direction the arm
moved; up and to the
man’s left.
4/15/2015
cg
cg
1
2
Center of gravity moves
in the opposite direction
of the removed leg; up
and to the man’s left.
Dr. Sasho MacKenzie HK 376
12
CG HEIGHT
1
2
3
GROUND
4/15/2015
Dr. Sasho MacKenzie HK 376
13
The previous slide shows three jumps with identical take-off
velocities. That means that the CofG of the jumps will all
reach the same height off the floor. However, by altering
body position at the top of the jump, the CofG changes it’s
position relative to the body, and reach height is also altered.
The following slide shows that while in the air, the CofG has a
set parabolic path determined by the take-off velocity. The
athlete can do nothing to change the position of his CofG
while in the air. Any muscular activity will only result in
relative movement about the CofG.
The illusion of hang time can be achieved by raising the limbs
on the way up to peak height, and then dropping the limbs after
peak vertical height is achieved.
4/15/2015
Dr. Sasho MacKenzie HK 376
14
cg
cg
cg
2
1
3
An individual watching this jump would usually focus on the
position of the dancer’s head. Even though the CofG has dropped a
considerable amount from 2 to 3, the head has remained at the same
vertical height, thus providing the illusion of “hang time”.
This is also the case in a well executed jump shot or dunk.
4/15/2015
Dr. Sasho MacKenzie HK 376
15
4/15/2015
Dr. Sasho MacKenzie HK 376
16
4/15/2015
Dr. Sasho MacKenzie HK 376
17
4/15/2015
Dr. Sasho MacKenzie HK 376
18
4/15/2015
Dr. Sasho MacKenzie HK 376
19
4/15/2015
Dr. Sasho MacKenzie HK 376
20
Balance
If your CofG lies within your base of support, you have the
potential for balance (equilibrium)
Base of Support
Equilibrium
Fx1
mg
mg
Fy1
4/15/2015
Fy2
Fy2
Manypoint = 0
The CofG has
passed outside the
base of the support
and equilibrium
has been lost.
Dr. Sasho MacKenzie HK 376
21
Headstand Example
Beginners move their head out from their hands increasing their
base of support, and thus their stability. However, this moves their
CofG out from their hands also. When a handstand is attempted
their base of support shrinks once the head leaves the floor. This
now places their CofG outside of the new base of support and
falling is likely.
CofG
CofG
head
Factors Influencing Stability
• Weight of the body
• Size of the base of support
• Height of the CofG of the base of support
4/15/2015
Dr. Sasho MacKenzie HK 376
22