5/19/2015dr. sasho mackenzie hk 3761 center of gravity the earth pulls down on each particle of an...

04/18/23 Dr. Sasho MacKenzie HK 376 1

Center of GravityCenter of Gravity

The earth pulls down on each particle of The earth pulls down on each particle of an object with a gravitational force that an object with a gravitational force that we call weight. Although individual we call weight. Although individual particles throughout an object all particles throughout an object all contribute to weight in this way, the net contribute to weight in this way, the net effect is as if the total weight of the effect is as if the total weight of the object were concentrated in a single object were concentrated in a single point - the object's center of gravity. point - the object's center of gravity.

Pages 129 - 143Pages 129 - 143


Why is Jordan reaching down with Why is Jordan reaching down with his left hand?his left hand?


Working DefinitionsWorking Definitions

1.1. Balance point of the body.Balance point of the body.

2.2. Point about which the body rotates while Point about which the body rotates while free in the air.free in the air.

3.3. Point of the body where we can consider Point of the body where we can consider all of the weight of the body to act.all of the weight of the body to act.

Near the surface of the earth, the force of Near the surface of the earth, the force of gravity is the same on all parts of the body. gravity is the same on all parts of the body. This means that the center of gravity and center This means that the center of gravity and center of mass of an object are the same location.of mass of an object are the same location.

Dr. Sasho MacKenzie

CM: Simple Example in 2D (7-iron) There are only two parts

XY

1. Shaft + Grip

2. Clubhead

Butt of the grip is the origin (0 cm, 0 cm)

Dr. Sasho MacKenzie

Where is the CM of the 7-iron?

XY

Grip + ShaftXshaft = 33 cmYshaft = 0 cmMshaft = 156 g

ClubheadXhead = 94 cmYhead = 4 cmMhead = 260 g

Xshaft*Mshaft + Xhead*Mhead

Mshaft + Mhead

XCM =Yshaft*Mshaft + Yhead*Mhead

Mshaft + Mhead

YCM =

Dr. Sasho MacKenzie

Where is the CM of the 7-iron?

XY

Grip + ShaftXshaft = 33 cmYshaft = 0 cmMshaft = 156 g

ClubheadXhead = 94 cmYhead = 4 cmMhead = 260 g

33*156 + 94*260

156 + 26071.125 cm =

0*156 + 4*260156 + 260

2.5 cm =


Formal/Mathematical Formal/Mathematical DefinitionDefinition

• i represents a particular particle in the bodyi represents a particular particle in the body• N represents the number of particles N represents the number of particles • m is the mass of the particlem is the mass of the particle• g = 9.81 m/sg = 9.81 m/s22

• r is the moment arm distance from the CG to the r is the moment arm distance from the CG to the particleparticle

01

N

iiigrmCG

01

iirmCMN

i

The equation for center of mass The equation for center of mass is conceptually the same. The is conceptually the same. The difference is the lack of g.difference is the lack of g.


Example: A uniform stickExample: A uniform stick

Center of gravity

There are numerous particles in the stick that each There are numerous particles in the stick that each have a mass. Gravity acts on all of these masses have a mass. Gravity acts on all of these masses producing forces which act at a distance from the producing forces which act at a distance from the center of gravity. Because these forces act at a center of gravity. Because these forces act at a distance, they produce a torque. The point where all distance, they produce a torque. The point where all these torques balance out is the center of gravity.these torques balance out is the center of gravity.

Center of Mass and TorqueCenter of Mass and Torque• A common working definition for the CG is:A common working definition for the CG is:

– The balance point of an object.The balance point of an object.

• The point where we can place a single finger The point where we can place a single finger and keep the object from rotating.and keep the object from rotating.

• A net torque produces rotationA net torque produces rotation

• Therefore, the CG is Therefore, the CG is the point the point about which about which the gravitational forces produce no net the gravitational forces produce no net torque.torque.

Dr. Sasho MacKenzie

Center of Mass and Torque Torque = Force x Moment Arm

Moment Arm: Perpendicular distance from the force to the point of rotation

22.875 cm

38.125 cm

156 g

260 g

Torque:5947.5

Torque:5947.5

The CoG is not the point that equally divides the weight on either side of an object.


The center of gravity is The center of gravity is notnot the point that equally the point that equally divides the weight on either side of a body or object. divides the weight on either side of a body or object. Consider the extreme example of a sledge hammer?Consider the extreme example of a sledge hammer?

C of G

There is much more weight to the right, but the moment There is much more weight to the right, but the moment arm is also much smaller on the right, therefore the arm is also much smaller on the right, therefore the torques will balance out.torques will balance out.

This is why you can safely remove weights from the bar on This is why you can safely remove weights from the bar on a squat rack.a squat rack.


Center of gravity moves Center of gravity moves in the direction the arm in the direction the arm moved; up and to the moved; up and to the man’s left.man’s left.

cg

1

cg cg

1 2

cg

Center of gravity moves Center of gravity moves in the opposite direction in the opposite direction of the removed leg; up of the removed leg; up and to the man’s left.and to the man’s left.

2


22 3311

GROUNDGROUND

CG

HE

IGH

TC

G H

EIG

HT


The previous slide shows three jumps with identical take-off The previous slide shows three jumps with identical take-off velocities. That means that the CofG of the jumps will all velocities. That means that the CofG of the jumps will all reach the same height off the floor. However, by altering reach the same height off the floor. However, by altering body position at the top of the jump, the CofG changes it’s body position at the top of the jump, the CofG changes it’s position relative to the body, and reach height is also altered.position relative to the body, and reach height is also altered.

The following slide shows that while in the air, the CofG has a The following slide shows that while in the air, the CofG has a set parabolic path determined by the take-off velocity. set parabolic path determined by the take-off velocity. The The athlete can do nothing to change the position of his CofG athlete can do nothing to change the position of his CofG while in the air.while in the air. Any muscular activity will only result in Any muscular activity will only result in relative movement about the CofG.relative movement about the CofG.

The illusion of hang time can be achieved by raising the limbs 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 on the way up to peak height, and then dropping the limbs after peak vertical height is achieved.peak vertical height is achieved.


An individual watching this jump would usually focus on the An individual watching this jump would usually focus on the position of the dancer’s head. Even though the CofG has dropped a 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 considerable amount from 2 to 3, the head has remained at the same vertical height, thus providing the illusion of “hang time”.vertical height, thus providing the illusion of “hang time”.

cgcg

11

cgcg

33

cgcg

22

This is also the case in a well executed jump shot or dunk.This is also the case in a well executed jump shot or dunk.


BalanceBalanceIf your CofG lies within your base of support, you have the If your CofG lies within your base of support, you have the potential for balance (equilibrium)potential for balance (equilibrium)

Base of SupportBase of Support

mgmg

Fy1Fy1 Fy2Fy2

mgmg

Fy2Fy2

Fx1Fx1

MManypointanypoint = 0 = 0

EquilibriumEquilibrium

The CofG has The CofG has passed outside the passed outside the base of the support base of the support and equilibrium and equilibrium has been lost.has been lost.


Factors Influencing StabilityFactors Influencing Stability• Weight of the bodyWeight of the body• Size of the base of supportSize of the base of support• Height of the CofG of the base of supportHeight of the CofG of the base of support

Headstand ExampleHeadstand ExampleBeginners move their head out from their hands increasing their Beginners move their head out from their hands increasing their base of support, and thus their stability. However, this moves their base of support, and thus their stability. However, this moves their CofG out from their hands also. When a handstand is attempted CofG out from their hands also. When a handstand is attempted their base of support shrinks once the head leaves the floor. This their base of support shrinks once the head leaves the floor. This now places their CofG outside of the new base of support and now places their CofG outside of the new base of support and falling is likely.falling is likely.

headhead

CofG CofGCofG

5/19/2015dr. sasho mackenzie hk 3761 center of gravity the earth pulls down on each particle of an...

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