5/19/2015Dr. Sasho MacKenzie HK 3761 Center of Gravity 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. Pages

5/19/2015Dr. Sasho MacKenzie HK 3762 Why is Jordan reaching down with his left hand?

5/19/2015Dr. Sasho MacKenzie HK 3763 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.

Dr. Sasho MacKenzie CM: Simple Example in 2D (7-iron)  There are only two parts X Y 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? X Y Grip + Shaft X shaft = 33 cm Y shaft = 0 cm M shaft = 156 g Clubhead X head = 94 cm Y head = 4 cm M head = 260 g X shaft *M shaft + X head *M head M shaft + M head X CM = Y shaft *M shaft + Y head *M head M shaft + M head Y CM =

Dr. Sasho MacKenzie Where is the CM of the 7-iron? X Y Grip + Shaft X shaft = 33 cm Y shaft = 0 cm M shaft = 156 g Clubhead X head = 94 cm Y head = 4 cm M head = 260 g 33* * cm = 0* * cm =

5/19/2015Dr. Sasho MacKenzie HK 3767 Formal/Mathematical Definition i represents a particular particle in the bodyi represents a particular particle in the body N represents the number of particlesN represents the number of particles m is the mass of the particlem is the mass of the particle g = 9.81 m/s 2g = 9.81 m/s 2 r is the moment arm distance from the CG to the particler is the moment arm distance from the CG to the particle The equation for center of mass is conceptually the same. The difference is the lack of g.

5/19/2015Dr. Sasho MacKenzie HK 3768 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.

Center 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 point where we can place a single finger and keep the object from rotating.The point where we can place a single finger and keep the object from rotating. A net torque produces rotationA net torque produces rotation Therefore, the CG is the point about which the gravitational forces produce no net torque.Therefore, the CG is the point about which the gravitational forces produce no net 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 cm cm 156 g 260 g Torque: Torque: The CoG is not the point that equally divides the weight on either side of an object.

5/19/2015Dr. Sasho MacKenzie HK 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.

5/19/2015Dr. Sasho MacKenzie HK Center of gravity moves in the direction the arm moved; up and to the man’s left. cg Center of gravity moves in the opposite direction of the removed leg; up and to the man’s left. 2

5/19/2015Dr. Sasho MacKenzie HK GROUND CG HEIGHT

5/19/2015Dr. Sasho MacKenzie HK 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.

5/19/2015Dr. Sasho MacKenzie HK 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”. cg 1 cg 3 cg 2 This is also the case in a well executed jump shot or dunk.

5/19/2015Dr. Sasho MacKenzie HK 37616

5/19/2015Dr. Sasho MacKenzie HK 37617

5/19/2015Dr. Sasho MacKenzie HK 37618

5/19/2015Dr. Sasho MacKenzie HK 37619

5/19/2015Dr. Sasho MacKenzie HK 37620

5/19/2015Dr. Sasho MacKenzie HK 37621Balance If your CofG lies within your base of support, you have the potential for balance (equilibrium) Base of Support mg Fy1 Fy2 mg Fy2 Fx1  M anypoint = 0 Equilibrium The CofG has passed outside the base of the support and equilibrium has been lost.

5/19/2015Dr. Sasho MacKenzie HK Factors 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 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. head CofGCofG