I) Newton's Laws and Angular Motion

A) With slight modification, Newton's laws of linear motion can  be applied to angular motion.

B) 1st Law: A body continues in a state of rest or uniform  rotation about its axis unless acted  upon by an external torque.
      1) Eccentric force.  A force which is applied off center. In other words, the direction of the force is not in line with the object’s center of gravity.  An eccentric force will result in rotation, provided the body is freely moving. External forces applied to the human body are typically eccentric.  Rotatory motion of a lever usually results when muscle pulls on bone, providing the external resistance is less than the amount of muscular force acting on the bone.
      2) When observing segmental motion of the human body, muscle force is considered an external force.  If you consider  the entire body undergoing general motion, muscle forces would be considered an internal force.

      3) Angular Inertia (I ; Moment of inertia) is the sum of all the masses (m)multiplied by the radius squared (r2).
            I =    (m)(r2)
If the mass is concentrated farther away from the axis  of rotation, the moment of inertia will  be greater, thus the system
(i.e., lever)  will be harder to start or stop.
The greater the moment of inertia, the more difficult it is for an external torque to change the state of rest or uniform motion of a rotating body.

     4) In regards to the human body, the mass distribution about an axis of rotation (i.e., joint) may be altered by changing the limb position (i.e., bringing the limb in closer to the axis of rotation by flexing at a joint).  As a human locomotes, angular inertia (moment of inertia) varies.  For example,  a jogger is able to recover the leg faster by tucking the foot close to the buttocks.  The jogger has concentrated the mass of the leg closer to the axis of rotation (hip joint) which decreases the moment of inertia and therfore increases the rate at which the leg is recovered.

 C) 2nd Law: The acceleration of a rotating body is directly  proportional to the torque causing it, is in the  same direction of the torque and is inversely proportional to the moment of inertia.
 

• Angular acceleration is the torque divided by  the moment of inertia.

• Angular acceleration is also the change in  angular velocity divided by time.

• Angular momentum is the force needed to start or stop rotational motion.
• Angular momentum is the product of angular  velocity and moment of inertia. The greater the angular momentum, the greater the force needed to stop the motion. Using a heavier bat will result in a greater angular momentum provided that angular velocity is maintained.  Also, increasing the angular velocity of a bat will increase theangular momentum.  Angular momentum of a limb is increased if the angular velocity is increased (i.e., kicking a ball).

Newton’s first law can be related to angular momentum.  Law of Conservation of Angular Momentum. The angular momentum associated with a  rotating body remains constant unless influenced by external torques.  Divers, dancers,  figure skaters make use of this law.  For example, a diver will change from a lay out position to a tucked position in order to increase angular rotation (angular velocity).  The tuck position results in a reduced moment of inertia. since angular momentum is conserved, angular velocity must increase.
 
 

D) 3rd Law:    When a torque is applied by one body to another, the  second body will  exert an equal and opposite torque  on the other body.

Body movements which serve to  regain balance are explained by Newton’s third law.  This is evident in gymnasts. If a gymnast lowers the left arm downward,  the right arm will react move downward (actually moving opposite the left arm) to maintain balance and therefore prevent falling from the balance beam.  Going from a tight tuck to a lay out position, the diver rotates the trunk back (extends the trunk). The reaction is for the lower extremities to rotate the opposite direction (extention at the hips).

 E) Transfer of momentum.

Angular momentum can be transferred from one body segment to the next.  Since body segments differ in mass, the moment of inertia of each body will vary.  Considering that momentum is conserved, a reduction in the moment of inertia of a body part will result in an increased angular velocity.  The latter can be applied to throwing and kicking movements. For example, throwing involves a series of angular rotations of progressively lighter body segments (leg/trunk--arm). A reduction in moment of inertia between the leg/trunk complex and the lighter arm, results in an increased velocity of the arm.

F) Centripetal and Centrifugal Forces.
Centripetal versus centrifugal force: Centripetal is an inward seeking force while centrifugal force is an outward pulling force.
The ground exerts centripetal force on a runner (via the foot) or cyclist (via the wheel) when they lean into a curve.  The amount of centripetal force necessary to prevent a runner from toppling outward (outward pulling force, centrifugal) is directly related to the mass and velocity of the runner.  The centrifugal force is producing an opposited pull to centripetal force.  In the case of both the runner and cyclist, the axis of rotation is either the foot or the bicycle wheel.

A hammer thrower exerts an inward directed force (centripetal) on the hammer via the wire. In addition, an outward-pulling  force (centrifugal) is exerted by the hammer on the thrower.
Centripetal force is also important in swinging moves in gymnastics, discuss throwing, or any throwing activities.  Once an implement (i.e., discuss, hammer) is released, its inertia will allow it to follow a linear path.

Remember, centripetal and centrifugal  force are exerted whenever a body moves on a curved path.  Because centripetal and centrifugal forces act opposite each other and possess the same  magnitude, the equation is the same for both.

                                     Forcecentripetal or centrifugal = (mass)(velocity)2/radius
                                         mass = mass of object
                                         velocity = velocity of object squared
                                         radius = distance of object’s center of gravity from axis of rotation

G)Torque

• Torque is the rotational equivalent of linear force.
• Torque is the product of the magnitude of force and the perpendicular distance to the axis of  rotation.
• Muscles of the body exert torque during segmental motion.
• Torques that are directed in a clockwise direction will be negative.
• Torques that are directed in a counterclockwise direction will be positive.

H)    Levers

• First Class: When the axis (fulcrum) of rotation is between the motive force(force which    instigates motion, also referred to as the effort) and resistive forces.
• Second Class: When the resistive force is between the axis of rotation and motive force.
• Third Class: When the motive force is between the axis of  rotation and resistive force.

I) Stabilizing and Rotary Components of Muscular Force.

• The stabilizing and rotary components are always  perpendicular to each other (See Diagram presented in lecture).
• Once a joint surpasses 90 degrees of flexion, the stabilizing component becomes a dislocating component.

Kinesiology. Worksheet on Rotatory Motion.
 

A. Levers. Indicate the type of lever, 1st class, 2nd class or 3rd class, for the following set of descriptions.

1.  The quadriceps complex acting at the knee joint in order to kick a soccer ball.

2.  The deltoid complex acting to abduct the upper extremity .

3.   Performing “latissimus” pulls while in a seated position.

4.   Leaning forward with a weight being supported on the shoulders, while executing a calf raise.
 

5.   The deltoid complex eccentrically acting to lower a heavy weight. Starting position-arm is abducted; Finishing position-arm is resting on trunk.

B. Application of Newton’s Laws of rotatory motion.
1. Why will a diver spin faster when they are in a tuck position versus a layout position?Explain.
 
 

2.  Describe the rotatory resistance (inertia) of a limb. What factors affect rotatory inertia of a limb?
 
 

3.  Why does the rotatory inertia of a sprinter’s leg decrease during the recovery phase?
 
 

4.   When a gymnast flexes at the hips, the trunk tends to rotate forward. Explain this occurrence.
 
 

5) If a hammer thrower exerts 750 N of force on the hammer, how much force will the hammer exert  on the thrower?
 
 

6) What happens to the hammer once it is released by the thrower ? (linear or rotational path)
                                  .   Explain your answer.
 
 
 

C) Muscular Torque
A. Calculate the torque for the biceps brachii given the following information: Show a diagram of the following infomation and all your calculations.
        1) It is exerting 125 lbs. of force.
        2) It inserts at the radial tuberosity at an angle of 90 degrees.
        3) The radial tuberosity is 2.0  inches from the elbow joint.
        4) In this particular example all of the muscular force is rotatory. Why?
 
 
 
 
 
 
 
 

B. What would happen to the muscular rotatory torque if:
1.  The angle of pull is decreased, while the distance of the radial tuberosity from the elbow remains the same, and the total force remains the same.

2. The radial tuberosity is 1.0 inch (vs. 2.0 inches) from the elbow joint, however angle of pull and total force remain the same.