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**Angular Kinetics Explaining the Causes of Angular Motion**

Chapter 7 Angular Kinetics Explaining the Causes of Angular Motion

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Resistance to Motion Inertia is a body’s tendency to resist acceleration. A body’s inertia is directly proportional to its mass.

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Resistance to Motion According to Newton’s second law, the greater a body’s mass, the greater its resistance to linear acceleration. Therefore, mass is a body’s inertial characteristic for considerations relative to linear motion.

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**Resistance to Angular Motion**

Resistance to angular acceleration is also a function of a body’s mass. The greater the mass, the greater the resistance to angular acceleration.

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**Resistance to Angular Acceleration**

However, the relative ease or difficulty of initiating or halting angular motion depends on an additional factor - the distribution of mass with respect to the axis of rotation.

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**Resistance to angular acceleration**

The more closely mass is distributed to the axis of rotation, the easier it is to initiate or stop angular motion.

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Moment of inertia Inertial property for rotating bodies that increases with both mass and the distance the mass is distributed from the axis of rotation. Swing leg when running. Body when somersaulting.

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**Moment of inertia The moment of inertia is represented by I = m r2**

m is the particle’s mass r is the particle’s radius of rotation. Defined as distance to axis of rotation

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Moment of inertia From this equation, it can be seen that the distribution of mass with respect to the axis of rotation is more significant than the total amount of body mass in determining resistance to angular acceleration because r is squared. A batter would have a more difficult time swinging a longer bat than a heavier bat.

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Moment of inertia Changes in joint angles of the human body cause changes in the moments of inertia of body limbs.

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Moment of inertia The fact that bone, muscle, and fat have different densities and are distributed dissimilarly in individuals complicates efforts to calculate human body segment moments of inertia.

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Moment of inertia Because there are formulas for calculating the moment of inertia of regularly shaped solids, some investigators have modeled the human body as a composite of various geometric shapes.

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Radius of gyration The radius of gyration is a length measurement that represents how far from the axis of rotation all of the object’s mass must be concentrated to create the same resistance to change in the angular motion as the object had in its original shape.

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Radius of gyration It is the distance from the axis of rotation to a point which the mass of the body can theoretically be concentrated without altering the inertial characteristics of the rotating body.

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Radius of gyration This point is not the same as the segmental center of gravity. The radius of gyration is always longer than the radius of rotation, the distance to the segmental CG.

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Radius of gyration The length of the radius of gyration changes as the axis of rotation changes.

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Angular Momentum The quantity of motion that an object possesses is referred to as its momentum. Linear momentum is the product of the linear inertial property (mass) and linear velocity.

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Angular Momentum The quantity of angular motion that a body possesses is likewise know as angular momentum. Angular momentum is the product of the angular inertial property (moment of inertia) and angular velocity. H = I

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Angular Momentum Three factors affect the magnitude of a body’s angular momentum: its mass (m) the distribution of that mass with respect to the axis of rotation (k) and the angular velocity of the body ().

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Angular Momentum If a body has no angular velocity, it has no angular momentum. As mass or angular velocity increases, angular momentum increases proportionally.

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Angular Momentum The factor that most dramatically influences angular momentum is the distribution of mass with respect to the axis of rotation because angular momentum is proportional to the square of the radius of gyration. H = m k2

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Angular Momentum For a multi-segmented object such as the human body, angular momentum about a given axis of rotation is the sum of the angular momenta of the individual body segments.

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Angular Momentum Whenever gravity is the only acting external force, angular momentum is conserved. The total angular momentum of a given system remains constant in the absence of external torques.

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Angular Momentum Gravitational force acting at a body’s CG produces no torque because the perpendicular distance to the axis of rotation equals 0 and therefore creates no change in angular momentum.

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Angular Momentum The magnitude and direction of the angular momentum vector for an airborne performer are established at the instant of takeoff.

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Angular Momentum When angular momentum is conserved, changes in body configuration produce a tradeoff between moment of inertia and angular velocity.

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**Transfer of angular momentum**

Although angular momentum remains constant in the absence of external torques, transferring angular velocity at least partially from one principle axis to another is possible.

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**Transfer of angular momentum**

This occurs when a diver changes from a primarily somersaulting rotation to one that is primarily twisting and vice versa. An airborne performer’s angular velocity vector does not necessarily occur in the same direction as the angular momentum vector.

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**Transfer of angular momentum**

It is possible for a body’s somersaulting angular momentum and its twisting angular momentum to be altered in midair, though the vector sum of the two (the total angular momentum) remains constant in magnitude and direction.

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**Change in angular momentum**

Changes in angular momentum depend not only on the magnitude and direction of acting external torques, but also on the length of the time interval over which each torque acts.

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Angular Impulse Change in angular momentum equal to the product of torque and the time interval over which the torque acts.

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Angular Impulse When a support surface reaction force is directed through the performer’s center of gravity, linear but not angular impulse is generated.

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Newton’s First Law The angular version of the first law of motion may be stated as follows: A rotating body will maintain a state of constant angular motion unless acted upon by an external torque.

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Newton’s Second Law A net torque produces angular acceleration proportional to the magnitude of the torque, in the same direction as the torque, and inversely proportional to the body’s moment of inertia.

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Newton’s Third Law For every torque exerted by one body on another, there is an equal and opposite torque exerted by the second body on the first.

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Centripetal force Force directed toward the center of rotation of any rotating body.

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Centrifugal force Reaction force equal in magnitude and opposite in direction to centripetal force.

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