Download Chapter 14 - Illinois State University

Chapter 7
Angular Kinetics
Explaining the Causes of
Angular Motion
Illinois State University
Resistance to Motion

Inertia is a body’s tendency to resist
acceleration.

A body’s inertia is directly proportional
to its mass.
Illinois State University
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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
Illinois State University
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.
Illinois State University
Moment of inertia

Changes in joint angles of the human
body cause changes in the moments of
inertia of body limbs.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
Radius of gyration

The length of the radius of gyration
changes as the axis of rotation changes.
Illinois State University
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.
Illinois State University
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
Illinois State University
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 ().
Illinois State University
Angular Momentum

If a body has no angular velocity, it has
no angular momentum.

As mass or angular velocity increases,
angular momentum increases
proportionally.
Illinois State University
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 
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
Angular Momentum

The magnitude and direction of the
angular momentum vector for an
airborne performer are established at
the instant of takeoff.
Illinois State University
Angular Momentum

When angular momentum is conserved,
changes in body configuration produce
a tradeoff between moment of inertia
and angular velocity.
Illinois State University
Illinois State University
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.
Illinois State University
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.

Illinois State University
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.
Illinois State University
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.
Illinois State University
Angular Impulse

Change in angular momentum equal to
the product of torque and the time
interval over which the torque acts.
Illinois State University
Angular Impulse

When a support surface reaction force
is directed through the performer’s
center of gravity, linear but not angular
impulse is generated.
Illinois State University
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.
Illinois State University
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.
Illinois State University
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.
Illinois State University
Centripetal force

Force directed toward the center of
rotation of any rotating body.
Illinois State University
Centrifugal force

Reaction force equal in magnitude and
opposite in direction to centripetal
force.
Illinois State University