Transcript Chapter 14 PPT lecture outline

Chapter 14
Angular Kinetics of
Human Movement
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
McGraw-Hill/Irwin
© 2014 The McGraw-Hill Companies, Inc. All rights reserved.
Resistance to Angular Acceleration
What is moment of inertia?
• The inertial property for rotating bodies
represents resistance to angular
acceleration based on both mass
and the distance the mass is
distributed from the axis of rotation
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-2
Resistance to Angular Acceleration
Axis of rotation
m
r
r
m
m
r
r
m
Moment of inertia is the sum of the
products of each particle’s mass (m)
and the radius of rotation (r) for that
particle squared. I = mr2
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-3
Resistance to Angular Acceleration
A
B
Although both bats have the same mass, bat A is
harder to swing than bat B because the weight ring
on it is positioned farther from the axis of rotation.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-4
Resistance to Angular Acceleration
What is the radius of gyration?
• Distance from the axis of rotation to a
point where the body’s mass could
be concentrated without altering its
rotational characteristics
• Used as the index for mass distribution
for calculating moment of inertia:
I = mk2
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-5
Resistance to Angular Acceleration
k1
k2
k3
k1
k3
k2
Knee angle affects the moment of inertia of the
swinging leg with respect to the hip because of
changes in the radius of gyration for the lower leg
(k2) and foot (k3).
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-6
Resistance to Angular Acceleration
During sprinting, extreme flexion at the knee
reduces the moment of inertia of the swinging leg.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-7
Resistance to Angular Acceleration
The ratio of muscular strength (ability to produce torque at a
joint) to segmental moments of inertia (resistance to rotation
at a joint) is important for performance in gymnastic events.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-8
Resistance to Angular Acceleration
Principal moments of inertia of the human body in different
positions with respect to different principal axes: (1) principal
axis; (2) moment of inertia (kg m 2 ).
.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-9
Angular Momentum
What is angular momentum?
• Quantity of angular motion possessed
by a body
• Measured as the product of moment of
inertia and angular velocity:
H = I
H = mk2
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-10
Angular Momentum
CG
CGs


s
g
Angular momentum is the sum of the local
term (Iss) and the remote term (mr2g).
H = Iss + mr2g
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-11
Angular Momentum
What is the principle of conservation of
angular momentum?
The total angular momentum of a given
system remains constant in the
absence of external torques.
H 1 = H2
(mk2)1 = (mk2)2
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-12
Angular Momentum
H = I
When angular momentum is conserved, there is a
tradeoff between moment of inertia and angular
velocity.
(Tuck position = small I, large )
(Extended position = large I, small )
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-13
Angular Momentum
During the airborne
execution of a spike in
volleyball, a
compensatory rotation of
the lower extremity
offsets the forceful
swinging arm so that
total body angular
momentum is conserved.
H = I
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-14
Angular Momentum
A skillful diver can rotate
180º or more in the air
with zero angular
momentum because
there is a large
discrepancy between the
radii of gyration for the
upper and lower
extremities with respect
to the longitudinal axes
of these two major body
segments.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
Axis 1
Axis 2
14-15
Angular Momentum
What produces change in angular
momentum?
Angular impulse - the product of torque
and the time interval over which the
torque acts:
T t = H
T t = (I)2 - (I)1
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-16
Angular Momentum
Backward
somersault
F
CG
d
Springboard reaction force (F) multiplied by its moment arm
from the diver’s CG (d ) creates a torque that generates the
angular impulse that produces angular momentum at takeoff.
Tt = H
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-17
Angular Momentum
The arm swing during takeoff contributes significantly to the
diver’s angular momentum.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-18
Angular Momentum
The surface reaction force is used by the dancer to generate
angular momentum during the takeoff in the tour jeté.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-19
Angular Analogues of Linear Kinematic
Quantities
What are the angular equivalents of linear
kinematic quantities?
Linear
Mass (m)
Force (F)
Momentum (M=mv)
Impulse (Ft)
Angular
Moment of inertia (I = mk2)
Torque (T = Fd )
Angular momentum (H=mk2)
Angular impulse (Fd t)
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-20
Angular Analogues of Newton’s Laws
What is the angular law of inertia?
A rotating body will maintain a state of
rest or constant rotational motion
unless acted on by an external
torque that changes the state.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-21
Angular Analogues of Newton’s Laws
What is the angular law of acceleration?
A net torque causes angular acceleration
of a body that is:
• of a magnitude proportional to the
torque
• in the direction of the torque
• and inversely proportional to the
body’s moment of inertia
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-22
Angular Analogues of Newton’s Laws
What is the angular law of acceleration?
T = I
T = mk2
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-23
Angular Analogues of Newton’s Laws
What is the angular law of reaction?
• For every angular action, there is an
equal and opposite angular reaction.
• When one body exerts a torque on a
second, the second body exerts a
reaction torque that is equal in
magnitude and opposite in direction
on the first body.
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-24
Centripetal Force
What is centripetal force?
(Force directed toward
the center of rotation
for a body in rotational
motion)
mv2
Fc = r
Fc
Fc = mr2
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-25
Centripetal Force
Cyclists and runners lean into a curve to offset the
torque created by centripetal force acting on the
base of support (tires).
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
14-26
Centripetal Force
Free body diagram of
a cyclist on a curve. R
is centripetal force.
When the cyclist is
balanced, summing
torques at the cyclist’s
CG,
RH
RV
(RV)(dR ) = (RH)(dR ).
V
Basic Biomechanics, 7th edition
By Susan J. Hall, Ph.D.
H
14-27