Angular Kinetics of Human
Movement
How do we quantify resistance to linear
acceleration?
Mass
1. Angular analogues of:
–
–
–
–
Resistance to Acceleration
mass
momentum
impulse
reaction force
How do we quantify resistance to angular
acceleration?
Moment of Inertia (I)
2. Angular versions of Newton’s Laws
of Motion
What directly influences I?
I = mk2; so mass and the distribution of
mass, relative to axis of rotation, affect I
Determining Moment of Inertia (I)
Determining Moment of Inertia
I equals the sum of the products of: (1) mass
element of an object and (2) square of the
distance between the mass element and axis of
rotation
A more practical approach:
IAXIS = (mBODY)(k2)
IAXIS = Σmiri2
r1
m1
axis
k indicates the radius of gyration, which
is an experimentally determined length
that applies to the whole object at
once
k depends on the
location of the axis
and location of the
mass
m2
r2
IAXIS = m1r12 + m2r22 + m3r32 + .... + mnrn2
Whole-body I
Whole-body I
Different body segments have corresponding moments of
inertia for each plane of motion
Different body segments have corresponding moments of
inertia for each plane of motion
Similarly, the whole human body, rotating free of external
force, also has a moment of inertia relative to each of the
cardinal axes
Similarly, the whole human body, rotating free of external
force, also has a moment of inertia relative to each of the
cardinal axes
A
B
C
Inside Collection Textbook, by Erik Christensen,
http://cnx.org/content/m42182/latest/?collection=col11435/latest
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Applications:
Applications:
Why would someone choke up on a bat, use an aluminum
bat, or illegally cork a wooden bat? Is there a legal
alternative?
Applications:
Tuck vs. Layout for a diver or gymnast
layout
tuck
A runner's leg during swing phase
hip
Newton’s First Law:
A rotating body will maintain a
state of constant rotational
motion unless acted on by an
external torque
This is the basis for the
principle of conservation of
angular momentum (H).
What is Angular Momentum (H)?
Momentum:
• For linear motion:
• For angular motion:
Or:
Angular analogues for Newton’s
Laws of Motion
L = mv
H=Iω
H = (mk2) ω
• Factors that affect angular momentum (H):
– mass of the object (m)
– location of mass relative to axis or rotation (k)
– angular velocity of the object (ω
ω)
Units for angular momentum: kg⋅⋅m2
s
Conservation of
Angular Momentum (H)
Similar to L, H for any given system remains
constant in the absence of a net external
torque
However, unlike the linear circumstances we
discussed, I and ω can change!
So, we cannot assume that
either I or ω are conserved
throughout flight.
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Conserving
Angular Momentum
Example Problem: A 60-kg diver is in a layout
position (k = 0.5 m), immediately after leaving the
board, with an angular velocity of 4 rad/s. What is
the diver’s angular velocity when he assumes a
tuck position and reduces his radius of gyration to
0.25 m?
Conservation of
Angular Momentum
First, find H when diver leaves the board:
H = mk2ω
H = (60 kg)(0.5 m)2(4 rad/s) = 60 kg⋅⋅m2/s
k = 0.25 m
H is constant, so now find ω when k is reduced
to 0.25 m:
k = 0.5 m
60 kg⋅⋅m2/s = (60 kg)(0.25 m)2 ω
ω = 16 rad/s
Falling cats?
Transfer of Angular Momentum
H, about the long
axis of the cat, is
initially zero and
must be
conserved
throughout flight;
however, the cat
always rights
itself. How?
Although total body
angular momentum is
conserved while the
body is airborne (no
external torques),
angular momentum can
be transferred between
body segments.
Video Link
Transfer of Angular Momentum
Also, although total body angular momentum is
conserved while the body is airborne (no
external torques), one can also change the
total body axis
of rotation (e.g.,
a forward roll
can become a
twist).
Finally, although angular momentum can be
transferred between body segments or the
whole-body axis of rotation may change…
the whole-body center of
mass must still follow a
parabolic trajectory while
the body is airborne.
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Newton’s Laws of Motion:
Angular Analogues
Net joint torque and knee pain
Newton’s Second Law:
Torque causes angular acceleration, as well as
∆H, for a body that is directly proportional to
the magnitude of torque, in the same direction
as the torque, and inversely proportional to
the body’s moment of inertia
ΣT = I α (compare with ΣF = m a)
A Practice Problem
The Answer
Angular Impulse & Momentum
Practice Problems
How does a person change H?
external torque
Force, torque, and linear and angular impulse
– Linear Impulse = force × time = F · ∆ t
– Angular Impulse = torque × time = T · ∆ t
Impulse-momentum relationship
– Linear:
– Angular:
F · ∆ t = ∆M F · t = (mv)2 – (mv)1
T · ∆ t = ∆H T · t = (I ω)2 – (I ω)1
To initiate a twisting jump, Miki applies a
force to the ice that results in a torque
about her long axis. Consequently, H
(about her long axis) increases from 0 to
50 kg·m2/s in 0.25 s. During this time, I
(about her long axis) is 2.2 kg·m2.
1.How large was the associated angular
impulse (AI)? Answer: 50 Nm·s
2.How large was the associated torque?
Answer: 200 Nm.
3.How fast is Miki’s angular velocity
(about her long axis) at the end of the
0.25 s? Answer 22.7 rad/s or 3.16 rev/s
Miki Ando is one of a few female
skaters to have landed a quadruple
jump (a salchow) in competition. She
first completed the jump at the 2002
ISU Junior Grand Prix Final in the
Netherlands at age 15.
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Practice Problems
A bit on centripetal force…
In order to rotate four times in the air
during a single jump, Miki must increase
ω, about her long axis.
Centripetal force (FC) keeps an object moving along a
curved path (rotating). Also, centripetal force produces
the centripetal component of acceleration.
1. How can Miki manipulate I (about her
long axis)?
Fc = mac = m(v2 / r) = m ω2 r
2. Is it beneficial for Miki to manipulate I
(about her long axis)? If so, why?
3. What might the overall effect of an
increased or decreased I (about her
long axis) be on her final score?
Japanese figure skater Miki Ando is the
only female skater to have landed a
quadruple jump (a salchow) in
competition. She first completed the
jump at the 2002 ISU Junior Grand Prix
Final in the Netherlands at age 15.
Newton’s Laws of Motion:
Angular Analogues
Newton’s Laws of Motion:
Angular Analogues
Newton’s Third Law:
Newton’s Third Law:
For every torque exerted by one
body (or body segment) on
another body (or body segment),
there is an equal and opposite
torque that is exerted by the
second body (or body segment)
on the first body (or body
segment)
Within our context, this is
helpful to consider as we
think about body segments
apply torques to one
another…
Summary
• Angular inertia (I) depends on (1) mass and (2)
mass location, relative to the axis of rotation
Practice…
Answer: each configuration has ~70 units of angular momentum
• Angular momentum: H = I · ω
• In the absence of external torques, H is conserved
• Angular impulse is required to change H
• ΣT = Iα
• For every torque, there is an equal and opposite
torque
• Centripetal force is required for an object to rotate
• See both tables online…
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A 7.27 kg shot makes seven
complete revolutions during its 2.5
second flight. If its radius of
gyration is 2.54 cm, what is its
angular momentum?
a.
b.
c.
d.
A 7.27 kg shot makes seven
complete revolutions during its 2.5
second flight. Its radius of gyration
is 2.54 cm.
What would happen to ω, if the ball
had more m, while conserving H?
0.0825 kg·m2/s
7.38 kg·m2/s
46.16 kg·m2/s
None of the above
A 7.27 kg hammer on a 1 m wire is
released with a linear velocity of 28
m/s. What reaction force is exerted
on the thrower by the hammer at
the instant before release?
a.
b.
c.
d.
0.5 N
1000 N
1604 N
None of the above
Answer: It would decrease.
What would happen to H, if ω and
k were increased?
Answer: It would increase.
A 7.27 kg hammer on a 1 m wire is
rotated at a linear velocity of 28 m/s.
What force and corresponding
component of acceleration are
associated with this change in the
direction of linear velocity? Answer:
Centripetal force and centripetal, or
radial, acceleration.
What component of acceleration is
associated with a potential change in
the linear speed? Answer: Tangential
acceleration.
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