Biomedical Engineering Reference
In-Depth Information
If the rigid object restriction is relaxed a bit to allow the moment of inertia I
33
in
(
1.4
) to vary, then the conservation of angular momentum about an axis may be
written in this special case as
M
3
¼
ð
I
33
o
3
Þ=
d
t
:
d
(1.5)
This is the form of the conservation of angular momentum that is employed to
explain why a figure skater spinning at one place on the surface of the ice can
increase or decrease his or her angular velocity by extending their arms out from the
torso or lowering the arms to the sides of the torso. If the skater is spinning, there is
no moment about the axis that is the intersection of the sagittal and frontal or
coronal planes, thus M
3
¼
0 and, from (
1.5
) above, the product I
33
o
3
must be a
constant. Since I
33
o
3
¼
constant, when the skater extends (lowers) the arms, the
moment of inertia of the skater increases (decreases) and the angular velocity of the
spin must decrease (increase).
For stationary objects, or objects moving with constant velocity, the conserva-
tion of linear and angular momentum reduce to the conditions that the sum of the
forces and the sum of the moments must be 0. These conditions provide six
equations in the case of a three-dimensional problem and three equations in the
case of a two-dimensional problem. The application of these equations is the topic
of an engineering course on the topic of statics.
Problem
1.6.1. A diver rotates faster when her arms and legs are tucked tightly in so that she
is almost like a ball rather than when the limbs are extended in the common
diving posture like a straight bar. Consider a diver with a mass of 63 kg, an
extended length of 2 m and a tucked length of 1 m. (a) Determine the factor
by which her angular velocity in the tucked configuration exceeds her angular
velocity in the extended configuration. It is reasonable to approximate the
body in the two configurations as cylinders, and to assume that the centroid of
the cylinder coincides with the center of mass of the diver. In the extended
configuration the cylinder has a length of two meters and an average radius of
0.1 m while in the tucked configuration the cylinder will have a length of 1 m
and an average radius of 0.1414 m. Recall that the mass moment of inertia of
a cylinder about an axis perpendicular to its long axis and passing through its
mass center is M(3
r
2
h
2
)/12, where
r
is the radius of the cylinder and
h
is
the height of the cylinder. (b) What is the parameter that predominates in the
determination of this ratio?
þ
1.7 The Deformable Continuum Model
The deformable continuum model differs from the particle and rigid object models
in that relative movement or motion is permitted between two points in the model.
Examples of deformable continua include the elastic solid used, for example, in the
Search WWH ::
Custom Search