Biomedical Engineering Reference
In-Depth Information
must cope are generally those due to gravity, that is to say the forces that manifest
themselves in the weight of objects. It is quite a different story for the cell
because the largest force system they experience is due to adhesion. The adhesive
forces on a cell are three to four orders of magnitude larger than the gravitational
forces on a cell. More importantly, cells control their adhesive forces; man does not
control gravity.
Problem
1.5.1. A ski jumper leaves the ski jump with a velocity
v
o
in a direction that is an
angle
a
above the horizon (see Fig.
1.4
). The final point on the ski jump is an
elevation
h
above the valley floor. If the drag of the wind is neglected, show
that the horizontal and vertical velocities,
v
x
and
v
y
, respectively, of the skier
as he reaches the flat valley floor are given by
v
x
¼
v
o
cos
a
and
v
y
¼
2
gh
)
1/2
. Find the time,
t
touch
, at which the skier touches valley
floor as a function of
v
o
,
)
2
((
v
o
sin
a
þ
a
,
h
and
g
, the acceleration of gravity.
1.6 The Rigid Object Model
The rigid object model differs from the particle model in that the rotational motion
as well as the translational motion of the object is considered. Deformations are
neglected, hence the adjective “rigid” modifying object. Thus, not only Newton's
second law of motion is involved, but also Euler's equations (after their creator, the
Swiss mathematician/engineer/physicist L
´
onard Euler, 1707-1783) for the rota-
tional motion. Euler's equations are special forms of the conservation of angular
momentum expressed in a reference coordinate system at the mass center of the
rigid object (or at a fixed point of rotation of the object), fixed to the rigid object,
and coincident with the principal axes of inertia. If I
11
,I
22
, and I
33
represent the
principal moments of inertia (see Appendix section A.8), and M
1
,M
2
, and M
3
represent the sums of the moments about the three axes, then Euler's equations may
be written in the form
M
1
¼
I
11
ð
d
o
1
=
d
t
Þþo
2
o
3
ð
I
33
I
22
Þ;
M
2
¼
I
22
ð
d
o
2
=
d
t
Þþo
3
o
1
ð
I
11
I
33
Þ;
(1.3)
M
3
¼
I
33
ð
d
o
3
=
d
t
Þþo
1
o
2
ð
I
32
I
11
Þ;
where
o
3
are the components of the angular velocity about the
respective coordinate axes. In the case when there is only one nonzero component
of the angular velocity
o
1
,
o
2
, and
o
3
and
o
1
¼ o
2
¼
0, then (
1.3
) reduces to
M
3
¼
I
33
a
3
;
(1.4)
where
o
3
/d
t
is the angular acceleration. This is the one-dimensional form of
the conservation of angular momentum, or of Euler's equations, a form that usually
appears in basic mechanics texts.
a
3
¼
d
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