Biomedical Engineering Reference
In-Depth Information
Vo
α
Fig. 1.4
An illustration of the particle model in biomechanics, determining the airborne trajectory
of a ski jumper. In the air the ski jumper is acted upon by the attraction of gravity, the drag of the
wind and the momentum established in the downhill run before contact with the ground ceased.
The ski jumper's trajectory is determined by the solution of Newton's second law with these
specified forces
the object,
p
x
3
e
3
, where
e
1
,
e
2
, and
e
3
are the Cartesian unit
base vectors. The position vector to the mass center of the object is denoted by
p
(mc)
¼
¼
x
1
e
1
þ
x
2
e
2
þ
x
(mc)3
e
3
. If the object is moving, then the location of
the point of the object is changing in the Euclidean space, and the Cartesian
coordinates
x
1
,
x
2
,
x
3
are all continuous, twice differentiable, functions of time,
x
1
¼
x
(mc)1
e
1
þ
x
(mc)2
e
2
þ
x
3
(
t
), and it is therefore possible to compute the veloc-
ity of the mass center of the object, or of any point on the object, as well as its
acceleration. The acceleration of the mass center is given by
x
1
(
t
),
x
2
¼
x
2
(
t
),
x
3
¼
d
2
x
ð
mc
Þ
1
=
d
t
2
d
2
x
ð
mc
Þ
2
=
d
t
2
d
2
x
ð
mc
Þ
3
=
d
t
2
a
mc
¼ð
Þ e
1
þð
Þ e
2
þð
Þ e
3
:
(1.1)
Denoting the total mass of the object by
m
and the sum of the forces acting on the
object by
F
, a statement of the second law of Newton can then be written in the form
F ¼
m
a
mc
:
(1.2)
As an illustration of the particle model, consider the question of determining the
airborne trajectory of a ski jumper. In the air the ski jumper is acted upon by
the attraction of gravity, the drag of the wind, and the momentum established in the
downhill run before contact with the ground ceased (Fig.
1.4
). The ski jumper's
trajectory is determined by the solution of Newton's second law with these
specified forces. The trajectory is obtained by an analysis that is completely
equivalent to that of an artillery shell or a sub-orbital rocket. While the particle
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