Biomedical Engineering Reference
In-Depth Information
where
x
0
denotes the position of the particle at
t
=
0 and
τ
is a constant,
characteristic time. The velocity of this particle is obtained from
(1
x
0
)
2
)
d
x
dt
=
d
dt
d
(1
+
(
t
/τ
)
2
)
2
)
v
=
+
(
t
/τ
=
x
0
=
(2
t
/τ
x
0
,
dt
while the acceleration follows from
d
v
dt
=
2
)
a
=
/τ
x
0
.
(2
The force on this particle equals
F
=
(2
m
/τ
2
)
x
0
.
2.4
Vector operations on the force vector
Suppose that a force vector is represented by
F
1
=
e
, (2.10)
then another force vector, say
F
2
may be obtained by multiplying the force by a
factor
F
1
α
, see Fig.
2.3
(a):
F
2
=
α
F
1
e
=
F
2
e
.
(2.11)
The force vector
F
2
has the same orientation in space as
F
1
, but if
α
=
1itwill
have a different length, and it may have a direction sense (if
0).
The net result of two force vectors, say
F
1
and
F
2
, acting on the same point P
is obtained by the vector sum, graphically represented in Fig.
2.3
(b):
F
3
=
F
1
+
F
2
. (2.12)
The vector
F
3
is placed along the diagonal of the parallelogram formed by the
vectors
F
1
and
F
2
. This implicitly defines the orientation, sense and magnitude of
the resulting force vector
F
3
.
α<
F
1
F
3
=
F
1
+
F
2
F
2
=
α
F
1
F
1
F
2
(a)
F
2
=
α
F
1
(b)
F
3
=
F
1
+
F
2
Figure 2.3
Graphical representation of the scalar multiplication of a force vector with
α<
0 (a) and the sum
of two force vectors (b).
