Biomedical Engineering Reference
In-Depth Information
Fig. 4.8
Solid cylinder of
radius
R
, length
L
, and mass
M
that rotates about the
y
-
axis passing through one of
the bases
y
M
R
L
Figs.
4.3
,
4.4
,
4.5
,
4.6
, and
4.7
are all about an axis passing through its center
of gravity. There is a very useful and simple relation between the body moment of
inertia about an axis passing through its center of gravity
I
C.G.
and the moment
of inertia
I
of this body about another axis parallel to the first. With
M
the total body
mass and
d
¼
L
/2, the distance between both parallel axes, the relation is given by
Md
2
I
¼
I
C
:
G
:
þ
:
(4.9)
Using (
4.9
) we can deduce several expressions of interest as, for example, to
obtain
I
y
of Fig.
4.8
from
I
y
in Fig.
4.7
which will be
I
C.G.
. In the case of Fig.
4.8
,
M
3
R
2
L
2
M
L
2
þ
Mk
2
I
y
¼
þ
4
¼
:
12
For a cylinder with radius
R
¼
0.05 m,
L
¼
0.20 m, and
M
¼
1 kg, we obtain
0.014 kg m
2
, hence, 3.5 times more difficult to rotate about this axis than about
another
y
-axis passing through its center of gravity.
I
y
¼
Example 4.1
Is it possible to explain, without calculating, why tightrope walkers
on ropes or steel cables at height use long poles?
The explanation comes from the fact that, in holding a long pole horizontally, the
moment of inertia of the body plus the pole about the
y
-axis passing through the center
of gravity increases, and consequently the rotational inertia increases, i.e., it becomes
more difficult to rotate the body; in other words we have increased the equilibrium of
the body. Of course, concentration is essential. In our daily life, we usually open our
arms outstretched to better equilibrate when we walk on a narrow wall.
Exercise 4.1
Determine the moment of inertia of a forearm plus a hand with a total
mass of 3.0 kg, about a rotation axis according to the figure of Exercise 4.1. Suppose
that the forearm plus the hand has the form of a cylinder with length equal to 0.38 m
and radius of 0.03 m. Find the value of radius of gyration and the moment of inertia
for each case.
