Biomedical Engineering Reference
In-Depth Information
velocity, also called linear velocity
v
, corresponds to the ratio between the
traveled distance
t
, it is possible to establish a relation
between the translational and angular velocities (
4.4
):
S
and the time interval
Δ
Δ
S
v
R
:
ω ¼
Δ
t
¼
(4.4)
R
Δ
If another particle of mass
m
, rotating about the same axis, but in a smaller
circumference, sweeps out the same displacement
t
, it means
that this particle has the same angular velocity, but its linear velocity is smaller.
Equation (
4.4
) shows that to maintain
Δ
θ
in the interval
Δ
ω
constant, if
v
is smaller, it is because
R
is
proportionally smaller.
The energy associated with a particle in motion is called the kinetic energy,
K
.
It depends on the mass
m
and on the linear velocity
v
of the particle, calculated
from (
4.5
):
mv
2
2
:
K
¼
(4.5)
R
obtained from (
4.4
), we can write
for the particle's rotational kinetic energy,
K
ROT
(
4.6
):
Substituting the velocity
v
by the product
ω
mR
2
2
ω
K
ROT
¼
:
(4.6)
2
Comparing (
4.5
) and (
4.6
), it is possible to establish an analogy between
rotational and translational motion. We note that, as we have substituted the
translational velocity
v
by the rotational angular velocity
, the translational mass
m
was substituted by the product
mR
2
for rotation. This product is called the
particle's moment of inertia, and its unit in the SI is kg m
2
.
The moment of inertia
I
of one single particle of mass
m
rotating about its axis at
distance
R
is, then, given by (
4.7
):
ω
mR
2
I
¼
:
(4.7)
The moment of inertia of an extensive solid body about an axis of rotation is the
sum of moments of inertia of each elementary mass or particle of mass
m
i
, which
composes the body about its axis of rotation:
X
m
1
R
1
þ
m
2
R
2
þ
m
3
R
3
þþ
m
n
R
n
¼
m
i
R
i
2
I
¼
:
(4.8)
i
Observe that the mass distribution is more significant than the total mass because
R
appears squared in (
4.7
) and (
4.8
).
