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Файл:A General Course of Physics. Mechanics. Textbook
.pdf
51
In order to determine the kinetic
N
i
ii
N
i
i
rm
WW
1
2
1
kk
2
N
i
ii
m
1
2
2
1
r
2
N
i
ii
mI
1
2
r
2
k
2
I
W
22
2
k
2
v
I
m
W
mi
i
r
O
O
i
v
Fig. 16
energy of the whole rotating body, it is
necessary to sum up the kinetic energies
of individual body elements:
. (44)
Let us introduce the notation:
.
This physical quantity is called the
moment of rigid body inertia with respect to the axis of rotation. The moment
of inertia is a measure of body inertia during the rotational motion and depends on the body mass, its size and form, and also on the rotation axis spatial attitude. The moment of inertia is measured in kilograms squared (kgm2)
in the SI system.
Thus, the kinetic energy of a body rotating about a fixed axis can be
expressed through the moment of the body inertia I and the angular velocity
of its rotation :
.
If a body is rotating and at the same time moving translationally,
then the kinetic energy of the body is the sum of the kinetic energy resultant from the translational motion of the center of mass and the kinetic
energy of the rotational motion:
.
52
4.2. The Moment of Inertia for Bodies with Simple Geometric Forms
2
iii
rmI
Vm
ii
Vm
i
i
i
VrI
2
i
V
dVrI
V
2
rdrbdV 2
Fig. 17
R r b
O
O
dr
The moment of body inertia equals the sum of its parts inertia
moments.
.
The mass distribution within the body is characterized by the
quantity called density.
If the body is homogeneous, its density is
throughout the whole volume V. Then,
and for the homoge-
and constant
neous rigid body the moment of inertia can be written in the following
way:
.
integral:
If the volume decreases
, the sum will change into the
. (45)
To sum up, the moments of inertia for homogeneous bodies
with simple forms can be determined
through integration using formula (45).
Now we will examine a few examples.
Example 1. Find the moment of
inertia for a homogeneous disk in relation to the axis, perpendicular to the disk
plane and crossing its center. The disk
radius is R, its thickness is b (Fig. 17).
Solution. Let us divide the disk into circular rings with the radius r and
thickness dr. The volume of such a ring is
. Then,
53
rdrbrdVrI
R
2
0
22
;
R
R
bdrrbI
0
4
3
4
22
mbR
2
2
2
mR
I
2222
mRmRRmrmI
i
i
i
iiii
2
mRI
SdхdV
O
O
d
Fig. 18
R
Fig. 19
l
x
dx
O
O
,
where
is the disk mass.
As a result, we get the formula for the
disk moment of inertia
.
This formula is true for the moment of inertia of a solid cylinder in
relation to the axis coinciding with the cylinder axis.
Example 2. Find the moment of inertia of a hoop in relation to the
axis perpendicular to the hoop plane and crossing its center (Fig. 18). The
hoop radius is R, the width is d<<R.
Solution. As d<<R, then ri = R.
So,
;
.
The same formula is true for a thin-walled cylinder.
Example 3. Find the moment of inertia of a thin rod in relation
to the axis perpendicular to the rod and crossing its center. The rod length
is l, the cross-sectional area is S
(Fig. 19).
Solution. We will select a
small volume dV, which is at a
distance x from the axis and has a
width dx,
.
Then,
54
2l
2l
22
dххSdVхI
12
l
8
l
8
l
333
S
Slm
12
2
ml
I
2
5
2
mRI
2
4
1
mRI
The moment of inertia in
relation to an unspecified axis can
be determined with the help of
Steiner’s theorem: The moment of
inertia about any axis О
1О1
equals
the sum of the moment of inertia I0
about the axis OO, parallel to that
axis and crossing the center of the
body inertia, and the product of the
2
0
maII
.
The rod mass is
. It follows that
.
Example 4. The moment of inertia of a ball with the radius R in
relation to the axis crossing its center is
Example 5. The moment of inertia of a disk with the radius R in
relation to the axis, coinciding with the disk diameter is
All the given formulas are true for the moments of inertia in relation to the axis crossing the center of mass (the center of inertia) of a rigid body.
body mass by the squared distance a between the axes:
With the help of this formula we will get the formula for
.
55
the moment of inertia in relation to the axis perpendicular to the rod and
12
2
0
ml
I
3
l
4
l
12
l
2
l
222
2
0
mmm
mII
2
52 MRI
6
2
aMI
crossing its end (Fig. 20). As it can be seen in the figure, it is clear that
а = l/2. Apart from that, the moment of inertia in relation to the axis
crossing the center of mass is
.
Therefore, according to Steiner’s theorem, we will have:
4.3. Principal Axes of Inertia
The moment of inertia of a rigid body with an unspecified form
and arbitrary distribution of mass depends on the spin axis orientation.
Assume that the axis crosses the center of mass (the center of inertia).
We will find such a spin axis orientation that has a maximum moment
of inertia. Then, as it is proved in theoretical mechanics, there is also an
axis which has the minimum moment of inertia of a rigid body. For the
third axis, orthogonal to the first ones, the moment of inertia in a general case has a value, intermediate between the maximum and minimum
values. The axes of rotation introduced are called the principal axes of
rotation. The moments of inertia about these axes do not necessarily
differ from each other in magnitude. In fact, if a rigid body homogeneous in density possesses whichever symmetry, then some principal moments of inertia may be equal to each other. For example, a ball homogeneous in density has three equal moments of inertia about the princip-
al axes, for each of them
, specifically M and R are the ball
mass and radius, respectively.
A homogeneous cube with a mass M and the edge length a has
also three equal moments of inertia in relation to the principal axes of
inertia
, which are perpendicular to the cube edge and cross the
cube center.
A thin disk homogenous in density has the moment of inertia
maximum in magnitude in relation to the axis crossing the disk center
56
perpendicular to its density, and also two other principal moments of
][ FrM
r
F
M
r
F
r
F
M
inertia equal to each other.
We will also give an example of a body where all three moments
of inertia in relation to the principal axes of inertia are different: a homogeneous in density parallelepiped with edges different in length.
The axis of rotation, whose position in space remains constant
without any external action, is called an axis of free rotation.
The principal axes of inertia are axes of free rotation. If external
forces do not act upon the body, then the rotation about the principal
axes, corresponding to the maximum and minimum values of the moment
of inertia, is steady. However, the rotation about the axis, equal to the
intermediate value of the moment of inertia, will be unsteady.
4.4. Moment of Force
1. The moment of force in relation to a point. Let us consider a
point particle with a mass m, which can rotate about a point O under the
action of a force F applied.
For the sake of simplicity and clarity, we will place the origin of
coordinates into the point O. Assume that the point particle with a mass
of m is in the plane хОу, and the force F is directed along the axis Ох
(Fig. 21).
The moment of force in relation to the point O is determined as
the vector product:
.
This vector is axial, its direction is connected with the direction
and
to the plane containing
through the right-hand screw rule, i.e. the vector
and
, and its direction coincides with the trans-
is perpendicular
lational motion of the corkscrew when rotating the corkscrew handle from
to
towards the smallest angle.
In this case, the vector
is directed along the axis Z.
If a body can rotate about the point О arbitrarily, then under the
action of the force F it will rotate about the axis, coinciding with the direction of the moment of force in relation to this point.
57
The moment of force is
k
rFsin FlrFM
sinrl
sin
k
FF
r
rF
||
M
M
,
specifically,
is the arm of force; Fk is the distance from the
point O to the straight line, being under the action of force lengthwise;
is the tangential component of the force perpendicular to
.
If = 0, i.e.
, then M = 0.
Consequently, the force crossing the point O does not cause rotation. Any force can be broken down into F
and F(F = Fк is a tangential
||
component). Thus, the moment of force is determined only by the com
component Fk, and it causes the body rotation.
2. Moment of force in relation to an axis. Now let us consider a
body, fixed at two stationary points О and О1 in a way that it can only
rotate about the axis crossing these points.
Assume that ОО1 coincides with the axis OZ. The moment of
force
is directed arbitrarily (Fig. 22). The vector
can be broken
down into components Mx, My, Mz, and each of them will tend to turn the
body about the axes x, y, z. The moment components Mx and My will be
compensated for the inertia reaction torque, occurring in the pinning
points. Therefore, the rotation will occur only upon the action of Mz com-
58
ponent. The vector component
,M
M
321
MMMM
directed along the axis of rotation, is
called the moment of force in relation to this axis.
The moment of force in relation to the axis is a scalar value.
However, if the moment of force applied to the body is directed along the
axis of rotation fixed in space (the axis z), then it will cause the rotation
itself (but not its projection) and it can be considered as a vector which
can have only two possible directions, corresponding to the clockwise or
counterclockwise rotation. If the moment of force causes a clockwise rotation, it is considered to be positive, if counterclockwise, it is considered
to be negative.
If a body fixed on an axis is acted upon by several moments of
force, their synergetic effect will be equivalent to the action of one mo-
ment of force
, equal to the sum of several individual moments:
Let us examine the synergetic effect of internal forces (Fig. 23).
We will select two elementary masses mi and mk within the
body.
The forces, whose two arbitrary elementary masses interact, are
located along one straight line. Their moments in relation to any axis O
59
(which is perpendicular to the figure) are equal in magnitude and opposite
kiik
FF
lFM
ikik
lFM
kiki
kiki
MM
0
kiik
MMM
ki
ki
M
,
internal
,
0
Let us consider a rigid body
which can rotate about an axis fixed in
space OO (Fig. 24). We will break down
the whole body into elementary masses
mi. In a general case, an external force
i
F
can be applied to each elementary
mass. The force component
i
F
causing
rotation must be directed along the tangential line to the circumference, along
which the elementary mass is moving.
The other two components of the external
force either cause deformation of the axis
of rotation fixed in space or determine the
load upon the supports of the rotation axis
without causing any rotation.
i
F
in direction:
;
;
;
;
.
In conclusion, the moments of internal forces mutually balance
each other and the sum of moments for any system of point particles al-
ways equal zero.
4.5. Fundamental Equation of Gyrodynamics of a Rigid Body
We will denote the total external force, applied to an elementary mass
mi which causes rotation as
the elementary masses of a solid body, will not be taken into considera-
(Fig. 24). The internal forces acting upon
60
tion, as being subsequently summed up both the forces themselves and
iiiii
rmamF
i
a
i
r
ii
ra
2
iiii
rmrF
2
iii
rmM
2
ii
ii
iz
rmMM
Irm
iii
2
IM
z
z
M
their moments become mutually reduced (mind Newton’s third law!).
According to Newton’s second law, we have
, (46)
where
and
are the linear acceleration and radius-vector of the ele-
mentary mass; is the angular acceleration of the body rotation as a single whole (
).
We will multiply both parts (46) by ri:
.
As Firi = Мi is by definition the moment of force, acting upon
the mass element mi in relation to the given axis of force, therefore,
.
Having summed up all elementary masses, which the body has
been broken down into, we will get
.
The sum value
is the moment of the body inertia in
relation to the given axis of rotation, so this equation can be written in the
following form:
, (47)
viz.
is the projection of the total moment of external forces, acting
upon the solid body, on the given axis of rotation; is the angular acceleration.
We are reminded that the moment of forces in relation to the
given axis of rotation is the projection of the moment vector onto the
axis of rotation. So, in a general case, when the axis of rotation is not
fixed in space, the rotation is caused not by one projection of the mo-
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