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Файл:A General Course of Physics. Mechanics. Textbook
.pdf
61
ment of forces, but by all three projections, i.e. by the vector of the
M
IM
i
i
i
i
IMM
amFF
i
i
F 0
i
i
M 0
moment of forces
. Consequently, equation (47) can be written in the
form:
. (48)
We are reminded that the moment of inertia of the solid body I depends on the rotation axis attitude.
Equation (48) is the fundamental equation of gyrodynamics of a ri-
gid body. According to its form, it is similar to the equation of Newton’s
second law and it is sometimes called the second law of dynamics for a rotational motion. It follows from comparison that during a rotational motion, the
role of force is performed by the moment of force; the role of mass – by the
moment of inertia and the role of the linear acceleration – by the angular acceleration.
The motion of a rigid body is determined by two equations:
, (49)
specifically Fi is the external force, Mi is the total force.
A body can stay at rest in cases when there are no reasons for occurring translational motion or rotation. In compliance with (49), two
conditions are necessary and sufficient for this:
1. The sum of all external forces, applied to the body:
fixed axis:
.
2. The resultant moment of external forces in relation to any
.
62
4.6. The Work of External Forces during the Rotation of a Solid
i
dl
iii
dlFdA
i
F
drdl
ii
drFdA
iii
ii
rF
dMdA
ii
i
i
MddMdA
MMddAA
0
Body
Let us consider a solid body which can rotate about an axis fixed
in space. Assume that Fi is a component of an external force causing rotation applied to an elementary mass ∆m
interval, the elementary mass will travel
of a rigid body. For a short time
i
distance and, as a result, the
work will be done by the force:
, (50)
where
(it is this force that causes the body to rotate);
is the force, tangent to the trajectory line of the motion of mass
is the displacement quantity; d is the elementary angle of rotation of the rigid body; ri
is the radius of the circumference along which the given mass is travel-
ling.
Then, formula (50) will have the form:
The product
is the moment of force Mi in relation to the axis
of rotation and acting upon the body element mi. Consequently, the
work of force is
.
Having summed up the work of the moments of force, applied to
all body elements, we will receive an elementary amount of energy, consumed by an elementary rotational motion d:
,
viz. M is the net moment of forces, acting upon the rigid body, rotating
about a given axis.
And finally we have an equation of the work during the limited
time interval t (when M = const):
.
63
4.7. Angular Momentum. The Law of Conservation of Angular
]Fr[M
][][ vmrprL
L
L
mvrprprLL
;sin
Momentum
We considered above (see Unit 4.4) that the moment of force can
be written as a vector product:
.
Similarly, it is possible to determine the moment of any vector.
We will examine the motion of a point particle with a mass of m
along the circumference with a radius r (Fig. 25). The angular momentum
of the point particle in relation to the circle center is
Vector
is perpendicular to the circumference plane, and its di-
rection is determined by the right-hand screw rule. The vector magnitude
is
;
L = mvr.
The angular momentum of the point equals the product of the
momentum by the arm in relation to the rotation axis.
64
We will consider the rotation of a rigid body about the axis OO
2
iiiiii
rmrvmL
i
L
2
iii
rmL
JrmrmLL
i
ii
i
ii
i
i
22
JL
J
dt
d
JJ
dt
d
dt
Ld
)(; M
dt
Ld
,dtMLd
dtMJd
)(
dtM
with an angular velocity of . (Fig. 26). Each elementary mass of the
body moving along the circumference possesses the angular momentum:
.
Direction
coincides with the vector direction
, so in the
vector form it is as follows:
.
As a result, while the impulse vectors of the elementary masses of a
rotating body have a wide variety of directions, the momentum vectors of the
elementary masses are directed equally.
. (51)
Consequently, the angular momentum of a rigid body in relation
to the rotation axis equals the product of the moment of the body inertia
in relation to this axis and the angular velocity vector.
Having used the time derivative from (51), we get:
or
where
is the angular momentum of external forces.
,
All in all, the angular momentum increment of a rigid body
equals the impulse of the net moment of external forces applied to the
body.
If the net moment of all external forces acting upon the body
equals zero, the angular momentum vector remains unchanged. This
statement expresses the law of the angular momentum conservation:
65
const;0,0
JL
dt
Ld
M
.
J
Translational motion
Rotational motion
Displacement vector
dr
Angle of rotation
d
Linear velocity
dtdrv
Angular velocity
dtd
Linear acceleration
dtà dv
Angular acceleration
dtdε
External force F
Angular force M
Body mass m
Moment of inertia
V
dVrJ
2
Impulse
vmð
Angular impulse
JL
It follows from this law that if the moment of inertia changes,
then the angular velocity changes respectively to the way that the product
remains constant. We will give a few examples.
1. In figure skating, there is an element called a spin. A figureskater, changing their arm positions, also changes their moment of inertia
at the same time, and as a result, their rotation velocity changes as well.
2. A gymnast, doing a somersault, bends their arms and legs to the
body. While doing so, their moment of inertia decreases, but the angular velocity increases. For the short time interval, that the gymnast is in the air,
they have time to flip over several times.
3. A ball is attached to a thread which is being wound around a stick.
While the thread length is decreasing, the ball’s moment of inertia is decreasing too and, as a result, the angular velocity rises.
4. Tests on a Zhukovsky bench with dumbbells and a wheel.
Let us compare the basic quantities and equations characterizing
rotation of a body about a fixed axis and its translational motion. Such
comparison is convenient for memorizing formulas and also for a deeper
understanding of the connections between these motion types.
66
Newton’s second law
maF
Fundamental equation of rotational mo-
tion
JM
The law of momentum change
Fdt dð
The law of angular impulse change
Mdt dL
Elementary work
)(FdrdA
Elementary work
)( MddA
Kinetic energy of translational mo-
tion
2
2
ê
vm
W
Kinetic energy of rotational motion
2
2
ê
JW
L
tML
M
4.8. Gyroscope. Gyrocompass
A gyroscope (or a spinning wheel) is a massive symmetrical body,
rotating at a high velocity about the axis of symmetry.
The main gyroscope property is the ability of the axis to keep the direction of its rotation constant without any action of an external moment.
This gyroscope property is based on the law of angular momentum conservation. The higher the gyroscope’s angular velocity of rotation and the moment
of inertia in relation to the rotation axis, the higher the gyroscope’s stability.
The operational principle of the gyroscopic stabilizers (such as
automatic pilot systems on planes) is based on the gyroscope’s ability to
maintain the direction of its rotation axis unchanged.
If a couple of forces are applied to a spinning gyroscope, tending
to turn it about its axis, perpendicular to the gyroscope’s rotational axis,
then it will rotate around its third axis which is perpendicular to the first
two. This effect is called the gyroscopic effect. The motion occurred under these circumstances is called precession.
Now let us examine this phenomenon in detail (Fig. 27). The gyroscope is spinning about the axis OO. We will apply a couple of forces
to it in the direction O2O2. The moment of these forces is directed along
O1O1. For the time interval t, the gyroscope’s moment of impulse
will receive the increment
as
. The gyroscope’s moment of impulse some time later t will be
, which also has the same direction
67
equal to the resultant
LLL
1
1
L
22
OO
M
1
L
M
L
M
, being located in the figure plane. The
direction of the moment of impulse coincides with the rotation axis of the
body. So, the vector direction
points at the new direction of the gyroscope’s rotational axis. Thus, the gyroscope’s axis will turn about the
straight line
, in a way that the angle between the vectors
and
will decrease.
If we act upon the gyroscope for a long time with the external
moment
, constant in direction, then the gyroscope’s axis will settle in
such a way that the axis and direction of its own rotation will coincide
with the rotation axis and direction of the external moment (vector
will coincide with vector
in direction).
The gyroscope’s mode of behavior thus described forms the basis
of the operational principle of the device called the gyroscopic compass
(gyrocompass).
This device is a gyroscope whose axis can rotate freely in the ho-
rizontal plane. The source of the external torque in this case is the Earth’s
daily rotation about its axis (Fig. 28). As a result, the gyroscope’s axis
68
turns in a way that the angle between the vector of the gyroscope’s mo-
L
ment of impulse
and the vector of the Earth’s angular velocity
should decrease. It continues until the angle between L and ω becomes
minimum, i.e. until the gyroscope’s axis settles in the meridian plane.
The gyroscope has an obvious advantage over the magnetic compass because it settles directly in the plane of the geographic meridian
rather than in the magnetic one; and moreover, there is no need to take
measures against the impact of iron objects on the compass needle.
4.9. Non-Inertial Reference Frames
The laws of dynamics (Newton’s laws) are only true for inertial
reference frames. The frame of reference moving with acceleration is
called non-inertial. Within these systems, Newton’s laws are inapplicable.
But any motion can be described both in an inertial and non-inertial reference frame. For example, when a train applies a hard brake, passengers
lean forward. An observer standing on the platform (being in a motionless inertial reference system) will describe this phenomenon using Newton’s laws. For a passenger sitting inside a wagon (within a non-inertial
reference system moving with acceleration), there is no acceleration in
the system, it is motionless to him. He will try to explain the observed
phenomenon using Newton’s laws, too. From the point of view of the
first observer, the passenger is acted upon by a force when he leans forward. However, the observer won’t be able to identify the bodies, which
could interact with the passenger, and the force which could result from
the interaction; in other words, the observer will face a discrepancy with
Newton’s third law.
It turns out that after abandoning Newton’s first and third laws it
is still possible to apply Newton’s second law. For this purpose, addition-
al forces are formally introduced in non-inertial reference systems, which
are called inertial forces (they say, the passenger leans forward under his
own inertia).
We will consider three cases of exposure to inertial forces.
69
1. Inertial forces during the accelerated translational motion of
gm
í
F
a
amFgm
í
am
0 amFgm
t
a reference system.
Assume that a ball is attached to a wagon ceiling with a thread. If
the wagon stands still (or moves uniformly and rectilinearly), the thread
hangs vertically. If the wagon moves rectilinearly with acceleration, the
thread deviates from the vertical line. We will see how it is possible to
explain it from the standpoint of an observer being in an inertial or noninertial reference system (Fig. 29).
Inertial reference frame.
The gravity force
ball. The total force communicates the acceleration
and the tensile stress on the thread
act upon the
to the ball (Fig.
29).
According to Newton’s second law, or having transposed
to the left-hand part of the equation, we have:
. (52)
Non-inertial reference frame.
70
Within the reference frame, which involves the wagon moving
gm
í
F
i
F
0
it
FFgm
amF
i
with acceleration, the ball is motionless. It is possible in cases when
the total real force
and
are balanced against an additional
force, equal in magnitude and with the opposite direction, which is
called the force of inertia
(Fig. 30).
Then, according to Newton’s second law
(53)
After comparing equations (52) and (53), we get the expression
for the force of inertia:
.
We can observe forces of inertia in everyday life: when using
public transport, after a severe brake application or hard acceleration and
also under start-up conditions and deboost of a spacecraft.
2. Inertia forces, acting upon a body which stays at rest within a
rotating reference system.
Let us consider a horizontal disk, rotating uniformly at an angular
velocity about a vertical axis. A ball attached with a thread hangs at the
edge of the disk. During the rotation of the disk, the thread diverges from
the vertical line at an angle. Let us examine this fact from the perspective
of an observer being within inertial and non-inertial systems.
Inertial reference frame.
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