where r is the radius vector drawn from point O to point A, the location of the material point, p=m v is the momentum of the material point. Momentum vector modulus:

where a is the angle between the vectors r and p, l is the shoulder of the vector p with respect to the point O. The vector L according to the definition of the cross product is perpendicular to the plane in which the vectors lie r and p(or v), its direction coincides with the direction of the translational movement of the right screw when it rotates from r to p

Angular moment about the axis is called a scalar value equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point on this axis.

The moment of force M of a material point relative to the point O is called a vector quantity determined by the vector product of the radius vector r, drawn from the point O to the point of application of the force, by the force F: .

Fig.2.

Moment of force vector modulus:

where a is the angle between the vectors r and F, d \u003d r * sina - the shoulder of the force - the shortest distance between the line of action of the force and point O. Vector M(as well as L) - perpendicular to the plane in which the vectors lie r and F, its direction coincides with the direction of the translational motion of the right screw when it rotates from r to F the shortest distance as shown in the figure.

Moment of force about the axis called a scalar quantity equal to the projection onto this axis of the vector of the moment of force M defined with respect to an arbitrary point on this axis.

The basic law of the dynamics of rotational motion

To clarify the purpose of the above concepts, we consider a system of two material points (particles) and then generalize the result to a system of an arbitrary number of particles (ie, to a rigid body). Let on particles with masses m 1 , m 2 , whose momenta p1 and p2, external forces act F1 and F2. Particles also interact with each other by internal forces f 12 and f 21 .

Fig.3.

Let's write down Newton's second law for each of the particles, as well as the connection between internal forces arising from Newton's third law:

Multiply the vector equation (1) by r1, and equation (2) – on r2 and add the resulting expressions:

Let us transform the left parts of equation (4), taking into account that

.

The vectors and are parallel and their cross product is zero, so we can write

. (5)

The first two terms on the right in (4) are equal to zero, i.e.

insofar as f 21 = -f 12, and the vector r1-r2 directed along the same straight line as the vector f 12.

Taking into account (5) and (6) from (4) we obtain

or

where L=L 1 +L 2; M=M1+M2. Generalizing the result to a system of n particles, we can write L=L 1 +L 2 +…+L n = M=M 1 +M 2 +M n=

Equation (7) is a mathematical record of the basic law of the dynamics of rotational motion: the rate of change of the angular momentum of the system is equal to the sum of the moments of external forces acting on it. This law is valid for any fixed or moving at a constant speed point in an inertial frame of reference. From this follows the law conservation of angular momentum: if the moment of external forces M is equal to zero, then the angular momentum of the system is conserved (L= const).

The angular momentum of a perfectly rigid body about a fixed axis.

Consider the rotation of an absolutely rigid body around a fixed axis z. A solid body can be represented as a system of n material points (particles). During rotation, some considered point of the body (we denote it by the index i, and i=1…n) moves along a circle of constant radius R i with a linear speed v i around the z axis (Fig. 4). Her speed v i and momentum m i v i perpendicular to the radius R i. Therefore, the modulus of the angular momentum of a particle of a body relative to the point O, located on the axis of rotation:

where r i is the radius vector drawn from the point О to the particle.

Using the relationship between the linear and angular velocity v i =wR i , where R i is the distance of the particle from the axis of rotation, we obtain

The projection of this vector onto the rotation axis z, i.e. the angular momentum of a body particle relative to the z-axis will be equal to:

The angular momentum of a rigid body about the axis is the sum of the angular momentum of all parts of the body:

The value I z , equal to the sum of the products of the masses of the particles of the body and the squares of their distances to the z axis, is called the moment of inertia of the body about this axis:

From expression (8) it follows that the angular momentum of the body does not depend on the position of the point O on the axis of rotation, therefore, we speak of the angular momentum of the body relative to some axis of rotation, and not relative to the point

There is a similarity between the formulations of the basic law of rotational motion, the definitions of the moment of momentum, and the force with the formulations of Newton's second law and the definitions of momentum for translational motion.

Free axes and principal axes of inertia of the body

In order to maintain a fixed position in space of the axis of rotation of a rigid body, it is mechanically fixed, usually using bearings, i.e. influenced by external forces. However, there are such axes of rotation of bodies that do not change their orientation in space without the action of external forces on them. These axes are called free axes. It can be proved that any body has three mutually perpendicular axes passing through its center of mass, which are free. These axes are also called main axes of inertia of the body.

Gyroscopes

Currently, gyroscopes are called a very wide class of devices that use more than a hundred different phenomena and physical principles. In this laboratory work, a classical gyroscope is studied, in the future, just a gyroscope.

A gyroscope (or top) is a massive symmetrical body rotating at a high angular velocity around its axis of symmetry. We will call this axis the axis of the gyroscope. The axis of the gyroscope is one of the main axes of inertia (free axis). The angular momentum of the gyroscope in this case is directed along the axis and is equal to L=I w.

Consider a horizontally balanced gyroscope (whose center of gravity is above the fulcrum). Since the moment of gravity for it is equal to zero, then, according to the law of conservation of angular momentum L=I w= const, i.e. the direction of its axis of rotation does not change its position in space.

When trying to cause the axis of the gyroscope to rotate, a phenomenon is observed called gyroscopic effect. The essence of the effect: under the action of a force F applied to the axis of a rotating gyroscope, the axis of the gyroscope rotates in a plane perpendicular to this force. For example, under the action of a vertical force, the axis of the gyroscope rotates in the horizontal plane. At first glance, this seems counterintuitive.

The gyroscopic effect is explained as follows (Fig. 5). Moment M strength F directed perpendicular to its axis, because M=, r is the radius vector from the center of mass of the gyroscope to the point of force application.

Fig.5.

During the time dt, the angular momentum of the gyroscope L will receive an increment d L=M*dt (according to the basic law of rotational motion), and directed in the same direction as M and become equal L+d L. Direction L+d L coincides with the new direction of the axis of rotation of the gyroscope. Thus, the axis of the gyroscope will rotate in a plane perpendicular to the force F at some angle dφ=|dL|/L=M*dt/L, with angular velocity

The angular velocity of rotation of the gyroscope axis W is called the angular velocity of precession, and such rotational motion of the gyroscope axis precession.

From (9) it follows

Vectors M, L, W mutually perpendicular, so we can write

M=.

This formula is obtained when the vectors M, L, W are mutually perpendicular, but it can be proved that it is valid in the general case.

Note that these arguments and the derivation of formulas are valid in the case when the angular velocity of rotation of the gyroscope is w>>W.

It follows from formula (9) that the precession velocity W is directly proportional to M and inversely proportional to the gyroscope angular momentum L. If the time of action of the force is short, the angular momentum L is large enough, then the precession velocity W will be small. Therefore, the short-term action of forces practically does not lead to a change in the orientation of the axis of rotation of the gyroscope in space. To change it, forces must be applied for a long time.

Practical application of gyroscopes

The properties of the gyroscope described above have found various practical applications. One of the first applications of the properties of gyroscopes was found in rifled weapons. After leaving the gun barrel, the air resistance force acts on the projectile, the moment of which can overturn the projectile and change its orientation relative to the trajectory in a random way, which negatively affects the flight range and accuracy of hitting the target. Screw rifling in the barrel of the gun imparts a rapid rotation around its axis to the emerging projectile. The projectile turns into a gyroscope and the external moment of the air resistance force causes only the precession of its axis around the direction of the tangent to the trajectory of the projectile. At the same time, a certain orientation of the projectile in space is preserved.

Another important application of gyroscopes is various gyroscopic instruments: gyrohorizon, gyrocompass, etc. Balanced gyroscopes are also used to maintain a given direction of aircraft movement (autopilot). To do this, the gyroscope is mounted on a cardan suspension, which reduces the effect of external moments of forces arising during aircraft maneuver. Due to this, the axis of the gyroscope maintains its direction in space, regardless of the movement of the aircraft. When the direction of aircraft movement deviates from the direction specified by the axis of the gyroscope, automatic commands appear that return to the specified direction.

The described behavior of the gyroscope is also the basis of the device called the gyroscopic compass (gyrocompass). This device is a gyroscope, the axis of which can freely rotate in a horizontal plane. If the axis of the gyroscope does not coincide with the direction of the meridian, then, due to the rotation of the Earth, a force arises that tends to rotate the axis in the direction perpendicular to the horizon. However, due to the gyroscopic effect, it rotates in a horizontal direction until the direction coincides with the meridian, pointing exactly north. A gyroscopic compass compares favorably with a compass with a magnetic needle in that its readings do not need to be corrected for the so-called magnetic declination (associated with the mismatch of the geographic and magnetic poles of the Earth), and it is also not necessary to take measures to compensate for the effects of magnetic interference from the body and equipment vessel.

Description of the experimental setup

The experimental setup (Fig. 6) consists of the following main units:

1. Gyro disk.

2. Lever with metric scale.

3. The load, by moving it along the lever 2, the value of the moment of force is set.

4. Disk with an angular scale for determining the angle of rotation of the gyroscope axis in the horizontal plane during precession.

5. Block of measurements and control.

1. Determine the modulus of the moment of gravity for several positions of the load z on the gyroscope lever:

,

where m is the mass of the load, z p is the coordinate of the load on the metric scale of the lever when the gyroscope is balanced.

2. For each position of the load, determine the time of rotation of the gyroscope axis Δ t to a given angle Δ φ and calculate the angular velocity of precession:

3. Calculate the value of the momentum of the gyroscope for each of the measurements:

4. Calculate the average value of the momentum of the gyroscope:

Where N is the number of measurements.

5. Calculate the moment of inertia of the gyroscope using the formula I = L/w (w is the angular velocity of the gyroscope, w = 2pn, n is the number of engine revolutions per unit time) and determine the absolute and relative errors in determining the moment of inertia of the gyroscope.

test questions

1. What is the angular momentum of a material point relative to a point?

2. The basic law of the dynamics of rotational motion.

3. What is the moment of force about a point?

4. Momentum of an absolutely rigid body.

5. Moment of inertia of a rigid body about a given axis.

6. Formulate the law of conservation of angular momentum.

7. What is a gyroscope?

8. What is the gyroscopic effect?

9. What is called gyroscope precession and under what conditions is it observed?

10. What is the angular velocity of precession?

Literature

1. Saveliev I.V. Course of general physics. Proc. allowance. In 3 volumes. T.1 Mechanics. Molecular physics. M.: Science. Chief editor phys.math. lit., 19873. -432 p.

2. Trofimova T.I. Physics course. Proc. allowance for universities. M.: Higher. Shk., 2003. -541 p.

GYROSCOPE(from the Greek gyreuo - I spin, I rotate and skopeo - I look, I observe) - a rapidly rotating symmetrical solid body, the axis of rotation (axis of symmetry) to-rogo can change its direction in space. Rotating celestial bodies, artillery shells, rotors of turbines installed on ships, aircraft propellers, and so on have properties of gravity. G.'s technique - basic. element of various gyroscopes. devices or instruments widely used for automatic control the movement of aircraft, ships, torpedoes, missiles, and in a number of other gyroscopic systems. stabilization, for navigation purposes (indicators of the course, turn, horizon, cardinal points, etc.), for measuring angular or incoming. speeds of moving objects (eg, rockets) and in many others. other cases (eg, during the passage of adit shafts, the construction of subways, while drilling wells).

So that the G.'s axis can freely rotate in space, G. is usually fixed in the so-called rings. gimbals (Fig. 1), in Krom axis vnutr. and ext. rings and G.'s axis intersect at one point, called. suspension center. A gimbal fixed in such a suspension has 3 degrees of freedom and can make any turn about the center of the suspension. If the center of gravity of G. coincides with the center of suspension, G. is called. balanced, or astatic. The study of the laws of motion of gravity is a task of the dynamics of a rigid body.

Rice. 1. Classic gimbals, a- outer ring b- inner ring in- rotor.

Rice. 2. Precession of the gyroscope. The angular velocity of the precession is directed so that the vector of intrinsic angular momentum H tends to coincide with the moment vector M pair acting on the gyroscope.

Basic properties of a gyroscope. If a pair of forces ( P-F) with moment ( h- the shoulder of the force) (Fig. 2), then (against expectations) the G. will begin to additionally rotate not around the axis X, perpendicular to the plane of the pair, and around the axis at, lying in this plane and perpendicular to the proper. body z axis. This will complement. movement called precession. G.'s precession will occur in relation to inertial reference frame(to the axes directed to the fixed stars) with an angular velocity

Fig 13. Direction gyroscope.

A number of devices also make use of the property of gas to precess uniformly under the action of constantly applied forces. So, if by means of supplement. load to cause G.'s precession with an angular velocity numerically equal and opposite to the vertical component of the angular velocity of the Earth's rotation (where U- the angular velocity of the Earth, - the latitude of the place), then the axis of such a G. with varying degrees of accuracy will maintain an unchanged direction relative to the cardinal points. During several hours, until an error of 1-2 ° accumulates, such a G., called the gyroazimuth, or G. direction (Fig. 13), can replace the compass (for example, on airplanes, in particular in polar aviation, where the readings of the magnetic compass unreliable). Similar to G., but with a significantly larger shift of the center of gravity from the axis of precession, it is possible to determine the flow. the speed of an object moving in the direction of an axis bb 1 , with any acceleration (Fig. 14). If we abstract from the influence of gravity, then we can assume that the moment of the transfer force of inertia acts on the G. Q, where t- mass G., l- shoulder. Then, according to formula (1), the gyroscope will precess around the axis bb 1 with angular velocity . After integrating the last equality, we obtain , where - beg. object speed. T. o., it turns out to be possible to determine the speed of an object v at any moment in time along the angle , on which the G. will turn around the axis by this moment bb one . To do this, the device must be equipped with a revolution counter and a device that subtracts from the total angle of rotation the angle by which the turbine will turn due to the action of the moment of gravity on it. Such a device (an integrator of longitudinal apparent accelerations) determines the vertical velocities. rocket takeoff; in this case, the rocket must be stabilized so that it does not have rotation around its axis of symmetry.

Rice. 14. Gyroscopic meter of rocket ascent rate. - lifting acceleration; g- acceleration of gravity; P - gravity, Q- force of inertia, - own kinetic moment.

In a number of modern structures use the so-called. float, or integrating, G. The rotor of such G. is placed in a casing - a float immersed in a liquid (Fig. 15). When the float rotates around its axis X moment will act on G. M x viscous friction, proportional to the angular velocity of rotation . Thanks to this, it turns out that if G. will force him to report. rotation around an axis at, then the angular velocity of this rotation in accordance with equality (1) will be proportional to . As a result, the angle of rotation of the float around the axis X will, in turn, be proportional to the integral over time of (which is why the G. is called integrating). Additional electric and electromechanical. Devices make it possible either to measure the angular velocity of this G., or to make it an element of a stabilizing device. In the first case, special electromagnets create a moment about the axis X, directed against the rotation of the float; the magnitude of this moment is adjusted so that the float stops. Then a moment M1 how to replace the moment M x forces of viscous friction and, therefore, according to f-le (1), the angular velocity will be proportional to the value M 1, determined by the strength of the current flowing through the windings of the electromagnet. In the second case, when stabilizing, for example, around a fixed axis at, the body of the integrating g. is placed on the platform, which can rotate around the axis at specialist. electric motor (Fig. 16). To explain the principle of stabilization, suppose that the base, on which the platform bearings are located, will itself rotate around the axis at to some corner. When the engine is off, the platform will turn in this case together with the base at the same angle, and the float will rotate around the axis X by an angle proportional to the angle . If now the engine will rotate the platform in the opposite direction until the float returns to its original position, then the platform will also return to its original position at the same time. You can continuously control the motor so that the angle of rotation of the float is reduced to zero, then the platform will be stabilized. The combination of two float floats in a common suspension with similarly controlled electric motors leads to the stabilization of a fixed direction, and three - to space. stabilization used, in particular, in inertial navigation schemes.

Rice. 15. Float integrating gyroscope: a- gyroscope rotor; b- float, in the body of which the bearing of the rotor axis is located; in- supporting fluid; G- frame; d- steel trunnions in stone supports; e- sensor of the angle of rotation of the float relative to the body; well- an electromagnetic device that applies a moment around the axis of the float.

Rice. 16. Stabilization around a fixed axis by means of a float gyroscope a- float gyroscope; b- amplifier, in- electric motor; G- platform, d- base.

Rice. 17. Power gyro frame: a- the actual frame; b- gyroscope; in- a partner; G- sensor of the angle of rotation of the gyroscope relative to the frame; d- sensor signal amplifier; e- stabilizing engine; well- torque sensor.

In the system of stabilization considered above, sensitivity plays a role. an element that detects deviations of an object from a given position, and the return to this position is performed by an electric motor that receives a corresponding signal. Similar gyroscopic systems. stabilization called. indicator (stabilizers of indirect action). Along with this, the so-called systems are used in technology. power gyroscope. stabilization (direct-acting stabilizers), in which the G. directly take on the efforts that interfere with the implementation of stabilization, and the engines play auxiliary. role, unloading partially or completely G. and thereby limiting the angles of their precession. Structurally, such systems are simpler than indicator ones. An example is a single-axis two-gyroscopic. frame (Fig. 17); rotors located in the frame rotate in different directions. Suppose that a force acts on the frame, tending to rotate it around the axis X and report the angular velocity . Then, according to the Zhukovsky rule, a pair will begin to act on casing 1, tending to align the rotor axis with the axis X. As a result, gravity will begin to precess around the axis y 2 with some angular velocity . casing 2 for the same reason will precess around the axis y 2 in the opposite direction. The angles of rotation of the casings will be the same, since the casings are connected by a gear clutch. Due to this precession on the casing bearings 1 a new pair will act, striving to align the rotor axis with the axis y one . The same pair will act on the casing bearings 2 . The moments of these pairs are directed oppositely (which follows from the Zhukovsky rule) and stabilize the frame, i.e. keep it from turning around the axis X. However, if the G.'s precessions are not limited, then, as can be seen from formula (3), when the casings are rotated around the axes y 1 , at 2 90° will stop stabilization. Therefore, on the axis of one of the casings there is a sensor that registers the angle of rotation of the casing relative to the frame and controls the stabilization engine. The torque arising from the engine is directed opposite to the moment tending to rotate the frame around the axis X; as a result, G.'s precession stops. The considered frame is stabilized with respect to rotation around the axis X. Rotate the frame around any axis perpendicular to X, you can freely, but the resulting gyroscopic. moment can cause mean. pressure on G. bearings and their casings. The combination of three such frames with mutually perpendicular axes leads to spaces. stabilization (eg, artificial satellite).

In power gyroscopic systems, in contrast to free G., due to the large moments of inertia of the stabilized masses, very noticeable oscillations arise. nutation movements. Specials must be accepted. measures to ensure that these oscillations are damped, otherwise self-oscillations occur in the system. In technology, other gyroscopes are also used. devices, the principles of operation of which are based on the properties of G.

Lit.: Bulgakov B.V., Applied theory of gyroscopes, 3rd ed., M., 1976; Nikolay E. L., Gyroscope in gimbals, 2nd ed., M., 1964; Maleev P. I., New types of gyroscopes, L., 1971; Magnus K., Gyroscope. Theory and Application, trans. from German, M., 1974; Ishlinsky A. Yu, Orientation, gyroscopes and inertial navigation, M., 1976; his, Mechanics of relative motion and inertia forces, M., 1981; Klimov D. M., Kharlamov S. A., Dynamics of a gyroscope in a gimbal suspension, M., 1978; Zhuravlev V. F., Klimov D. M., Wave solid-state gyroscope, M., 1985; Novikov L. 3., Shatalov M. Yu., Mechanics of dynamically tuned gyroscopes, M., 1985.

A. Yu. Ishlinsky.

Fig.91

Fig.90

Fig.89

Gyroscopes. Free gyroscope.

The study of these issues is necessary in the discipline "Machine parts".

A gyroscope is a massive axially symmetrical body rotating at a high angular velocity around its axis of symmetry.

In this case, the moments of all external forces, including the force of gravity, relative to the center of mass of the gyroscope are equal to zero. This can be realized, for example, by placing the gyroscope in the gimbals shown in Fig. 89.

and the angular momentum is conserved:

The gyroscope behaves in the same way as a freer body of revolution. Depending on the initial conditions, two options for the behavior of the gyroscope are possible:

1. If the gyroscope is spun around the axis of symmetry, then the directions of the angular momentum and angular velocity coincide:

and the direction of the axis of symmetry of the gyroscope remains unchanged. This can be verified by turning the stand on which the gimbal is located - with arbitrary turns of the stand, the axis of the gyroscope retains the same direction in space. For the same reason, the top, "launched" on a sheet of cardboard and thrown up (Fig. 90), retains the direction of its axis during the flight, and, falling with its tip onto the cardboard, continues to rotate steadily until its kinetic energy is used up.

A free gyroscope, spun around the axis of symmetry, has a very significant stability. It follows from the basic equation of moments that the change in angular momentum

If the time interval is small, then it is also small, that is, with short-term influences of even very large forces, the movement of the gyroscope changes insignificantly. The gyroscope, as it were, resists attempts to change its angular momentum and seems to be "hardened".

Let us take a cone-shaped gyroscope resting on a support rod at its center of mass O (Fig. 91). If the body of the gyroscope does not rotate, then it is in a state of indifferent equilibrium, and the slightest push moves it from its place. If this body is brought into rapid rotation around its axis, then even strong blows with a wooden hammer will not be able to significantly change the direction of the gyroscope axis in space. The stability of a free gyroscope is used in various technical devices, for example, in an autopilot.

2. If the free gyroscope is spun in such a way that the instantaneous angular velocity vector and the axis of symmetry of the gyroscope do not coincide (as a rule, this mismatch is insignificant during fast rotation), then a movement described as "free regular precession" is observed. In relation to the gyroscope, it is called nutation. In this case, the axis of symmetry of the gyroscope, the vectors and lie in the same plane, which rotates around the direction with an angular velocity equal to where is the moment of inertia of the gyroscope relative to the main central axis perpendicular to the axis of symmetry. This angular velocity (let's call it the nutation rate) during the rapid proper rotation of the gyroscope turns out to be quite large, and the nutation is perceived by the eye as a small jitter of the gyroscope's symmetry axis.

Nutational motion can be easily demonstrated using the gyroscope shown in Fig. 91 - it occurs when a hammer strikes the rod of a gyroscope rotating around its axis. At the same time, the more the gyroscope is spun, the greater its angular momentum - the greater the nutation rate and the "smaller" the jitter of the figure's axis. This experience demonstrates another characteristic feature of nutation - over time, it gradually decreases and disappears. This is a consequence of the inevitable friction in the gyroscope bearing.

Our Earth is a kind of gyroscope, and nutation movement is also characteristic of it. This is due to the fact that the Earth is somewhat flattened from the poles, due to which the moments of inertia about the axis of symmetry and about the axis lying in the equatorial plane are different. At the same time a. In the reference frame associated with the Earth, the axis of rotation moves along the surface of the cone around the Earth's axis of symmetry with angular velocity , that is, it completes one revolution in about 300 days. In fact, due to the non-absolute rigidity of the Earth, this time turns out to be longer - it is about 440 days. At the same time, the distance of the point on the earth's surface through which the axis of rotation passes, from the point through which the axis of symmetry passes (the North Pole), is only a few meters. The nutational motion of the Earth does not die out - apparently, it is supported by seasonal changes occurring on the surface

Let us now consider the situation when a force is applied to the axis of the gyroscope, the line of action of which does not pass through the anchor point. Experiments show that in this case the gyroscope behaves in a very unusual way.

If a spring is attached to the axis of a gyroscope hinged at point O (Fig. 92) and pulled upwards with a force, then the axis of the gyroscope will not move in the direction of the force, but perpendicular to it, sideways. This movement is called the precession of the gyroscope under the action of an external force.

The purpose of the work: to study the features of the movement of a gyroscope under the action of the moment of external forces, the measurement of the angular velocity of the precession and the angular momentum of the gyroscope

A gyroscope is a symmetrical rigid body that rapidly rotates around an axis of symmetry, which can change its direction in space.

For demonstration purposes, gyroscopes of the design, which is schematically shown in Fig. 6.1. Gyro wheel To(rotor) is mounted on an axis that can rotate both around a horizontal axis and around a vertical axis, i.e. can take any position in space. (The vertical axis deviations in this design are limited to not very large angles). In order for the moment of gravity relative to the three axes of the gyroscope to be equal to zero, the center of gravity of the gyroscope must coincide with the intersection point of the three axes of rotation. The gyroscope rotor is driven into rapid rotation by an electric motor.

Rice. 6.1. Experience Scheme

Since the moment of gravity relative to the point O is equal to zero, the axis of the rotating gyroscope remains stationary in the absence of any other external forces. The gyroscope has a constant angular momentum directed along the fixed axis of rotation of the gyroscope. If external forces begin to act on the gyroscope, then the axis of the gyroscope begins to move - rotation around other axes appears. Then it no longer coincides with the axis of the gyroscope, but always remains close to it. Therefore, knowing how the vector changes, we can tell how approximately the axis of the gyroscope moves.

The rotation of a rigid body is determined by the equation

Here is the moment of external forces, = I, where I is the moment of inertia of the gyroscope, and is its angular velocity. Equation (6.1) shows that the vector changes only when the torque is acting. Therefore, the axis of the gyroscope can move appreciably only as long as there is a moment that changes direction. Changes in short time intervals according to equation (6.1) are determined by the relation

With a short-term action of external forces (a sharp blow), it is small, therefore, and? little - almost does not change. Consequently, the direction of the gyroscope axis should also change very little. Indeed, with a sharp impact, the axis of the gyroscope does not go far, but trembles, remaining almost in place. ceases to change after impact. But the axis of the gyroscope should not coincide with the direction, but should only be close to it. It can make small movements around the direction. Such movements of the gyroscope axis about the direction are called nutations. . The trembling of the axis of the gyroscope after the impact is one of the types of nutations.

If the gyroscope rotates around its axis at a very high speed, then even in the presence of slow rotations around other axes, the angular momentum vector practically coincides with the axis of the gyroscope. In what follows, we will assume that the direction coincides with the axis of the gyroscope.

With prolonged exposure to external forces, the vector will change its direction in space. Together with it, it will change its direction and axis of the gyroscope. Direction? coincides with the direction, i.e. not with the direction of the force, but with the direction of the moment of force relative to the axis O. If you press on the gyroscope from the side with some force (Fig. 6.1), then its axis will move not in the direction of the force, but in the direction of the moment of force.

If a force acts on the gyroscope, creating a constant moment , then the direction will change for the same time intervals by the same amount? = ?t. If at the same time lies in the plane of motion of the axis of the gyroscope, then? lies in the same plane; the vector will remain in the same plane and rotate at a constant speed. Together with it, the axis of the gyroscope will also rotate. This movement of the axis is called precession.

The precession of a gyroscope can be demonstrated by hanging a small weight on the axis of the gyroscope m(Fig. 6.1) at a distance r. The force of gravity will create a moment, all the time lying in a horizontal plane. In the presence of a load, the axis of the gyroscope rotates in a horizontal plane at a constant speed.

Let us calculate the angular velocity of rotation of the gyroscope axis.

During? t the axis of the gyroscope is rotated by an angle

Taking into account relation (6.2), for the angular velocity of rotation of the axis (precession velocity) we obtain

Since, a, we rewrite relation (6.3) in the form

It follows from the resulting expression that the smaller the moment of external forces acting on the gyroscope, and the greater the momentum of the gyroscope, the lower the speed of its precession.

If you push the precessing gyroscope in the direction of precession, then the end of the axis on which the weight hangs will rise. On the contrary, if you press on the gyroscope against the direction of precession, then the end of the axis with the load will fall. External forces preventing precession cause the load to sink. During precession, friction forces in the bearing act on the vertical axis, preventing precession, so the axis of the precessing gyroscope does not remain in the horizontal plane - the end of the axis, on which the load hangs, gradually lowers.

The precession of the gyroscope occurs at a constant speed while the external moment is acting, and stops immediately as soon as the external moment disappears. The movement of the gyroscope axis has no inertia. This is due to the fact that the speed of rotation of the axis is determined by the acting forces. Inertia is a manifestation of the fact that accelerations are determined by forces.

In all the described experiments, not only external forces act on the gyroscope, but the gyroscope also acts on those bodies that are the source of these forces. When we press our hand on the axis of the gyroscope, the gyroscope with the same force presses on the hand. If the gyroscope is rigidly connected to a certain body, then with any movement of this body, accompanied by a change in the direction of the axis of the gyroscope, forces arise that act on the body from the side of the gyroscope. These forces often play a prominent role.

For example, the rotating parts of the ship's machines are a gyroscope with a large moment of momentum. When the ship pitches (when the bow of the ship rises and falls), the direction of the angular momentum of the machine changes. As a result, there are pressure forces from the shaft on the bearings. These forces lie in the horizontal plane and rotate the ship around the vertical axis. This "course orientation" is noticeable in small vessels with powerful machines (tugs).

The forces that arise when the direction of the axis of rotation of the gyroscope changes can be used to stabilize the ship (reduce rolling). For this purpose, huge gyroscopes with high speed are used.

All the described properties of the gyroscope are explained by the fact that the movement of the gyroscope axis obeys equation (6.1). The movement of the axis of the gyroscope is determined not by the direction of the force, but by the direction of the moment of external forces. But this moment is determined by the forces acting from the outside on the entire device as a whole, only when the gyroscope is completely free, i.e. when the device design allows any position of the gyroscope axis. If the gyroscope is not completely free, then it is necessary to take into account the moments of those forces that can act on the axis of the gyroscope from the side of the bearings in which it is fixed.

These moments of force can completely change the behavior of the gyroscope under the action of external forces. For example, if you fix the vertical axis and make it possible to rotate the axis of the gyroscope only in the horizontal plane, then it becomes completely "obedient". Under the action of a force applied to the gyroscope in the horizontal plane, the axis of the gyroscope begins to rotate in the direction of the force. This change in the behavior of the gyroscope is explained by the fact that, along with the moment of force, a moment of forces also acts on the axis from the side of the stand in which it is fixed. The occurrence of this moment is easy to explain. Initially, while no force acts on the gyroscope, moments from the side of the stand also do not act on it. The gyroscope "does not know" that it is fixed. Therefore, at first it behaves like a completely free gyroscope: under the action of a force that creates a moment directed vertically upwards, the end of the gyroscope axis begins to rise.

The vertical axis, with which the axis of the gyroscope is rigidly connected, is slightly bent, and there is a moment of elastic forces acting on the axis of the gyroscope. Under the action of this moment, the axis of the gyroscope will move in the horizontal plane just in the direction in which the force acts. Therefore, a non-free gyroscope is “obedient”: its axis turns in the direction where an external force seeks to turn it . In a free gyroscope, the axis rotates in a plane perpendicular to the force.

If a couple of forces are applied to a rotating gyroscope, tending to rotate it about an axis perpendicular to the axis of rotation, then the gyroscope will indeed rotate, but only around the third axis, perpendicular to the first two.

A more detailed analysis of phenomena similar to those described above shows that the gyroscope tends to position its axis of rotation in such a way that it forms the smallest possible angle with the axis of forced rotation, and that both rotations take place in the same direction.

This property of the gyroscope is used in the gyroscopic compass, which is widely used, especially in the navy. The gyrocompass is a rapidly rotating top (three-phase current motor, making 25,000 rpm), which floats on a special float in a vessel with mercury and whose axis is set in the plane of the meridian. In this case, the source of external torque is the daily rotation of the Earth around its axis. Under its action, the axis of rotation of the gyroscope tends to coincide in direction with the axis of rotation of the Earth, and since the rotation of the Earth acts on the gyroscope continuously, the axis of the gyroscope takes this position, i.e. is established along the meridian and continues to remain in it in exactly the same way as an ordinary magnetic needle. Gyroscopic compasses have a number of advantages over magnetic compasses. Their readings are not affected by nearby masses of iron, they are not sensitive to magnetic storms, etc.

Gyroscopes are often used as stabilizers. They are installed to reduce pitching on ocean-going ships. Stabilizers for single-rail railways were also designed; a massive, rapidly rotating gyroscope placed inside a single-rail car prevents the car from tipping over. Rotors for gyroscopic stabilizers are manufactured from 1 to 100 or more tons.

1. Free axes of rotation. Let us consider two cases of rotation of a rigid rod about an axis passing through the center of mass.

If the rod is rotated about the axis OO and leave it to itself, that is, release the axis of rotation from the bearings, then in the case of Fig. 71-a, the orientation of the axis of free rotation relative to the rod will change, since the rod, under the action of a pair of centrifugal forces of inertia, will turn into a horizontal plane. In the case of Fig. 71-b, the moment of a pair of centrifugal forces is equal to zero, so the untwisted rod will continue to rotate around the axis OO and after her release.

The axis of rotation, the position of which in space is maintained without the action of any forces from the outside, is called the free axis of the rotating body. Therefore, the axis perpendicular to the rod and passing through its center of mass is the free axis of rotation of the rod.

Any rigid body has three mutually perpendicular free rotation axes intersecting at the center of mass. The position of the free axes for homogeneous bodies coincides with the position of their geometric axes of symmetry (Fig. 72).

The parallelepiped has all three axes fixed. The cylinder has only one fixed axis, coinciding with the geometric axis. All three axes of a ball are not fixed.

Free axes of rotation are also called main axes of inertia. With free rotation of bodies around the main axes of inertia, only rotations around those axes that correspond to the maximum and minimum values ​​of the moment of inertia are stable. If external forces act on the body, then rotation is stable only around that main axis, which corresponds to the maximum moment of inertia.

2. Gyroscope(from Greek gyreuo- rotate and skopeo- I see) - a homogeneous body of revolution rapidly rotating around the axis of symmetry, the axis of which can change its position in space.

When studying the motion of a gyroscope, we consider that:

a. The center of mass of the gyroscope coincides with its fixed point O. Such a gyroscope is called balanced.

b. Angular velocity w the rotation of the gyroscope around the axis is much greater than the angular velocity W of the movement of the axis in space, that is w >> W.

B. Gyroscope angular momentum vector L coincides with the angular velocity vector w , since the gyroscope rotates around the main axis of inertia.

Let the force act on the axis of the gyroscope F during time D t. According to the second law of dynamics for rotational motion , so that the change in the angular momentum of the gyroscope over this time , (26.1)

where r is the radius vector drawn from the fixed point O to the point of action of the force (Fig. 73).

The change in the angular momentum of the gyroscope can be considered as the rotation of the gyroscope axis by an angle with the angular velocity . (26.2)

Here, is the component of the force acting on it, normal to the axis of the gyroscope.

Under the force F applied to the axis of the gyroscope, the axis rotates not in the direction of the force, but in the direction of the moment of force M relative to a fixed point O. At any moment of time, the rate of rotation of the gyroscope axis is proportional in magnitude to the moment of force, and with a constant arm of the force, it is proportional to the force itself. Thus, movement of the gyroscope axis without inertia. This is the only case of inertial motion in mechanics.

The movement of the gyroscope axis under the action of an external force is called forced precession gyroscope (from the Latin praecessio - moving ahead).

3. Impact action on the axis of the gyroscope. Let us determine the angular displacement of the gyroscope axis as a result of a short-term action of a force on the axis, that is, an impact. Let for a short time dt on the axis of the gyroscope at a distance r from the center O force is acting F . Under the influence of this force F dt the axis rotates (Fig. 74) in the direction of the impulse of the moment of force created by it M dt to some angle

dq = W dt=(rF/Iw)dt. (26.3)

If the point of application of the force does not change, then r= const and when integrating we get. q = .(26.4)

The integral in each case depends on the type of function ( t). Under normal conditions, the angular velocity of rotation of the gyroscope is very high, so the numerator is often much less than the denominator, and therefore the angle q- small value. A rapidly spinning gyroscope is shock-resistant, the more so the greater its angular momentum.

4. Interestingly, the force under which the axis of the gyroscope precesses does not do work. This is because the point of the gyroscope, to which the force is applied, at any moment is displaced in the direction perpendicular to the direction of the force. Therefore, the scalar product of the force and the small displacement vector is always zero.

Forces in this manifestation are called gyroscopic. So, the Lorentz force is always gyroscopic, acting on an electrically charged particle from the magnetic field in which it moves.

5. The equilibrium condition for TP. For the HP to be in equilibrium, it is necessary that the sum of the external forces and the sum of the moments of the external forces be equal to zero:

. (26.5)

There are 4 types of balance: stable, unstable, saddle-shaped and indifferent.

a. The equilibrium position of the TT is stable if, with small deviations from equilibrium, forces begin to act on the body, tending to return it to the equilibrium position.

Figure 75 shows situations of stable equilibrium of bodies in the field of gravity. The forces of gravity are mass forces, therefore the resultant of the forces of gravity acting on the point elements of the HP is applied to the center of mass. In such situations, the center of mass is called the center of gravity.

A stable position of equilibrium corresponds to a minimum potential energy of the body.

b. If, with small deviations from the equilibrium position, forces begin to act on the body in the direction from the equilibrium, then the equilibrium position is unstable. The unstable position of equilibrium corresponds to the relative maximum potential energy of the body (Fig. 76).

in. Saddle-shaped is such an equilibrium when, when moving along one degree of freedom, the balance of the body is stable, and when moving along another degree of freedom, it is unstable. In the situation shown in Figure 77, the position of the body with respect to the coordinate x is stable, and with respect to the coordinate y- unstable.

G. If, when the body deviates from the equilibrium position, no forces arise that tend to displace the body in one direction or another, then the equilibrium position is called indifferent. For example, a ball in the field of gravity on an equipotential surface, a rigid body suspended at the point of the center of mass (at the point of the center of gravity) (Fig. 78).

In those cases when the body rests on a support, then the larger the area of ​​\u200b\u200bthe support and the lower the center of gravity, the more stable the balance of the body (Fig. 79).