« Physics - Grade 10 "

Why, to increase the angular speed of rotation, the skater stretches along the axis of rotation.
Should the helicopter rotate when its propeller rotates?

The questions asked suggest that if external forces do not act on the body or their action is compensated and one part of the body begins to rotate in one direction, then the other part should rotate in the other direction, just as when fuel is ejected from a rocket, the rocket itself moves in the opposite direction.

Moment of impulse.

If we consider a rotating disk, it becomes obvious that the total impulse of the disk is equal to zero, since any particle of the body corresponds to a particle moving with a velocity equal in magnitude, but in the opposite direction (Fig. 6.9).

But the disk is moving, the angular velocity of rotation of all particles is the same. However, it is clear that the farther a particle is from the axis of rotation, the greater its momentum. Consequently, for rotary motion, it is necessary to introduce one more characteristic similar to an impulse - the angular momentum.

The moment of momentum of a particle moving in a circle is called the product of the momentum of a particle by the distance from it to the axis of rotation (Fig. 6.10):

The linear and angular velocities are related by the relation v = ωr, then

All points of a solid matter move relative to a fixed axis of rotation with the same angular velocity. A solid body can be represented as a collection of material points.

The moment of momentum of a rigid body is equal to the product of the moment of inertia and the angular velocity of rotation:

The angular momentum is a vector quantity, according to the formula (6.3) the angular momentum is directed in the same way as the angular velocity.

The basic equation of the dynamics of rotational motion in impulse form.

The angular acceleration of a body is equal to the change in angular velocity divided by the time interval during which this change occurred: Substitute this expression into the basic equation of the dynamics of rotational motion hence I (ω 2 - ω 1) = MΔt, or IΔω = MΔt.

Thus,

ΔL = MΔt. (6.4)

The change in the angular momentum is equal to the product of the total moment of forces acting on a body or system by the time of action of these forces.

The law of conservation of angular momentum:

If the total moment of forces acting on a body or a system of bodies with a fixed axis of rotation is equal to zero, then the change in the angular momentum is also equal to zero, i.e., the angular momentum of the system remains constant.

ΔL = 0, L = const.

The change in the impulse of the system is equal to the total impulse of the forces acting on the system.

The rotating skater spreads his arms to the sides, thereby increasing the moment of inertia in order to reduce the angular speed of rotation.

The law of conservation of angular momentum can be demonstrated using the following experiment, called the "experiment with the Zhukovsky bench." A person stands on a bench with a vertical axis of rotation passing through its center. A man holds dumbbells in his hands. If the bench is made to rotate, then the person can change the speed of rotation by pressing the dumbbells to the chest or lowering the arms, and then spreading them. By spreading his arms, he increases the moment of inertia, and the angular speed of rotation decreases (Figure 6.11, a), lowering his arms, he decreases the moment of inertia, and the angular speed of rotation of the bench increases (Figure 6.11, b).

A person can also make the bench spin by walking along the edge. In this case, the bench will rotate in the opposite direction, since the total angular momentum should remain equal to zero.

The principle of operation of devices called gyroscopes is based on the law of conservation of angular momentum. The main property of a gyroscope is the preservation of the direction of the axis of rotation if external forces do not act on this axis. In the XIX century. gyroscopes were used by sailors for orientation at sea.

Kinetic energy of a rotating solid.

The kinetic energy of a rotating solid is equal to the sum of the kinetic energies of its individual particles. Let's divide the body into small elements, each of which can be considered a material point. Then the kinetic energy of the body is equal to the sum of the kinetic energies of the material points of which it consists:

The angular velocity of rotation of all points of the body is the same, therefore,

The value in brackets, as we already know, is the moment of inertia of a rigid body. Finally, the formula for the kinetic energy of a rigid body with a fixed axis of rotation has the form

In the general case of motion of a rigid body, when the axis of rotation is free, its kinetic energy is equal to the sum of the energies of translational and rotational motions. So, the kinetic energy of a wheel, the mass of which is concentrated in the rim, rolling along the road at a constant speed, is equal to

The table compares the formulas of the mechanics of the translational motion of a material point with similar formulas for the rotational motion of a rigid body.

If m.t. rotates in a circle, then a force acts on it, then when turning through a certain angle, elementary work is performed:

(22)

If the acting force is potential, then

then (24)

Rotating power

Instantaneous power generated by rotating the body:

Kinetic energy of a rotating body

Kinetic energy of a material point. Kinetic energy sis of material points ... Because , we obtain the expression for the kinetic energy of rotation:

In a plane motion (the cylinder rolls along an inclined plane), the total speed is:

where is the velocity of the center of mass of the cylinder.

The total is equal to the sum of the kinetic energy of the translational motion of its center of mass and the kinetic energy of the rotational motion of the body relative to the center of mass, i.e.:

(28)

Conclusion:

And now, having considered all the lecture material, let us summarize, compare the values ​​and equations of the rotational and translational motion of the body:

Annex 1:

A person stands in the center of Zhukovsky's bench and rotates with it by inertia. Rotation frequency n 1 = 0.5 s -1. Moment of inertia j o the human body is

Specifically, the axis of rotation is 1.6 kg m 2. In arms outstretched to the sides, a person holds a weight on a mass m= 2 kg each. Distance between weights l 1 = l, 6 m. Determine the speed n 2 , benches with a man when he lowers his arms and distance l 2 between the weights becomes equal to 0.4 m. Neglect the moment of inertia of the bench.

Symmetry properties and conservation laws.

Energy saving.

The conservation laws considered in mechanics are based on the properties of space and time.

Conservation of energy is associated with the homogeneity of time, conservation of momentum - with the homogeneity of space, and, finally, conservation of angular momentum is associated with the isotropy of space.

We start with the law of conservation of energy. Let the system of particles be in constant conditions (this is the case if the system is closed or exposed to a constant external force field); connections (if any) are ideal and stationary. In this case time, due to its homogeneity, cannot appear explicitly in the Lagrange function. Really homogeneity means the equivalence of all points in time. Therefore, replacing one moment in time with another without changing the values ​​of the coordinates and velocities of the particles should not change the mechanical properties of the system. This is, of course, true if the replacement of one moment in time by another does not change the conditions in which the system is located, that is, in the case of the time independence of the external field (in particular, this field may be absent).

So for a closed system located in a closed force field,.

Rotational work. Moment of power

Consider the work performed when a material point rotates around a circle under the action of the projection of the acting force on the displacement (tangential component of the force). In accordance with (3.1) and Fig. 4.4, passing from the parameters of the translational motion to the parameters of the rotary motion (dS = R dcp)

Here the concept of the moment of force about the axis of rotation OOi is introduced as the product of the force F s on the shoulder of the force R:

As can be seen from relation (4.8), moment of force in rotational motion is analogous to force in translational motion, since both parameters, when multiplied by analogs dcp and dS give work. Obviously, the moment of force must also be specified vector, and with respect to the point O, its definition is given through the vector product and has the form

Finally: work in rotary motion is equal to the scalar product of the moment of force and angular displacement:

Rotational kinetic energy. Moment of inertia

Consider an absolutely rigid body rotating about a fixed axis. Let us mentally break this body into infinitely small pieces with infinitesimal dimensions and masses mi, m2, W3 ... located at a distance R b R 2, R3 ... from the axis. We find the kinetic energy of a rotating body as the sum of the kinetic energies of its small parts

where Y is the moment of inertia of a rigid body, relative to a given axis OOj.

From a comparison of the formulas for the kinetic energy of translational and rotational motions, it can be seen that the moment of inertia in rotary motion is analogous to mass in translational motion. Formula (4.12) is convenient for calculating the moment of inertia of systems consisting of individual material points. To calculate the moment of inertia of solid bodies, using the definition of the integral, one can transform (4.12) to the form

It is easy to see that the moment of inertia depends on the choice of the axis and changes with its parallel translation and rotation. Let us give the values ​​of the moments of inertia for some homogeneous bodies.

It is seen from (4.12) that moment of inertia of a material point is equal to

where T- point mass;

R- distance to the axis of rotation.

It is easy to calculate the moment of inertia and for hollow thin-walled cylinder(or a special case of a cylinder with a low height - thin ring) radius R relative to the axis of symmetry. The distance to the axis of rotation of all points for such a body is the same, equal to the radius and can be taken out from under the sum sign (4.12):

Solid cylinder(or a special case of a cylinder with a low height - disk) radius R for calculating the moment of inertia about the axis of symmetry requires calculating the integral (4.13). The mass in this case is, on average, concentrated somewhat closer than in the case of a hollow cylinder, and the formula will be similar to (4.15), but a coefficient less than one will appear in it. Let's find this coefficient.

Let a solid cylinder have density R and height h. Let's break it down into

hollow cylinders (thin cylindrical surfaces) thick dr(Figure 4.5) shows a projection perpendicular to the axis of symmetry). The volume of such a hollow cylinder of radius G equal to the surface area times the thickness: weight: and the moment

inertia according to (4.15): Total moment

of inertia of a solid cylinder is obtained by integrating (summing) the moments of inertia of hollow cylinders:

... Taking into account that the mass of a solid cylinder is related to

density by the formula T = 7iR 2 hp we finally have the moment of inertia of a solid cylinder:

Similarly, it is searched for moment of inertia of a thin bar length L and masses T, if the axis of rotation is perpendicular to the bar and passes through its middle. We split such a rod in accordance with Fig. 4.6

into pieces thick dl. The mass of such a piece is dm = m dl / L, and the moment of inertia according to Gender

the moment of inertia of a thin rod is obtained by integrating (summing) the moments of inertia of the pieces:

Consider an absolutely rigid body rotating around a fixed axis. If you mentally break this body into n points by masses m 1, m 2, ..., m n located at distances r 1, r 2,…, r n from the axis of rotation, then during rotation they will describe circles and move at different linear speeds v 1, v 2, ..., v n... Since the body is absolutely rigid, the angular velocity of rotation of the points will be the same:

The kinetic energy of a rotating body is the sum of the kinetic energies of its points, i.e.

Considering the relationship between angular and linear velocities, we get:

Comparison of formula (4.9) with the expression for the kinetic energy of a body moving translationally with speed v, shows that the moment of inertia is a measure of the body's inertia in rotational motion.
If a rigid body moves translationally with speed v and simultaneously rotates with an angular velocity ω around an axis passing through its center of inertia, then its kinetic energy is determined as the sum of two components:

(4.10)

where v c- the speed of the center of mass of the body; J c- moment of inertia of a body about an axis passing through its center of mass.
The moment of force about a fixed axis z called the scalar quantity M z equal to the projection on this axis of the vector M moment of force, defined relative to an arbitrary point 0 of the given axis. The value of the moment M z does not depend on the choice of the position of point 0 on the axis z.
If the axis z coincides with the direction of the vector M, then the moment of force is represented as a vector coinciding with the axis:

M z = [ rF] z
Let's find an expression for work when the body rotates. Let the force F applied to point B located at a distance from the axis of rotation r(fig. 4.6); α is the angle between the direction of the force and the radius vector r... Since the body is absolutely solid, the work of this force is equal to the work spent on turning the whole body.

When turning the body through an infinitely small angle dφ application point B passes the path ds = rdφ, and the work is equal to the product of the projection of the force and the direction of displacement by the amount of displacement:

dA = Fsinα * rdφ
Considering that Frsinα = M z can be written dA = M z dφ, where M z- moment of force about the axis of rotation. Thus, the work during the rotation of the body is equal to the product of the moment of the acting force and the angle of rotation.
Work during the rotation of the body goes to increase its kinetic energy:

dA = dE k
(4.11)

Equation (4.11) is the equation of the dynamics of the rotational motion of a rigid body about a fixed axis.

Work and power when rotating a rigid body.

Let's find an expression for work when the body rotates. Let the force be applied at a point located at a distance from the axis - the angle between the direction of the force and the radius vector. Since the body is absolutely solid, the work of this force is equal to the work spent on turning the whole body. When the body is rotated through an infinitely small angle, the point of application passes the path and the work is equal to the product of the projection of the force by the direction of displacement by the amount of displacement:

The modulus of the moment of force is:

then we get the following formula for calculating the work:

Thus, the work during the rotation of a rigid body is equal to the product of the moment of the acting force and the angle of rotation.

Kinetic energy of a rotating body.

Moment of inertia mat.t. called physical value numerically equal to the product of the mass of the mat.t. by the square of the distance of this point to the axis of rotation W ki = mi V 2 i / 2 V i -Wr i Wi = miw 2 r 2 i / 2 = w 2/2 * miri 2 I i = mir 2 i moment of inertia of a rigid body is equal to the sum of all mat.t I = S imir 2 i the moment of inertia of a rigid body is called. physical quantity equal to the sum of products of mat.t. into the squares of the distances from these points to the axis. W i -I i W 2/2 W k = IW 2/2

W k = S i W ki moment of inertia during rotary motion yavl. analogue of the mass in translational motion. I = mR 2/2

21. Non-inertial frames of reference. Forces of inertia. Equivalence principle. Equation of motion in non-inertial frames of reference.

Non-inertial frame of reference- an arbitrary frame of reference that is not inertial. Examples of non-inertial reference systems: a system moving in a straight line with constant acceleration, as well as a rotating system.

When considering the equations of motion of a body in a non-inertial frame of reference, it is necessary to take into account additional forces of inertia. Newton's laws are fulfilled only in inertial frames of reference. In order to find the equation of motion in a non-inertial frame of reference, it is necessary to know the laws of transformation of forces and accelerations during the transition from an inertial frame to any non-inertial one.

Classical mechanics postulates the following two principles:

time is absolute, that is, the time intervals between any two events are the same in all arbitrarily moving frames of reference;

space is absolute, that is, the distance between any two material points is the same in all arbitrarily moving frames of reference.

These two principles make it possible to write down the equation of motion of a material point relative to any non-inertial frame of reference in which Newton's First Law is not fulfilled.

The basic equation of the dynamics of the relative motion of a material point has the form:

where is the mass of the body, is the acceleration of the body relative to the non-inertial frame of reference, is the sum of all external forces acting on the body, is the translational acceleration of the body, is the Coriolis acceleration of the body.

This equation can be written in the usual form of Newton's Second Law, if we introduce fictitious forces of inertia:

Transportable inertia force

Coriolis force

Force of inertia- a fictitious force that can be introduced in a non-inertial frame of reference so that the laws of mechanics in it coincide with the laws of inertial systems.

In mathematical calculations, the introduction of this force occurs by transforming the equation

F 1 + F 2 +… F n = ma to the form

F 1 + F 2 +… F n –ma = 0 Where F i is the actual force, and –ma is the “force of inertia”.

Among the forces of inertia, the following are distinguished:

simple force of inertia;

centrifugal force, which explains the tendency of bodies to fly away from the center in rotating frames of reference;

the Coriolis force, which explains the tendency of bodies to leave the radius during radial motion in rotating frames of reference;

From the point of view of general relativity, gravitational forces at any point are the forces of inertia at a given point in Einstein's curved space

Centrifugal force- the force of inertia, which is introduced in a rotating (non-inertial) frame of reference (to apply Newton's laws, calculated only for inertial FRs) and which is directed from the axis of rotation (hence the name).

The principle of equivalence of the forces of gravity and inertia- a heuristic principle used by Albert Einstein in the derivation of the general theory of relativity. One of the versions of its presentation: “The forces of gravitational interaction are proportional to the gravitational mass of the body, while the forces of inertia are proportional to the inertial mass of the body. If the inertial and gravitational masses are equal, then it is impossible to distinguish which force acts on a given body - gravitational or inertial force. "

Einstein's formulation

Historically, the principle of relativity was formulated by Einstein as follows:

All phenomena in a gravitational field occur in the same way as in the corresponding field of inertial forces, if the intensities of these fields coincide and the initial conditions for the bodies of the system are the same.

22. Galileo's principle of relativity. Galileo's transformations. The classical theorem of addition of velocities. Invariance of Newton's laws in inertial reference frames.

Galileo's principle of relativity- this is the principle of physical equality of inertial reference systems in classical mechanics, manifested in the fact that the laws of mechanics in all such systems are the same.

Mathematically, Galileo's principle of relativity expresses the invariance (invariance) of the equations of mechanics with respect to transformations of coordinates of moving points (and time) during the transition from one inertial system to another - Galileo's transformations.
Let there are two inertial reference frames, one of which, S, we agree to consider at rest; the second system, S ", moves with respect to S with a constant velocity u as shown in the figure. Then Galileo's transformations for the coordinates of a material point in the S and S" systems will have the form:
x "= x - ut, y" = y, z "= z, t" = t (1)
(Shaded quantities refer to the S "frame, unhatched ones - to S.) Thus, time in classical mechanics, like the distance between any fixed points, is considered the same in all reference frames.
From Galileo's transformations, one can obtain the relationship between the velocities of a point and its accelerations in both systems:
v "= v - u, (2)
a "= a.
In classical mechanics, the motion of a material point is determined by Newton's second law:
F = ma, (3)
where m is the mass of a point, and F is the resultant of all forces applied to it.
In this case, forces (and masses) are invariants in classical mechanics, that is, quantities that do not change when passing from one frame of reference to another.
Therefore, under the Galileo transformations, Eq. (3) does not change.
This is the mathematical expression of the Galilean principle of relativity.

GALILEY TRANSFORMATION.

In kinematics, all frames of reference are equal to each other and motion can be described in any of them. When studying motions, sometimes it is necessary to switch from one frame of reference (with the coordinate system ОХУZ) to another - (O`X`Y`Z`). Let us consider the case when the second frame of reference moves relative to the first uniformly and rectilinearly with a speed V = сnst.

To facilitate the mathematical description, we assume that the corresponding coordinate axes are parallel to each other, that the velocity is directed along the X-axis, and that at the initial moment of time (t = 0) the origin of coordinates of both systems coincided with each other. Using the assumption, which is valid in classical physics, about the same flow of time in both systems, it is possible to write down the relations connecting the coordinates of some point A (x, y, z) and A (x`, y`, z`) in both systems. Such a transition from one frame of reference to another is called Galileo's transformations):

ОХУZ О`Х`У`Z`

x = x` + V x t x` = x - V x t

x = v` x + V x v` x = v x - V x

a x = a` x a` x = a x

Acceleration in both systems is the same (V = const). The deep meaning of Galileo's transformations will be clarified in dynamics. The transformation of Galileo's velocities reflects the principle of the independence of displacements taking place in classical physics.

Addition of speeds in service stations

The classical law of addition of velocities cannot be true, because it contradicts the statement about the constancy of the speed of light in a vacuum. If the train is moving at a speed v and a light wave propagates in the car in the direction of the train movement, then its speed is relative to the Earth anyway c, but not v + c.

Consider two frames of reference.

In system K 0 body moves with speed v 1 . Regarding the system K it moves with speed v 2. According to the law of addition of speeds in the service station:

If v<<c and v 1 << c, then the term can be neglected, and then we get the classical law of addition of velocities: v 2 = v 1 + v.

At v 1 = c speed v 2 is equal c, as required by the second postulate of the theory of relativity:

At v 1 = c and at v = c speed v 2 is again equal to speed c.

A remarkable property of the law of addition is that at any speed v 1 and v(not more c), the resulting speed v 2 does not exceed c... The speed of movement of real bodies is more than the speed of light is impossible.

Speed ​​addition

When considering a complex motion (that is, when a point or a body moves in one frame of reference, and it moves relative to another), the question arises about the relationship of velocities in two frames of reference.

Classic mechanics

In classical mechanics, the absolute speed of a point is equal to the vector sum of its relative and portable speeds:

In simple language: The speed of movement of a body relative to a stationary frame of reference is equal to the vector sum of the speed of this body relative to the moving frame of reference and the speed of the most moving frame of reference relative to the stationary frame.