Momentum material point relative to some center O is equal to vector product radius vector of a moving point by the amount of motion, i.e.
Obviously, the modulus of angular momentum is equal to
where is the arm of the vector v relative to the center O (Fig. 167).
Projecting the vector equality (153) onto coordinate axes, passing through the center O, we obtain formulas for the moments of momentum of a material point relative to these axes:
IN vector form the theorem on angular momentum is expressed as follows: the time derivative of the angular momentum of a material point relative to any fixed center O is equal to the angular momentum acting force relative to the same center, i.e.
Projecting vector equality (156) onto any of the coordinate axes passing through the center O, we obtain an equation expressing the same theorem in scalar form:
i.e., the time derivative of the moment of momentum of a material point relative to any fixed axis equal to the moment of the acting force relative to the same axis.
This theorem is of great importance when solving problems in the case of a point moving under the influence of a central force. A central force is a force whose line of action always passes through the same point, called the center of this force. If a material point moves under the influence of a central force F with a center at point O, then
and therefore . Thus, the angular momentum in in this case remains constant in magnitude and direction. It follows that a material point under the action of a central force describes a flat curve located in a plane passing through the center of the force.
If the trajectory that a point describes under the action of a central force is known, then, using the angular momentum theorem, one can find this force as a function of the distance from the point to the center of the force.
Indeed, since the angular momentum relative to the center of force remains constant, then, denoting h the arm of the vector relative to the center of force, we have:
(158)
To determine this constant, the speed of a point at some point on the trajectory must be known. On the other hand, we have (Fig. 168):
where is the radius of curvature of the trajectory, is the angle between the radius vector of the point and the tangent to the trajectory at this point.
So, we have two equations (158) and (159) with two unknowns v and F; the remaining quantities included in these equations, i.e., being elements of a given trajectory, can easily be found. Thus, v and F can be found as functions of .
Example 129. Point M describes an ellipse under the action of a central force F (Fig. 169). The speed at vertex A is . Find the speed at vertex B if and .
Solution. Since in this case
Example 130. Point M of mass describes a circle of radius a, being attracted by point A of this circle (Fig. 170).
angular momentum
MOMENTUM OF MOMENTUM (kinetic torque, angular momentum, angular momentum) measure mechanical movement a body or system of bodies relative to some center (point) or axis. To calculate the moment of momentum K of a material point (body), the same formulas are valid as for calculating the moment of force, if you replace the force vector in them with the vector of momentum mv, in particular K0 = . The sum of the angular momentum of all points of the system relative to the center (axis) is called the principal angular momentum of the system (kinetic moment) relative to this center (axis). During rotational movement solid main point momentum relative to the axis of rotation z of the body is expressed by the product of the moment of inertia Iz by angular velocity? bodies, i.e. КZ = Iz?.
Momentum
kinetic moment, one of the measures of mechanical motion of a material point or system. Especially important role MKD plays in the study of rotational motion. As with the moment of force, a distinction is made between mechanical action relative to the center (point) and relative to the axis.
To calculate the mechanical efficiency k of a material point relative to the center O or the z axis, all the formulas given for calculating the moment of force are valid if the vector F is replaced by the momentum vector mv. Thus, ko = , where r ≈ radius vector of the moving point drawn from the center O, and kz is equal to the projection of the vector ko onto the z axis passing through the point O. The change in the M. efficiency of the point occurs under the influence of the moment mo (F) of the applied force and is determined by the theorem on the change in M. efficiency, expressed by the equation dko/dt = mo(F). When mo(F) = 0, which, for example, is the case for central forces, the movement of a point obeys the area law. This result is important for celestial mechanics, theories of motion artificial satellites Earth, space aircraft and etc.
Main M. k. d. (or kinetic moment) mechanical system relative to the center O or the z axis is equal to the geometric or algebraic sum M. coefficient of all points of the system relative to the same center or axis, i.e. Ko = Skoi, Kz = Skzi. Vector Ko can be determined by its projections Kx, Ky, Kz onto the coordinate axes. For a body rotating around a stationary axis z with angular velocity w, Kx = ≈ Ixzw, Ky = ≈Iyzw, Kz = Izw, where lz ≈ axial, and Ixz, lyz ≈ centrifugal moments of inertia. If the z axis is main axis inertia for the origin O, then Ko = Izw.
A change in the main M. efficiency of the system occurs under the influence only external forces and depends on their main point Moe. This dependence is determined by the theorem on the change in the main M. efficiency of the system, expressed by the equation dKo/dt = Moe. A similar equation relates the moments Kz and Mze. If Moe = 0 or Mze = 0, then, respectively, Ko or Kz will be constant quantities, i.e., the law of conservation of the MQD holds (see Conservation laws). That., internal forces cannot change the M. k.d. system, but M. k. d. individual parts systems or angular velocities can change under the influence of these forces. For example, when rotating around vertical axis z of a figure skater (or ballerina), the value Kz = Izw will be constant, since practically Mze = 0. But by changing the value of the moment of inertia lz with the movement of his arms or legs, he can change the angular velocity w. Dr. An example of the fulfillment of the law of conservation of mechanical efficiency is the appearance of a reactive torque in an engine with a rotating shaft (rotor). The concept of mechanical dynamics is widely used in rigid body dynamics, especially in the theory of the gyroscope.
Dimension of M. k.d. ≈ L2MT-1, units of measurement ≈ kg×m2/sec, g×cm2/sec. MKDs also have electromagnetic, gravitational, etc. physical fields. Most elementary particles is inherent in its own, internal M. k.d. ≈ spin. Great importance M. Q.D. has in quantum mechanics.
Lit. see under art. Mechanics.
The amount of motion (mV) is a vector quantity, i.e. has a certain direction relative to a selected reference point (for example, a coordinate axis) or an axis of rotation. Basic equation for the dynamics of rotational motion
can also be written in the form
Here C/oo) has the meaning of analogue physical quantity (mV) amount of movement. Power moment M = Ph then, taking into account (7.14)
Size L can be considered as angular momentum (mV) relative to a given point or axis. It is called kinetic moment. Here h- shortest distance from the line of action of the vector mV clockwise. IN general case
The “-” sign is taken in case of vector rotation mV clockwise.
For a spatial system, the angular momentum of a material point relative to an axis perpendicular to a given plane and passing through given point 0, equal to the projection angular momentum. For example, for the z axis: Lz = L 0 cos a, where a is the angle between a given plane and the radius vector of a given point (the distance from the material point to the center “0”).
Magnitude L relative to the rectangular coordinate axes is determined by the projections of velocities on these axes and the coordinates of the moving material
Rice. 7.1.
no point. For example, in the plane xOy(Fig. 7.1) angular momentum about the axis z(perpendicular to this plane)
here L, and L 2 - moments created by momentum projections mV relative to point 0.
By physical meaning derivative - the sum of moments of forces,
acting on a material point relative to the selected coordinate axis. When J M i= 0, value L= const, i.e. if the moment of the resultant force equal to zero , then the angular momentum relative to the selected axis remains constant.
Rice. 7.2.
For example, for point body M with mass T magnitude Lz= 0, if the body is acted upon by a force P directed to the origin of coordinates, since the moments of force R and gravity mg(parallel to the z axis, Fig. 7.2) are equal to zero. Here L z = mxV = const.
If the direction of speed V 0 all the time perpendicular to the radius r, the value of which when moving the point M 2 decreases, then from the equality Lz= const follows an increase in point speed M when approaching point O.
By analogy with the main moment of forces, we can derive the concept: principal moment of momentum i 0 mechanical system(or kinetic moment), relative to a given center, which is equal to geometric sum quantities L 0j all material points of a given system relative to this center, i.e.
Kinetic moment of a mechanical system relative to an axis(for example axes G) equal to the algebraic sum of the moments of momentum of all points of a given system: L 0 = X Liz.
It is obvious that the derivative of the kinetic moment with respect to time is equal to the main moment of external forces acting on a given mechanical system (relative to the selected center), i.e.
This implies the law of conservation of the angular momentum of a mechanical system relative to the axis
those. the kinetic moment in this case remains constant.
Changes in the kinetic moment of a mechanical system upon impact follows as a consequence of the consideration of the above concepts about force impulse and angular momentum and is determined by expressions (7.17) and (7.18). So, for example, during an impact, the change in the kinetic moment of the system relative to any axis is equal to the sum of the moments of external force impulses relative to a given axis. If only internal force impulses are applied to the points of a mechanical system, then the kinetic moment of the system upon impact does not change.
To calculate M. efficiency. k material point relative to the center ABOUT or axes z All formulas given for calculating the moment of force are valid if the vector is replaced in them F momentum vector mv. That., k o = [ r · mυ], Where r- radius vector of a moving point drawn from the center ABOUT,a k z equals the projection of the vector k o per axis z, passing through the point ABOUT. The change in the M. efficiency of a point occurs under the influence of the moment m o(F) of the applied force and is determined by the theorem on the change in mechanical efficiency, expressed by the equation dk o /dt = m o(F). When m o(F) = 0, which, for example, is the case for central forces, the motion of a point obeys the Area law.
Chief M.K.D. (or kinetic moment) of a mechanical system relative to the center ABOUT or axes z equal, respectively, to the geometric or algebraic sum of the M. efficiency of all points of the system relative to the same center or axis, i.e. K o = Σ k oi, K z = Σ k zi. Vector K o can be determined by its projections K x , K y , K z to the coordinate axes. For a body rotating around a fixed axis z with angular velocity ω, K x = - I xz ω, K y = - I yz ω, K z = I z ω, where l z- axial, and I xz, l yz- centrifugal moments of inertia.
If the axis z is the principal axis of inertia for the origin ABOUT, That K o = I z ω.
A change in the main mechanical efficiency of a system occurs under the influence of only external forces and depends on their main moment M o e. This dependence is determined by the theorem on the change in the main M. efficiency of the system, expressed by the equation dK o /dt = M o e. A similar equation relates the moments K z And M z e. If M o e= 0 or M z e= 0, then accordingly K o or K z will be constant quantities, i.e., the law of conservation of magnetic efficiency holds.
Ticket 20
General equation speakers.
General equation of dynamics– when the system moves with ideal connections in each this moment times the sum of elementary works of all applied active forces and all inertia forces on any possible movement of the system will be equal to zero. The equation uses the principle of possible displacements and D'Alembert's principle and allows you to compose differential equations of motion of any mechanical system. Gives general method solving dynamics problems. Sequence of compilation: a) the specified forces acting on it are applied to each body, and forces and moments of pairs of inertial forces are also conditionally applied; b) inform the system of possible movements; c) draw up equations for the principle of possible movements, considering the system to be in equilibrium.
Potential power. Job potential strength on final movement.
Potential strength- a force whose work depends only on the initial and final position of the point of its application and does not depend on either the type of trajectory or the law of motion of this point
Potential force work equal to the difference between the values of the force function in the final and starting points does not depend on the path or type of trajectory of the moving point.
The main property of potential force field and the fact is that the work of field forces when a material point moves in it depends only on the initial and final positions of this point and does not depend on the type of its trajectory or on the law of motion.
Ticket 21
The principle of virtual (possible) movements.
There are two different formulations of the principle of possible movements. One formulation states that for equilibrium material system it is necessary that the sum of the elementary works of all external forces applied to the system be equal to zero at any possible displacement.
Another formulation, on the contrary, says that the system must be in equilibrium so that the sum of the elementary works of all forces equals zero. This definition of this principle is given, for example, in the work: “The virtual work of given forces applied to a system with ideal connections and in equilibrium is equal to zero.”
Mathematically, the principle of possible movements is presented as:
, (1)
where is the scalar product of the force vector and the virtual displacement vector.
Couple power
A pair of forces is a system of two equal in magnitude, parallel and directed in opposite sides forces acting on an absolutely rigid body.
Power couple power:
,
where omega Z is the projection of angular velocity onto the axis of rotation.
Ticket 22
1. The principle of virtual movements
Consider the virtual movement of a system point with number i. Virtual movement δr i is the mental infinitesimal movement of a point allowed by connections without their destruction at a given fixed instant of time.
If there is only one connection and is described by equation (2), it is physically clear that the connection will not be broken when the virtual displacement vector
Where grad f- gradient of function (2) at a fixed t, perpendicular to the surface connection at the location of the point, equal to
In the calculus of variations, infinitesimal quantities δr i , δx i , δy i , δz i are called variations of functions r i, x i, y i, z i. Changes in the coordinates of points or communication equations at constant time are found by synchronous variation, which is carried out according to the left sides of formulas (4) and (6).
That is, projections δx i , δy i , δz i virtual point movement δr vanish the first variation of the coupling equation, provided that time does not vary (synchronous variation):
| (7) |
Consequently, the virtual movement of a point does not characterize its movement, but determines the connection or, in the general case, connections imposed on the point of the system. Thus, virtual movements allow us to take into account the effect of mechanical connections without introducing the reaction of connections, as we did before, and obtain equations of equilibrium or motion of the system in analytical form, not containing unknown bond reactions.
2.Elementary work
Elementary work of forces, acting on an absolutely rigid body, is equal to the algebraic sum of two terms: the work of the main vector of these forces on the elementary translational movement of the body together with an arbitrarily chosen pole and the work of the main moment of forces, taken relative to the pole, on the elementary rotational movement bodies around the pole. [ 1
]
Elementary work of force equal to scalar product force on the differential of the radius vector of the point of application of the force. [ 2 ]
Elementary work of forces it depends on the choice of possible movement of the system. [ 3 ]
Elementary work of force during rotation of a body on which a force acts
Ticket 23
1. The principle of virtual movements in generalized coordinates.
Let us write down the principle, expressing virtual work active forces of the system in generalized coordinates:
Since holonomic constraints are imposed on the system, variations of generalized coordinates are independent of each other and cannot simultaneously be equal to zero. Therefore, the last equality is satisfied only when the coefficients of δ j (j = 1 ÷ s) simultaneously vanish, that is
2. Work of force on final displacement
Job force on a final displacement is defined as the integral sum of elementary Job and when moving M 0 M 1 is expressed curvilinear integral:
Ticket 24
1.Lagrange equation of the second kind.
To derive the equations, we write the D'Alembert-Lagrange principle in generalized coordinates in the form -Q j u = Q j (j = 1 ÷ s).
Taking into account that Ф i = -m i a i = -m i dV i / dt, we get:
(1)
(2)
Substituting (2) into (1) we obtain the differential equation of motion of the system in generalized coordinates, which is called the Lagrange equation of the second kind:
(3)
that is, a material system with holonomic connections is described by Lagrange equations of the second kind for all s generalized coordinates.
Note important features obtained equations.
1. Equations (3) are a system of ordinary differential equations second order with respect to s unknown functions q j (t), which completely determine the motion of the system.
2. The number of equations is equal to the number of degrees of freedom, that is, the motion of any holonomic system is described the smallest number equations.
3. In equations (3) there is no need to include reactions of ideal bonds, which allows, by finding the law of motion of a non-free system, by choosing generalized coordinates to eliminate the problem of determining unknown reactions of bonds.
4. Lagrange equations of the second kind make it possible to specify a unified sequence of actions for solving many problems of dynamics, which is often called the Lagrange formalism.
2. The condition for the relative rest of a material point is obtained from the dynamic Coriolis equation by substituting the values relative acceleration and Coriolis inertia force equal to zero:
Momentum moment of momentum
(kinetic moment, angular momentum, angular momentum), a measure of the mechanical motion of a body or system of bodies relative to some center (point) or axis. To calculate angular momentum K material point (body), the same formulas are valid as for calculating the moment of force, if you replace the force vector in them with the vector of momentum mv, i.e. K = [r· mv], Where r- distance to the axis of rotation. The sum of the angular momentum of all points of the system relative to the center (axis) is called the principal angular momentum of the system (kinetic moment) relative to this center (axis). In the rotational motion of a rigid body, the main angular momentum relative to the axis of rotation is z I z on the angular velocity ω of the body, i.e. K z = I zω.
TORQUE OF MOTIONMOMENT OF MOTION (kinetic moment, angular momentum, angular momentum), a measure of the mechanical movement of a body or system of bodies relative to some center (point) or axis. To calculate angular momentum TO material point (body), the same formulas are valid as for calculating the moment of force (cm. MOMENT OF POWER), if you replace the force vector in them with the momentum vector mv, in particular K 0 = [r· mv]. The sum of the angular momentum of all points of the system relative to the center (axis) is called the principal angular momentum of the system (kinetic moment) relative to this center (axis). In the rotational motion of a rigid body, the main angular momentum relative to the axis of rotation z of a body is expressed by the product of the moment of inertia (cm. MOMENT OF INERTIA) I z by the angular velocity w of the body, i.e. TO Z= I z w.
encyclopedic Dictionary . 2009 .
See what “momentum” is in other dictionaries:
- (kinetic momentum, angular momentum), one of the measures of mechanical. movement of a material point or system. MKD plays a particularly important role in the study of rotation. movements. As for the moment of force, a distinction is made between mechanical action relative to the center (point) and... ... Physical encyclopedia
- (kinetic moment, moment of impulse, angular moment), a measure of the mechanical movement of a body or system of bodies relative to any center (point) or axis. To calculate the angular momentum K of a material point (body), the same applies... ... Big Encyclopedic Dictionary
Angular momentum (kinetic momentum, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational motion. A value that depends on how much mass rotates, how it is distributed relative to the axis... ... Wikipedia
angular momentum- kinetic moment, one of the measures of mechanical motion of a material point or system. Angular momentum plays a particularly important role in the study of rotational motion. As for the moment of force, a distinction is made between the moment... ... Encyclopedic Dictionary of Metallurgy
angular momentum- judesio kiekio momentas statusas T sritis Standartizacija ir metrologija apibrėžtis Dydis, lygus dalelės padėties vektoriaus iš tam tikro taško į dalelę ir jos judesio kiekio vektorinei sandaugai, t. y. L = r p; čia L – judesio kiekio momento… …
angular momentum- judesio kiekio momentas statusas T sritis Standartizacija ir metrologija apibrėžtis Materialiojo taško arba dalelės spindulio vektoriaus ir judesio kiekio vektorinė sandauga. Dažniausiai apibūdina sukamąjį judesį taško arba ašies, iš kurios yra… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
angular momentum- judesio kiekio momentas statusas T sritis fizika atitikmenys: engl. angular moment; moment of momentum; rotation moment vok. Drehimpuls, m; Impulse moment, n; Rotationsmoment, n rus. angular momentum, m; moment of momentum, m; angular momentum … Fizikos terminų žodynas
Kinetic moment, one of the measures of mechanical motion of a material point or system. MKD plays a particularly important role in the study of rotational motion (See. Rotational movement). As for the moment of force (See Moment of force), ... ... Big Soviet encyclopedia
- (kinetic moment, angular momentum, angular momentum), a measure of mechanical. movement of a body or system of bodies relative to a cosmic l. center (point) or main. To calculate the M. efficiency K of a material point (body), the same formulas are valid as for calculating the moment ... Natural science. encyclopedic Dictionary
Same as angular momentum... Big Encyclopedic Polytechnic Dictionary
Books
- Theoretical mechanics. Dynamics of metal structures eBook
- Theoretical mechanics. Dynamics and analytical mechanics, V. N. Shinkin. The main theoretical and practical issues dynamics of the material system and analytical mechanics on the following topics: geometry of masses, dynamics of a material system and solid...
