Newton's third laws for rotational motion. Dynamics of a material point and translational motion of a rigid body. Material point and rigid body

1. The time derivative of the amount of motion K of a material point or a system of material points relative to a fixed (inertial) frame of reference is equal to the main vector F of all external forces applied to the system:
dK/dt = F or mac = F

where ac is the acceleration of the center of inertia of the system, and m is its mass.
In the case of translational motion of a rigid body with an absolute velocity v, the velocity of the center of inertia is vc = v. Therefore, when considering the translational motion of a rigid body, this body can be mentally replaced by a material point coinciding with the center of inertia of the body, possessing its entire mass and moving under the action of the main driver of external forces applied to the body.
In projections on the axes of a fixed rectangular Cartesian coordinate system, the equations of the basic law of the dynamics of the translational motion of the system have the form:
Fx = dK/dt, Fy = dK/dt, Fz = dK/dt

or
macx=Fx, macy=Fy, macz=Fz

2. The simplest cases of translational motion of a rigid body.
a) Coasting (F = 0):
mv = const, a=0.

b) Motion under the action of a constant force:
d/dt (mv) = F = const, mv = Ft + mv0,

where mv0 is the amount of motion of the body at the initial time t = 0.
c) Movement under the action of a variable force. The change in the momentum of the body over a period of time from t1 to t2 is
mv2 - mv1 = Fcp(t2 - t1)

where Fcp is the average value of the force vector in the time interval from t1 to t2.

Other entries

06/10/2016. Newton's first law

1. Newton's first law: any material point maintains a state of rest or uniform and rectilinear motion until the impact from other bodies brings her out of this state. This…

06/10/2016. Strength

1. Force - a vector quantity, which is a measure of mechanical action on a material point or body from other bodies or fields. A force is fully specified if its numerical value, direction are indicated ...

06/10/2016. Newton's third law

1. The actions of two material points on each other are numerically equal and directed in opposite directions: Fij = - Fji, where i is not equal to j. These forces are applied to different points and can be mutually balanced ...

Progressive movement is mechanical movement system of points (body), in which any line segment associated with a moving body, the shape and size of which do not change during the movement, remains parallel to its position at any previous moment in time. If the body moves forward, then to describe its movement it is sufficient to describe the movement of its arbitrary point (for example, the movement of the center of mass of the body).

One of the most important characteristics of the movement of a point is its trajectory, which in the general case is a spatial curve, which can be represented as conjugate arcs of various radii, each emanating from its center, the position of which can change in time. In the limit, a straight line can also be considered as an arc whose radius is equal to infinity.

In this case, it turns out that during translational motion at each given moment of time, any point of the body makes a turn around its instantaneous center of rotation, and the length of the radius at the given moment is the same for all points of the body. The velocity vectors of the points of the body, as well as the accelerations they experience, are the same in magnitude and direction.

Progressively moves, for example, the elevator car. Also, in the first approximation, the cabin of the Ferris wheel performs forward movement. However, strictly speaking, the movement of the Ferris wheel cabin cannot be considered progressive.

The basic equation of the dynamics of the translational motion of an arbitrary system of bodies

The rate of change of the momentum of the system is equal to the main vector of all external forces acting on this system.

Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the mechanical motion of a material point (body) changes under the action of forces applied to it. Considering the action of various forces on a given material point (body), the acceleration acquired by the body is always directly proportional to the resultant of these applied forces:

Under the action of the same force on bodies with different masses, the accelerations of the bodies turn out to be different, namely

Taking into account (1) and (2) and the fact that force and acceleration are vector quantities, we can write

Relation (3) is Newton's second law: the acceleration acquired by a material point (body), proportional to the force causing it, coincides with it in direction and is inversely proportional to the mass of the material point (body). In the SI measurement system, the coefficient of proportionality k \u003d 1. Then

Given that the mass of a material point (body) in classical mechanics is constant, in expression (4) the mass can be brought under the sign of the derivative:

Vector quantity

numerically equal to the product of the mass of a material point and its speed and having the direction of speed, is called the momentum (momentum) of this material point. Substituting (6) into (5), we obtain

This expression is a more general formulation of Newton's second law: the rate of change of momentum of a material point is equal to the force acting on it.

The main characteristics of the translational movement:

1.path - any movement along the trajectory

2. moving - the shortest way.

As well as force, momentum, mass, speed, acceleration, etc.

The number of degrees of freedom is the minimum number of coordinates (parameters), the setting of which completely determines the position of the physical system in space.

In translational motion, all points of the body at each moment of time have the same speed and acceleration.

The law of conservation of angular momentum (the law of conservation of angular momentum) is one of the fundamental laws of conservation. It is expressed mathematically in terms of the vector sum of all angular momenta about the chosen axis for a closed system of bodies and remains constant until external forces act on the system. In accordance with this, the angular momentum of a closed system in any coordinate system does not change with time.

The law of conservation of angular momentum is a manifestation of the isotropy of space with respect to rotation. It is a consequence of Newton's second and third laws.

Experimental studies of the interactions of various bodies - from planets and stars to atoms and elementary particles- showed that in any system of bodies interacting with each other, in the absence of the action of forces from other bodies that are not included in the system, or if the sum is equal to zero active forces the geometric sum of the momenta of the bodies remains unchanged.

A system of bodies that do not interact with other bodies that are not included in this system is called a closed system.

P-Pulse

(with vectors)

14. Differences between rotational and translational motion. Kinematics of rotary motion. Rotary motion is a type of mechanical motion. During the rotational motion of an absolutely rigid body, its points describe circles located in parallel planes. Translational motion is the mechanical motion of a system of points (body), in which any line segment associated with a moving body, the shape and dimensions of which do not change during movement, remains parallel to its position at any previous moment in time .[ There is a close and far-reaching analogy between the motion of a rigid body around a fixed axis and the motion of an individual material point (or the translational motion of a body). Each linear quantity from the kinematics of a point corresponds to a similar quantity from the kinematics of the rotation of a rigid body. Coordinate s corresponds to angle φ, linear velocity v - angular velocity w, linear (tangential) acceleration a - angular acceleration ε. Comparative movement parameters:

translational movement	rotational movement
Moving S	Angular displacement φ
Line speed	Angular speed
Acceleration	Angular acceleration
	Moment of inertia I
	angular momentum
	Moment M

Work:	Work:
Kinetic energy	Kinetic energy
Law of Conservation of Momentum (FSI)	Law of Conservation of Momentum (LSM)

When describing the rotational motion of a rigid body relative to a fixed one in a given frame of reference, it is customary to use vector quantities of a special kind. In contrast to the above polar vectors r (radius vector), v (velocity), a (acceleration), the direction of which follows naturally from the nature of the quantities themselves, the direction of the vectors characterizing the rotational motion coincides with axis of rotation, therefore they are called axial (lat. axis - axis).

The elementary rotation dφ is an axial vector, the module of which is equal to the rotation angle dφ, and the direction along the rotation axis OO" (see Fig. 1.4) is determined by the rule of the right screw. (angle of rotation of a rigid body).

Fig.1.4. To determine the direction of the axial vector

The linear displacement dr of an arbitrary point A of a rigid body is associated with the radius vector r and the rotation dφ by the relation dr=rsinα dφ or in vector form through the cross product:

dr= (1.9)

Relation (1.9) is valid precisely for an infinitely small rotation dφ.

Angular velocity ω is an axial vector determined by the derivative of the rotation vector with respect to time:

The vector ω, like the vector dφ, is directed along the axis of rotation according to the rule of the right screw (Fig. 1.5).

Fig.1.5. To determine the direction of the vector

Angular acceleration β is an axial vector determined by the time derivative of the angular velocity vector:

β=dω/dt=d2φ/dt2=ω"=φ""

During accelerated motion, the vector β coincides in direction with ω (Fig. 1.6, a), and during slow motion, the vectors β and ω are directed opposite to each other (Fig. 1.6, b).

Fig.1.6. Relationship between directions of vectors ω and β

Important note: the solution of all problems on the rotation of a rigid body around a fixed axis is similar in form to problems on rectilinear motion of a point. It suffices to replace the linear quantities x, vx, ax with the corresponding angular quantities φ, ω and β, and we will obtain equations similar to (1.6) -(1.8).

Treatment period-

(The time it takes the body to make one revolution)

Frequency (number of revolutions per unit of time) -

Dynamics is a branch of mechanics that studies the movement of material bodies together with the physical causes that cause this movement.

Dynamics is based on Newton's laws.

1. Law of inertia. There are such RMs in which any body can be at rest or uniform rectilinear motion, until the impact of elastic forces changes its state.

This law considers the body as a material point and is fulfilled only in ISO.

Strength - physical quantity, which characterizes the impact on this body from other bodies, causing change body movements.

2. The law of motion of a material point. body momentum. The rate of change of momentum of a material point is equal to the force F acting on it:

The change in momentum at a point over time dt is equal to the resultant forces.

3. Law of interaction. If one body acts on another with some force, then the second one acts on the first with the same force.

These forces are always of the same nature, equal in modulus, opposite in direction and applied to different bodies.

Dynamics of a material point. Basic equations of motion of a material point in differential form.

Dynamics of a system of particles, the center of inertia of the system, the law of motion of the center of inertia.

Consider a system of points with masses m1,m2…m n .

Center of mass– point, for the radius vector of which the following is true:

The center of mass of an isolated system is at rest or in uniform rectilinear motion.

Law of motion of the center of mass- in inertial reference systems, the center of mass of the system moves as a material point, where the mass of the entire system is located and on which the force acts, equal to the geometric sum of all external forces acting on the system.

Dynamics of a system of particles, the law of conservation of momentum in a closed system.

In the absence of forces, the momentum of a material point remains unchanged in modulus and direction (a consequence of Newton's second law).

Let's rewrite it for a system of N particles:

where the summation is over all forces acting on nth particle from the m-th side. According to Newton's third law, forces of the form and will be equal in absolute value and opposite in direction, that is, Then after substituting the result obtained into expression (1), the right side will be equal to zero, that is:

As you know, if the derivative of some expression is equal to zero, then this expression is constant with respect to the differentiation variable, which means:

(constant vector).

That is, the total momentum of a system of particles is a constant value.

The rotation of the body through a certain angle can be specified as a segment, the length of which is equal to j, and the direction coincides with the axis around which the rotation is performed. The direction of rotation and the segment depicting it is connected by the rule of the right screw.

In mathematics, it is shown that very small rotations can be considered as vectors, denoted by the symbols or . The direction of the rotation vector is associated with the direction of rotation of the body; - the vector of the elementary rotation of the body - is a pseudovector, since it does not have an application point.

During the rotational motion of a rigid body, each point moves along a circle, the center of which lies on a common axis of rotation (Fig. 6). In this case, the radius vector R, directed from the axis of rotation to a point, rotates in time Dt to some angle DJ. To characterize the rotational motion, the angular velocity and angular acceleration are introduced.

angular velocity is called a vector quantity equal to the first derivative of the angle of rotation of the body with respect to time:

An angle of 1 radian is a central angle whose arc length is equal to the radius of the circle; 360 o \u003d 2p rad.

The direction of the angular velocity is given right screw rule: the angular velocity vector is co-directed with the vector , that is, with the translational movement of the screw, the head of which rotates in the direction of movement of the point along the circle.

The linear velocity of a point is related to the angular velocity:

In vector form.

If during rotation the angular velocity changes, then angular acceleration occurs.

Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time. The vector of the angular velocity is co-directed with the vector of the elementary change in the angular velocity that occurred during the time dt:

With accelerated motion, the vector is parallel (Fig. 7), with slow motion, it is opposite (Fig. 8).

Angular acceleration occurs in the system only when there is a change in the angular velocity, that is, when the linear speed of movement changes in magnitude. The change in velocity characterizes the tangential acceleration in magnitude.

Let's find the relationship between the angular and tangential accelerations:

A change in the direction of speed during curvilinear motion is characterized by normal acceleration:

Thus, the relationship between linear and angular quantities is expressed the following formulas:

Rotary motion types:

but) variable- a movement in which and change:

b) equally variable- rotary motion with constant angular acceleration:

in) uniform– rotational movement with constant angular velocity:

Uniform rotational motion can be characterized by period and frequency of rotation.

Period is the time it takes for the body to complete one revolution.

Rotation frequency is the number of revolutions per unit of time.

For one turn:

, .

Newton's laws. The basic equation of the dynamics of translational motion.

Dynamics studies the movement of bodies, taking into account the causes that cause this movement.

Dynamics is based on Newton's laws.

I law. Exist inertial systems reference (ISO), in which the material point (body) maintains a state of rest or uniform rectilinear motion, until the impact from other bodies takes it out of this state.

The property of a body to maintain a state of rest or uniform rectilinear motion in the absence of influence of other bodies on it is called inertia.

ISO is a frame of reference in which a body, free from external influences, is at rest or moves uniformly in a straight line.

An inertial reference frame is one that is at rest or moves uniformly in a straight line with respect to any IFR.

The frame of reference, moving with acceleration relative to the IFR, is non-inertial.

Newton's first law, also called the law of inertia, was first formulated by Galileo. Its content boils down to 2 statements:

1) all bodies have the property of inertia;

2) there are ISO.

Galileo's principle of relativity: all mechanical phenomena in all ISOs occur in the same way, i.e. it is impossible to establish by any mechanical experiments inside the IFR whether the given IFR is at rest or moves uniformly in a straight line.

In most practical problems, the frame of reference, rigidly connected with the Earth, can be considered as ISO.

From experience it is known that under the same influences, different bodies change their speed unequally, i.e. acquire various accelerations, the acceleration of bodies depends on their mass.

Weight- a measure of the inertial and gravitational properties of the body. With the help of precise experiments, it has been established that the inertial and gravitational masses are proportional to each other. By choosing units in such a way that the proportionality factor becomes equal to one, we get that , therefore, they simply talk about the mass of the body.

[m]=1kg - mass of platinum-iridium cylinder, diameter and height of which are h=d=39mm.

To characterize the action of one body on another, the concept of force is introduced.

Strength- a measure of the interaction of bodies, as a result of which the bodies change their speed or deform.

Strength is characterized numerical value, direction, application point. The line along which the force acts is called line of force.

The simultaneous action of several forces on a body is equivalent to the action of one force, called resultant or the resulting force and equal to their geometric sum:

Newton's second law - the basic law of the dynamics of translational motion - answers the question of how the motion of a body changes under the action of forces applied to it.

The section of mechanics that studies the movement of material bodies together with the physical causes that cause this movement is called dynamics. The basic ideas and quantitative laws of dynamics have arisen and are developing on the basis of centuries-old human experience: observations of the movement of terrestrial and celestial bodies, industrial practice and specially designed experiments.

The great Italian physicist Galileo Galilei experimentally established that a material point (body) sufficiently remote from all other bodies (that is, not interacting with them) will maintain its state of rest or uniform rectilinear motion. This position of Galileo was confirmed by all subsequent experiments and constitutes the content of the first basic law of dynamics, the so-called law of inertia. In this case, rest should be considered as a special case of uniform and rectilinear motion, when .

This law is equally valid both for the movement of giant celestial bodies and for the movement of the smallest particles. The property of material bodies to maintain a state of uniform and rectilinear motion is called inertia.

The uniform and rectilinear motion of a body in the absence of external influences is called inertia motion.

The frame of reference, in relation to which the law of inertia is fulfilled, is called the inertial frame of reference. The inertial frame of reference is almost exactly the heliocentric frame. In view of the enormous distance to the stars, their movement can be neglected, and then the coordinate axes directed from the Sun to three stars that do not lie in the same plane will be fixed. Obviously, any other frame of reference moving uniformly and rectilinearly relative to the heliocentric frame will also be inertial.

The physical quantity characterizing the inertia of a material body is its mass. Newton defined mass as the amount of matter contained in a body. This definition cannot be considered exhaustive. Mass characterizes not only the inertia of a material body, but also its gravitational properties: the force of attraction experienced by a given body from another body is proportional to their masses. Mass determines the total energy supply of a material body.

The concept of mass allows us to refine the definition of a material point. A material point is a body, in the study of the motion of which one can abstract from all its properties, except for mass. Each material point, therefore, is characterized by the magnitude of its mass. In Newtonian mechanics, which is based on Newton's laws, the mass of a body does not depend on the position of the body in space, its speed, the action of other bodies on the body, etc. Mass is an additive quantity, i.e. The mass of a body is equal to the sum of the masses of all its parts. However, the additivity property is lost at speeds close to the speed of light in vacuum, i.e. in relativistic mechanics.

Einstein showed that the mass of a moving body depends on the speed

, (2.1)

where m0 - mass of the resting body,  - the speed of the body, c - the speed of light in vacuum.

From (2.1) it follows that when bodies move with low velocities c, the mass of the body is equal to the rest mass, i.e. m=m0; at c the mass is m.

Summarizing the results of Galileo's experiments on the fall of heavy bodies, Kepler's astronomical laws on the motion of planets, and the data of his own research, Newton formulated the second fundamental law of dynamics, which quantitatively linked the change in the motion of a material body with the forces that cause this change in motion. Let us dwell on the analysis of this most important concept.

IN general case strength - is a physical quantity that characterizes the action exerted by one body on another. This vector quantity is determined by the numerical value or module
, direction in space and application point.

If two forces act on a point And , then their action is equivalent to the action of one force

obtained from the well-known triangle of forces (Fig. 2.1). If n-forces act on the body, the total action is equivalent to the action of one resultant, which is the geometric sum of forces:

. (2.2)

The dynamic manifestation of force consists in the fact that under the action of force the material body experiences acceleration. The static action of a force leads to the fact that elastic bodies (springs) are deformed under the action of forces, gases are compressed.

Under the action of forces, the movement ceases to be uniform and rectilinear and acceleration appears ( ), its direction coincides with the direction of the force. Experience shows that the acceleration received by a body under the action of a force is inversely proportional to the value

its masses:

or
. (2.3)

Equation (2.3) represents the mathematical notation of the second basic law of dynamics:

the vector of force acting on a material point is numerically equal to the product of the mass of the point and the acceleration vector arising from the action of this force.

Since the acceleration

where
- unit vectors,
are projections of acceleration on the coordinate axes, then

. (2.4)

If we denote , then expression (2.4) can be rewritten in terms of projections of forces on the coordinate axes :

The SI unit of force is the newton.

According to (2.3), a newton is such a force that imparts an acceleration of 1 m / s 2 to a mass of 1 kg. It is easy to see that

Newton's second law can be written differently if we introduce the concept of the momentum of the body (m) and the momentum of the force (Fdt). Substitute in

(2.3) expression for acceleration

we get

or
. (2.5)

Thus, the elementary impulse of force acting on a material point during the time interval dt is equal to the change in the momentum of the body over the same time interval.

Denoting the momentum of the body

we obtain the following expression for Newton's second law:

In relativistic mechanics, for c, the basic law of dynamics and the momentum of the body, taking into account the dependence of mass on velocity (2.1.), will be written in the following form

Until now, we have considered only one side of the interaction between bodies: the influence of other bodies on the nature of the movement of a given selected body (material point). Such influence cannot be one-sided, the interaction must be mutual. This fact is reflected by the third law of dynamics, formulated for the case of the interaction of two material points: if the material point m 2 experiences from the side of the material point m 1 force equal to , then m 1 experiencing from the side m2 strength equal in magnitude and opposite in direction :