Normal acceleration characterizes the change in speed. Natural axes and natural trihedron

i.e., it is equal to the first derivative with respect to time of the speed modulus, thereby determining the rate of change of speed in modulus.

The second component of acceleration, equal to

called normal component of acceleration and is directed along the normal to the trajectory to the center of its curvature (therefore it is also called centripetal acceleration).

So, tangential acceleration component characterizes speed of change of speed modulo(directed tangentially to the trajectory), and normal acceleration component - speed of change of speed in direction(directed towards the center of curvature of the trajectory).

Depending on the tangential and normal components of acceleration, motion can be classified as follows:

1) , and n = 0 - rectilinear uniform motion;

2) , and n = 0 - rectilinear uniform motion. With this type of movement

If the initial time t 1 =0, and the initial speed v 1 =v 0 , then, denoting t 2 =t And v 2 =v, we get where from

By integrating this formula over the range from zero to arbitrary moment time t, we find that the length of the path traveled by a point in the case of uniformly variable motion

· 3) , and n = 0 - linear motion with variable acceleration;

· 4) , and n = const. When the speed does not change in absolute value, but changes in direction. From the formula a n =v 2 /r it follows that the radius of curvature must be constant. Therefore, the circular motion is uniform;

· 5) , - uniform curvilinear movement;

· 6) , - curvilinear uniform motion;

· 7) , - curvilinear movement with variable acceleration.

2) A rigid body moving in three-dimensional space, can have a maximum of six degrees of freedom: three translational and three rotational

Elementary angular displacement is a vector directed along the axis according to the rule of the right screw and numerically equal to the angle

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

The unit is radian per second (rad/s).

Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time:

When a body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of angular velocity. When the movement is accelerated, the vector is codirectional to the vector (Fig. 8), when it is slow, it is opposite to it (Fig. 9).

Tangential component of acceleration

Normal component of acceleration

When a point moves along a curve, the linear speed is directed

tangent to the curve and modulo equal to the product

angular velocity to the radius of curvature of the curve. (connection)

3) Newton's first law: every material point (body) maintains a state of rest or uniform rectilinear motion until the influence of other bodies forces it to change this state. The desire of a body to maintain a state of rest or uniform rectilinear motion is called inertia. Therefore, Newton's first law is also called law of inertia.

Mechanical motion is relative, and its nature depends on the frame of reference. Newton's first law is not satisfied in every frame of reference, and those systems in relation to which it is satisfied are called inertial reference systems. An inertial reference system is a reference system relative to which the material point, free from external influences, either at rest or moving uniformly and in a straight line. Newton's first law states the existence of inertial frames of reference.

Newton's second law - basic law of dynamics forward motion - answers the question of how mechanical motion changes material point(body) under the influence of forces applied to it.

Weight bodies - physical quantity, which is one of the main characteristics of matter, determining its inertial ( inert mass) and gravitational ( gravitational mass) properties. At present, it can be considered proven that the inertial and gravitational masses are equal to each other (with an accuracy of at least 10–12 of their values).

So, strength is a vector quantity that is a measure of the mechanical impact on a body from other bodies or fields, as a result of which the body acquires acceleration or changes its shape and size.

Vector quantity

numerically equal to the product of the mass of a material point by its speed and having the direction of speed is called impulse (amount of movement) this material point.

Substituting (6.6) into (6.5), we get

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 expression is called equation of motion of a material point.

Newton's third law

The interaction between material points (bodies) is determined Newton's third law: every action of material points (bodies) on each other is in the nature of interaction; the forces with which material points act on each other are always equal in magnitude, oppositely directed and act along the straight line connecting these points:

F 12 = – F 21, (7.1)

where F 12 is the force acting on the first material point from the second;

F 21 - force acting on the second material point from the first. These forces are applied to different material points (bodies), always act in pairs and are forces of the same nature.

Newton's third law allows for the transition from dynamics separate material point to dynamics systems material points. This follows from the fact that for a system of material points, the interaction is reduced to the forces of pair interaction between material points.

Elastic force is a force that arises during deformation of a body and counteracts this deformation.

In the case of elastic deformations, it is potential. The elastic force is of an electromagnetic nature, being a macroscopic manifestation of intermolecular interaction. In the simplest case of tension/compression of a body, the elastic force is directed opposite to the displacement of the particles of the body, perpendicular to the surface.

The force vector is opposite to the direction of deformation of the body (displacement of its molecules).

Hooke's law

In the simplest case of one-dimensional small elastic deformations, the formula for the elastic force has the form: where k is the rigidity of the body, x is the magnitude of the deformation.

GRAVITY, force P acting on any body nearby earth's surface, and defined as the geometric sum of the Earth's gravitational force F and centrifugal force inertia Q, taking into account the effect of the Earth's daily rotation. The direction of gravity is vertical at a given point on the earth's surface.

existence friction forces, which prevents sliding of contacting bodies relative to each other. Friction forces depend on the relative velocities of the bodies.

There are external (dry) and internal (liquid or viscous) friction. External friction is called friction that occurs in the plane of contact of two contacting bodies during their relative movement. If the contacting bodies are motionless relative to each other, they speak of static friction, but if there is a relative movement of these bodies, then, depending on the nature of their relative motion, they speak of sliding friction, rolling or spinning.

Internal friction is called friction between parts of the same body, for example between different layers of liquid or gas, the speed of which varies from layer to layer. Unlike external friction, there is no static friction here. If bodies slide relative to each other and are separated by a layer of viscous liquid (lubricant), then friction occurs in the lubricant layer. In this case they talk about hydrodynamic friction(the lubricant layer is quite thick) and boundary friction (the thickness of the lubricant layer is »0.1 microns or less).

experimentally established the following law: sliding friction force F tr is proportional to force N normal pressure with which one body acts on another:

F tr = f N ,

Where f- coefficient of sliding friction, depending on the properties of the contacting surfaces.

f = tga 0.

Thus, the friction coefficient is equal to the tangent of the angle a 0 at which the body begins to slide along inclined plane.

For smooth surfaces, intermolecular attraction begins to play a certain role. For them it is applied sliding friction law

F tr = f ist ( N + Sp 0) ,

Where r 0 - additional pressure caused by intermolecular attractive forces, which quickly decrease with increasing distance between particles; S- contact area between bodies; f ist - true coefficient of sliding friction.

The rolling friction force is determined according to the law established by Coulomb:

F tr = f To N/r , (8.1)

Where r- radius of the rolling body; f k is the rolling friction coefficient, having the dimension dim f k =L. From (8.1) it follows that the rolling friction force is inversely proportional to the radius of the rolling body.

Liquid (viscous) friction between solid body and a liquid or gaseous medium or layers thereof.

where is the momentum of the system. Thus, the time derivative of the momentum of a mechanical system is equal to the geometric sum of the external forces acting on the system.

The last expression is law of conservation of momentum: The momentum of a closed-loop system is conserved, that is, it does not change over time.

Center of mass(or center of inertia) of a system of material points is called an imaginary point WITH, the position of which characterizes the mass distribution of this system. Its radius vector is equal to

Where m i And r i- mass and radius vector, respectively i th material point; n- number of material points in the system; – mass of the system. Center of mass speed

Considering that pi = m i v i, a there is an impulse r systems, you can write

that is, the momentum of the system is equal to the product of the mass of the system and the speed of its center of mass.

Substituting expression (9.2) into equation (9.1), we obtain

that is, the center of mass of the system moves as a material point in which the mass of the entire system is concentrated and on which a force acts equal to the geometric sum of all external forces applied to the system. Expression (9.3) is law of motion of the center of mass.

In accordance with (9.2), it follows from the law of conservation of momentum that the center of mass of a closed system either moves rectilinearly and uniformly or remains stationary.

5) Moment of force F relative fixed point ABOUT is a physical quantity determined by the vector product of the radius vector r drawn from the point ABOUT to the point A application of force, force F(Fig. 25):

Here M - pseudovector, its direction coincides with the direction of translational motion of the right propeller as it rotates from r to F. Modulus of the moment of force

where a is the angle between r and F; r sina = l- the shortest distance between the line of action of the force and the point ABOUT -shoulder of strength.

Moment of force about a fixed axis z called scalar magnitude Mz, equal to the projection onto this axis of the vector M of the moment of force determined relative to an arbitrary point ABOUT given z axis (Fig. 26). Torque value M z does not depend on the choice of point position ABOUT on the z axis.

If the z axis coincides with the direction of the vector M, then the moment of force is represented as a vector coinciding with the axis:

We find the kinetic energy of a rotating body as the sum of the kinetic energies of its elementary volumes:

Using expression (17.1), we obtain

Where J z - moment of inertia of the body relative to the z axis. Thus, kinetic energy rotating body

From a comparison of formula (17.2) with expression (12.1) for the kinetic energy of a body moving translationally (T=mv 2 /2), it follows that the moment of inertia is measure of body inertia during rotational motion. Formula (17.2) is valid for a body rotating around a fixed axis.

In case flat movement of a body, for example a cylinder, rolling down an inclined plane without sliding, the energy of motion consists of the energy of translational motion and the energy of rotation:

Where m- mass of the rolling body; vc- speed of the body's center of mass; Jc- moment of inertia of a body relative to an axis passing through its center of mass; w - angular velocity bodies.

6) To quantitatively characterize the process of energy exchange between interacting bodies, the concept is introduced in mechanics work of force. If the body moves straight forward and it is acted upon by a constant force F, which makes a certain angle  with the direction of movement, then the work of this force is equal to the product of the projection of the force F s to the direction of movement ( F s= F cos), multiplied by the displacement of the point of application of the force:

In the general case, the force can change both in magnitude and direction, so formula (11.1) cannot be used. If, however, we consider the elementary displacement dr, then the force F can be considered constant, and the movement of the point of its application can be considered rectilinear. Elementary work force F on displacement dr is called scalar magnitude

where  is the angle between vectors F and dr; ds = |dr| - elementary path; F s - projection of vector F onto vector dr (Fig. 13).

Work of force on the trajectory section from the point 1 to the point 2 equal to the algebraic sum of elementary work on individual infinitesimal sections of the path. This sum is reduced to the integral

To characterize the rate of work done, the concept is introduced power:

During time d t force F does work Fdr, and the power developed by this force is at the moment time

i.e., it is equal to the scalar product of the force vector and the velocity vector with which the point of application of this force moves; N- magnitude scalar.

Unit of power - watt(W): 1 W is the power at which 1 J of work is performed in 1 s (1 W = 1 J/s).

Kinetic energy of a mechanical system is the energy of mechanical movement of this system.

Force F, acting on a body at rest and causing it to move, does work, and the energy of a moving body increases by the amount of work expended. Thus, work d A force F on the path that the body has passed during the increase in speed from 0 to v, goes to increase the kinetic energy d T bodies, i.e.

Using Newton's second law and multiplying by the displacement dr we get

Potential energy- mechanical energy of a system of bodies, determined by their relative position and the nature of the interaction forces between them.

Let the interaction of bodies be carried out through force fields (for example, a field of elastic forces, a field of gravitational forces), characterized by the fact that the work done active forces when a body moves from one position to another, does not depend on the trajectory along which this movement occurred, but depends only on the initial and final positions. Such fields are called potential, and the forces acting in them are conservative. If the work done by a force depends on the trajectory of the body moving from one point to another, then such a force is called dissipative; an example of this is the force of friction.

The specific form of the function P depends on the nature of the force field. For example, the potential energy of a body of mass T, raised to a height h above the Earth's surface is equal to

where is the height h is counted from the zero level, for which P 0 =0. Expression (12.7) follows directly from the fact that potential energy is equal to the work done by gravity when a body falls from a height h to the surface of the Earth.

Since the origin is chosen arbitrarily, the potential energy can have a negative value (kinetic energy is always positive!). If we take the potential energy of a body lying on the surface of the Earth as zero, then the potential energy of a body located at the bottom of the mine (depth h"), P= -mgh".

Let's find the potential energy of an elastically deformed body (spring). The elastic force is proportional to the deformation:

Where F x pack p - projection of elastic force onto the axis X;k- elasticity coefficient(for a spring - rigidity), and the minus sign indicates that F x UP p is directed in the direction opposite to the deformation x.

According to Newton’s third law, the deforming force is equal in magnitude to the elastic force and directed oppositely to it, i.e.

Elementary work d A, done by force F x at infinitesimal deformation d x, equal to

a full job

goes to increase the potential energy of the spring. Thus, the potential energy of an elastically deformed body

The potential energy of a system is a function of the state of the system. It depends only on the configuration of the system and its position in relation to external bodies.

When the system transitions from the state 1 to some state 2

that is, the change in the total mechanical energy of the system during the transition from one state to another is equal to the work done by external non-conservative forces. If there are no external non-conservative forces, then from (13.2) it follows that

d ( T+P) = 0,

that is, the total mechanical energy of the system remains constant. Expression (13.3) is law of conservation of mechanical energy: in a system of bodies between which only conservative forces act, the total mechanical energy is conserved, that is, it does not change with time.

Point acceleration for all 3 ways to accelerate movement

The acceleration of a point characterizes the speed of change in the magnitude and direction of the point's velocity.

1. Acceleration of a point when specifying its movement in a vector way

the acceleration vector of a point is equal to the first derivative of the velocity or the second derivative of the radius vector of the point with respect to time. The acceleration vector is directed towards the concavity of the curve

2. Acceleration of a point when specifying its movement using the coordinate method

The magnitude and direction of the acceleration vector are determined from the relations:

3. Determination of acceleration when specifying its movement in a natural way

Natural axes and natural trihedron

Natural axes. Curvature characterizes the degree of curvature (curvature) of a curve. Thus, a circle has a constant curvature, which is measured by the value K, the reciprocal of the radius,

The larger the radius, the smaller the curvature, and vice versa. A straight line can be considered as a circle with an infinitely large radius and a curvature of zero. The point represents a circle of radius R = 0 and has infinitely large curvature.

An arbitrary curve has variable curvature. At each point of such a curve, you can select a circle with a radius whose curvature is equal to the curvature of the curve at a given point M (Fig. 9.2). The quantity is called the radius of curvature at a given point on the curve. The axis directed tangentially in the direction of movement and the axis directed radially to the center of curvature and called the normal form natural coordinate axes.

Normal and tangential acceleration of a point

In the natural way of defining motion, the acceleration of a point is equal to the geometric sum of two vectors, one of which is directed along the main normal and is called normal acceleration, and the second is directed along a tangent and is called the tangential acceleration of the point.

The projection of the acceleration of a point onto the main normal is equal to the square of the modulus of the velocity of boredom divided by the radius of curvature of the trajectory at the corresponding point. The normal acceleration of a point is always directed towards the center of curvature of the trajectory and is equal in magnitude to this projection.

The change in speed modulo is characterized by tangential (tangential) acceleration.

those. the projection of the acceleration of a point onto the tangent is equal to the second derivative of the arc coordinate of the point with respect to time or the first derivative of the algebraic value of the speed of the point with respect to time.

This projection has a plus sign if the directions of the tangential acceleration and the unit vector coincide, and a minus sign if they are opposite.

Thus, in the case of a natural method of specifying movement, when the trajectory of a point and, consequently, its radius of curvature are known? at any point and the equation of motion, you can find the projections of the point’s acceleration onto the natural axes:

If a > 0 and > 0 or a< 0 и < 0, то движение ускоренное и вектор а направлен в сторону вектора скорости. Если а < 0 и >0 or a > 0 and< 0, то движение замедленное и вектор а направлен в сторону, противоположную вектору скорости

Special cases.

1. If a point moves rectilinearly and unevenly, then = , and, consequently, = 0, a = a.

2. If a point moves rectilinearly and uniformly, = 0, a = 0 and a = 0.

3. If a point moves uniformly along a curved path, then a = 0 and a = . With uniform curvilinear motion of a point, the law of motion has the form s = t. It is advisable to assign a positive reference direction in tasks depending on specific conditions. In the case when 0 = 0, we get = gt and. Often in problems the formula is used (when a body falls from a height H without an initial speed)

Conclusion: normal acceleration exists only at curvilinear

32. Classification of the movement of a point by its acceleration

if during a certain period of time the normal and tangential accelerations of a point are equal to zero, then during this interval neither the direction nor the magnitude of the velocity will change, i.e. the point moves uniformly in a straight line and its acceleration is zero.

if for a certain period of time the normal acceleration is not zero and the tangential acceleration of a point is zero, then the direction of the velocity changes without changing its module, i.e. the point moves curvilinearly uniformly and the acceleration module.

If at a separate moment in time, then the point does not move uniformly, and at this moment in time the modulus of its speed has a maximum, minimum or the smallest rate of monotonic change.

if for a certain period of time the normal acceleration of a point is zero and the tangent acceleration is not zero, then the direction of the velocity does not change, but its magnitude changes, i.e. the point moves unevenly in a straight line. Point acceleration module in this case

Moreover, if the directions of the velocity vectors coincide, then the motion of the point is accelerated, and if they do not coincide, then the motion of the point is slow.

If at some point in time, then the point does not move rectilinearly, but passes the inflection point of the trajectory or the modulus of its velocity becomes zero.

If for a certain period of time neither the normal nor the tangential acceleration are equal to zero, then both the direction and the magnitude of its velocity change, i.e. the point makes a curvilinear uneven movement. Point acceleration module

Moreover, if the directions of the velocity vectors coincide, then the movement is accelerated, and if they are opposite, then the movement is slow.

If the tangential acceleration module is constant, i.e. , then the modulus of the point’s velocity changes proportionally to time, i.e. the point undergoes uniform motion. And then

Formula for the speed of uniformly variable motion of a point;

Equation of uniform motion of a point

Tangential acceleration characterizes the change in speed in absolute value (magnitude) and is directed tangentially to the trajectory:

,

Where  derivative of the velocity module,  unit tangent vector, coinciding in direction with the speed.

Normal acceleration characterizes the change in speed in direction and is directed along the radius of curvature to the center of curvature of the trajectory at a given point:

,

where R is the radius of curvature of the trajectory,  unit normal vector.

The magnitude of the acceleration vector can be found using the formula

.

1.3. The main task of kinematics

The main task of kinematics is to find the law of motion of a material point. For this, the following relationships are used:

;
;
;
;

.

Special cases of rectilinear motion:

1) uniform linear motion: ;

2) uniform linear motion:
.

1.4. Rotational motion and its kinematic characteristics

During rotational motion, all points of the body move in circles, the centers of which lie on the same straight line, called the axis of rotation. To characterize the rotational motion, the following kinematic characteristics are introduced (Fig. 3).

Angular movement
 vector numerically equal to the angle of rotation of the body
in time
and directed along the axis of rotation so that, looking along it, the rotation of the body is observed to occur clockwise.

Angular velocity  characterizes the speed and direction of rotation of the body, is equal to the derivative of the angle of rotation with respect to time and is directed along the axis of rotation as angular displacement.

P For rotational motion, the following formulas are valid:

;
;
.

Angular acceleration characterizes the rate of change in angular velocity over time, equal to the first derivative of the angular velocity and directed along the axis of rotation:

;
;
.

Addiction
expresses the law of body rotation.

With uniform rotation:  = 0,  = const,  = t.

With uniform rotation:  = const,
,
.

To characterize uniform rotational motion, the rotation period and rotation frequency are used.

Rotation period T is the time of one revolution of a body rotating at a constant angular velocity.

Rotational speed – the number of revolutions made by a body per unit of time.

Angular velocity can be expressed as follows:

.

Relationship between angular and linear kinematic characteristics (Fig. 4):

2. Dynamics of translational and rotational movements

1. Newton's laws Newton's first law: every body is in a state of rest or uniform rectilinear motion until the influence of other bodies takes it out of this state.

Bodies that are not subject to external influences are called free bodies. The reference system associated with a free body is called an inertial reference system (IRS). In relation to it, any free body will move uniformly and rectilinearly or be at rest. From the relativity of motion it follows that a reference system moving uniformly and rectilinearly with respect to the ISO is also an ISO. ISOs play an important role in all branches of physics. This is due to Einstein's principle of relativity, which states that the mathematical form of any physical law must have the same form in all inertial frames of reference.

The basic concepts used in the dynamics of translational motion include force, body mass, and momentum of the body (system of bodies).

By force is a vector physical quantity that is a measure of the mechanical action of one body on another. Mechanical action occurs both through direct contact of interacting bodies (friction, support reaction, weight, etc.), and through force field existing in space (gravity, Coulomb forces, etc.). Strength characterized by module, direction and point of application.

Simultaneous action of several forces on the body ,,...,can be replaced by the action of the resultant (resultant) force :

=++...+=.

Mass body is a scalar quantity that is a measure inertia bodies. Under inertia refers to the property of material bodies to maintain their speed unchanged in the absence of external influences and change it gradually (i.e. with finite acceleration) under the influence of force.

Impulse body (material point) is a vector physical quantity equal to the product of the body’s mass and its speed:
.

The momentum of a system of material points is equal to the vector sum of the momentum of the points that make up the system:
.

Newton's second law: the rate of change of momentum of a body is equal to the force acting on it:

.

If the mass of the body remains constant, then the acceleration acquired by the body relative to the inertial frame of reference is directly proportional to the force acting on it and inversely proportional to the mass of the body:

.

Types of accelerations in service stations.

So, we have shown that there are two types of measurable speeds. In addition, speed, measured in the same units, is also very interesting. At small values, all these speeds are equal.

How many accelerations are there? What acceleration should be constant at uniformly accelerated motion relativistic rocket, so that the astronaut always exerts the same force on the floor of the rocket, so that he does not become weightless, or so that he does not die from overloads?

Let's introduce definitions different types accelerations.

Coordinate acceleration d v/dt is the change coordinate velocity, measured by synchronized coordinate clock

d v/dt=d 2 r/dt 2 .

Looking ahead, we note that d v/dt = 1 d v/dt = g 0 d v/dt.

Coordinate-natural acceleration d v/dt is the change coordinate speed measured by own watch

d v/dt=d(d r/dt)/dt = gd 2 r/dt 2 .
d v/dt = g 1 d v/dt.

Proper coordinate acceleration d b/dt is the change own speed measured from synchronized coordinate clock, placed along the direction of motion of the test body:

d b/dt = d(d r/dt)/dt = g 3 v(v d v/dt)/c 2 + gd v/dt.
If v|| d v/dt, then d b/dt = g 3 d v/dt.
If v perpendicular to d v/dt, then d b/dt = gd v/dt.

Proper-intrinsic acceleration d b/dt is the change own speed measured by own watch associated with a moving body:

d b/dt = d(d r/dt)/dt = g 4 v(v d v/dt)/c 2 + g 2 d v/dt.
If v|| d v/dt, thend b/dt = g 4 d v/dt.
If v perpendicular to d v/dt, then d b/dt = g 2 d v/dt.

Comparing the indicators for the coefficient g in the four types of accelerations written above, we notice that in this group there is no term with a coefficient g 2 for parallel accelerations. But we have not yet taken derivatives of speed. This is also speed. Let's take the time derivative of speed using the formula v/c = th(r/c):

dr/dt = (c·arth(v/c))" = g 2 dv/dt.

And if we take dr/dt, we get:

dr/dt = g 3 dv/dt,

or dr/dt = db/dt.

Therefore we have two measurable speeds v And b, and one more, immeasurable, but most symmetrical, speed r. And six types of accelerations, two of which dr/dt and db/dt are the same. Which of these accelerations is proper, i.e. a perceived accelerating body?

We will return to our own acceleration below, but for now let’s find out what acceleration is included in Newton’s second law. As is known, in relativistic mechanics the second law of mechanics, written in the form f=m a turns out to be wrong. Instead, force and acceleration are related by the equation

f= m(g 3 v(va)/c 2 + g a),

which is the basis for engineering calculations of relativistic accelerators. If we compare this equation with the equation we just derived for the acceleration d b/dt:

d b/dt = g 3 v(v d v/dt)/c 2 + gd v/dt

then we note that they differ only in the factor m. That is, we can write:

f= m d b/dt.

The last equation returns mass to the status of a measure of inertia in relativistic mechanics. The force acting on the body is proportional to the acceleration d b/dt. The proportionality coefficient is the invariant mass. Force vectors f and acceleration d b/dt are codirectional for any vector orientation v And a, or b and d b/dt.

Formula written in terms of acceleration d v/dt does not give such proportionality. Force and coordinate-coordinate acceleration generally do not coincide in direction. They will be parallel only in two cases: if the vectors v andd v/dt are parallel to each other, and if they are perpendicular to each other. But in the first case the force f= mg 3 d v/dt, and in the second - f=mgd v/dt.

So in Newton's law we must use the acceleration d b/dt, that is, change own speed b, measured by synchronized clocks.

Perhaps with equal success it will be possible to prove that f= md r/dt, where d r/dt is the vector of its own acceleration, but speed is an immeasurable quantity, although it is easily calculated. I cannot say whether the vector equality will be true, but the scalar equality is true due to the fact that dr/dt=db/dt and f=md b/dt.

Acceleration is a quantity that characterizes the rate of change in speed.

For example, when a car starts moving, it increases its speed, that is, it moves faster. At first its speed is zero. Once moving, the car gradually accelerates to a certain speed. If a red traffic light comes on on its way, the car will stop. But it will not stop immediately, but over time. That is, its speed will decrease down to zero - the car will move slowly until it stops completely. However, in physics there is no term “slowdown”. If a body moves, slowing down, then this will also be an acceleration of the body, only with a minus sign (as you remember, speed is a vector quantity).

Average acceleration

Average acceleration> is the ratio of the change in speed to the period of time during which this change occurred. The average acceleration can be determined by the formula:

Where - acceleration vector.

The direction of the acceleration vector coincides with the direction of change in speed Δ = - 0 (here 0 is the initial speed, that is, the speed at which the body began to accelerate).

At time t1 (see Fig. 1.8) the body has a speed of 0. At time t2 the body has speed . According to the rule of vector subtraction, we find the vector of speed change Δ = - 0. Then you can determine the acceleration like this:

Rice. 1.8. Average acceleration.

In SI acceleration unit– is 1 meter per second per second (or meter per second squared), that is

A meter per second squared is equal to the acceleration of a point moving in a straight line, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s 2, then this means that the speed of the body increases by 5 m/s every second.

Instant acceleration

Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity equal to the limit to which the average acceleration tends as the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:

The direction of acceleration also coincides with the direction of change in speed Δ for very small values ​​of the time interval during which the change in speed occurs. The acceleration vector can be specified by projections onto the corresponding coordinate axes in a given reference system (projections a X, a Y, a Z).

With accelerated straight motion the speed of the body increases in absolute value, that is

V 2 > v 1

and the direction of the acceleration vector coincides with the velocity vector 2.

If the speed of a body decreases in absolute value, that is

V 2< v 1

then the direction of the acceleration vector is opposite to the direction of the velocity vector 2. In other words, in this case what happens is slowing down, in this case the acceleration will be negative (and< 0). На рис. 1.9 показано направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

Rice. 1.9. Instant acceleration.

When moving along a curved path, not only the speed module changes, but also its direction. In this case, the acceleration vector is represented as two components (see the next section).

Tangential acceleration

Tangential (tangential) acceleration– this is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the motion trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.

Rice. 1.10. Tangential acceleration.

The direction of the tangential acceleration vector τ (see Fig. 1.10) coincides with the direction of linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis with the tangent circle, which is the trajectory of the body.

Normal acceleration

Normal acceleration is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.

Full acceleration

Full acceleration during curvilinear motion, it consists of tangential and normal accelerations along vector addition rule and is determined by the formula:

(according to the Pythagorean theorem for a rectangular rectangle).

The direction of total acceleration is also determined vector addition rule:

= τ + n

For example, a car that starts moving moves faster as it increases its speed. At the point where the motion begins, the speed of the car is zero. Having started moving, the car accelerates to a certain speed. If you need to brake, the car will not be able to stop instantly, but over time. That is, the speed of the car will tend to zero - the car will begin to move slowly until it stops completely. But physics does not have the term “slowdown”. If a body moves, decreasing speed, this process is also called acceleration, but with a “-” sign.

Medium acceleration is called the ratio of the change in speed to the period of time during which this change occurred. Calculate the average acceleration using the formula:

where is this . The direction of the acceleration vector is the same as that of the direction of change in speed Δ = - 0

where 0 is the initial speed. At a moment in time t 1(see figure below) at the body 0. At a moment in time t 2 the body has speed. Based on the rule of vector subtraction, we determine the vector of speed change Δ = - 0. From here we calculate the acceleration:

.

In the SI system unit of acceleration called 1 meter per second per second (or meter per second squared):

.

A meter per second squared is the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in 1 second. In other words, acceleration determines the rate of change in the speed of a body in 1 s. For example, if the acceleration is 5 m/s2, then the speed of the body increases by 5 m/s every second.

Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity that is equal to the limit to which the average acceleration tends as the time interval tends to 0. In other words, this is the acceleration developed by the body in a very short period of time:

.

Acceleration has the same direction as the change in speed Δ in extremely short periods of time during which the speed changes. The acceleration vector can be specified using projections onto the corresponding coordinate axes in given system reference (projections a X, a Y, a Z).

With accelerated linear motion, the speed of the body increases in absolute value, i.e. v 2 > v 1 , and the acceleration vector has the same direction as the velocity vector 2 .

If the speed of a body decreases in absolute value (v 2< v 1), значит, у вектора ускорения направление противоположно направлению вектора скорости 2 . Другими словами, в таком случае наблюдаем slowing down(acceleration is negative, and< 0). На рисунке ниже изображено направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

If movement occurs along a curved path, then the magnitude and direction of the speed changes. This means that the acceleration vector is depicted as two components.

Tangential (tangential) acceleration they call that component of the acceleration vector that is directed tangentially to the trajectory at a given point of the motion trajectory. Tangential acceleration describes the degree of change in speed modulo during curvilinear motion.

U tangential acceleration vectorτ (see figure above) the direction is the same as that of linear speed or opposite to it. Those. the tangential acceleration vector is in the same axis with the tangent circle, which is the trajectory of the body.