Types of accelerations in SRT.

So, we have shown that there are two kinds of measurable speeds. In addition, speed, measured in the same units, is also very interesting. For 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 us introduce definitions of different types of accelerations.

Coordinate-coordinate acceleration d v/dt is a change coordinate speed, measured by synchronized coordinate clock

d v/dt=d2 r/dt 2 .

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

Coordinate own acceleration d v/dt is a 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.

Self-coordinate acceleration d b/dt is a change own speed measured by synchronized coordinate clock, placed in 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.

Own-own acceleration d b/dt is a change own speed measured by own watch associated with the 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, then d 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 performance at the coefficient g in the four types of accelerations written above, we notice that in this group there is no member with the coefficient g 2 at parallel accelerations. But we haven't taken derivatives of speed yet. It's also speed. Let's take the time derivative of the 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.

Hence we have two measurable speeds v And b, and one more, immeasurable, but most symmetric, speed r. And six kinds of accelerations, two of which dr/dt and db/dt are the same. Which of these accelerations is proper, i.e. felt by an accelerating body?

We will return to our own acceleration below, but for now we will find out what kind of 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 just obtained 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 by the factor m. That is, you can write:

f= m d b/dt.

The last equation returns to mass 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 coefficient of proportionality is the invariant mass. Force vectors f and acceleration d b/dt are co-directional for any orientation of the vectors 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 and d v/dt are parallel to each other, and if they are perpendicular to each other. But in the first case the power f=mg 3 d v/dt, and in the second - f=mgd v/dt.

Thus, in Newton's law, we must use the acceleration d b/dt, i.e. change own speed b, measured by synchronized clock.

Perhaps with the same success it will be possible to prove that f= md r/dt, where d r/dt is the vector of intrinsic acceleration, but the speed is an immeasurable value, although it is easily calculated. Whether the vector equality will be true, I can’t say, but the scalar equality is true due to the fact that dr/dt=db/dt and f=md b/dt.

Acceleration is a value that characterizes the rate of change of speed.

For example, a car, moving away, increases the speed of movement, that is, it moves at an accelerated pace. Initially, its speed is zero. Starting from a standstill, the car gradually accelerates to a certain speed. If a red traffic light lights up on its way, the car will stop. But it will not stop immediately, but after some 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 "deceleration". If the body is moving, slowing down, then this will also be the 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 time interval 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 the 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 Figure 1.8) the body has a speed of 0 . At time t2 the body has a speed . According to the vector subtraction rule, we find the vector of speed change Δ = - 0 . Then the acceleration can be defined as follows:

Rice. 1.8. Average acceleration.

in SI unit of acceleration 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 in one second the speed of this point increases by 1 m / s. 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 Boost

Instantaneous acceleration of a body (material point) at this point in time is physical quantity, equal to the limit to which the average acceleration tends when 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 set by projections onto the corresponding coordinate axes in a given reference system (projections a X, a Y , a Z).

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

V2 > v1

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

If the modulo velocity of the body decreases, 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, deceleration, while 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 curvilinear trajectory, not only the modulus of speed 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 is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the 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 the linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis as the tangent circle, which is the trajectory of the body.

Normal acceleration

Normal acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. 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 the 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 in 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 full acceleration is also determined vector addition rule:

= τ + n

Linear movement, linear speed, linear acceleration.

moving(in kinematics) - a change in the location of a physical body in space relative to the selected reference system. Also, displacement is a vector that characterizes this change. It has the additivity property. The length of the segment is the displacement modulus, measured in meters (SI).

You can define displacement as a change in the radius vector of a point: .

The modulus of movement coincides with the distance traveled if and only if the direction of movement does not change during movement. In this case, the trajectory will be a straight line segment. In any other case, for example, with curvilinear motion, it follows from the triangle inequality that the path is strictly longer.

Vector D r = r -r 0 , drawn from the initial position of the moving point to its position at a given time (increment of the radius-vector of the point over the considered time interval), is called moving.

With rectilinear motion, the displacement vector coincides with the corresponding section of the trajectory and the displacement modulus |D r| equal to the distance traveled D s.
Linear velocity of a body in mechanics

Speed

To characterize the movement of a material point, a vector quantity is introduced - the speed, which is defined as rapidity movement, as well as direction at this point in time.

Let the material point move along some curvilinear trajectory so that at the moment of time t it corresponds to the radius vector r 0 (Fig. 3). For a short period of time D t point will pass path D s and will receive an elementary (infinitely small) displacement Dr.

Average speed vector is the ratio of the increment Dr of the radius vector of the point to the time interval D t:

The direction of the average velocity vector coincides with the direction of Dr. With an unlimited decrease in D t average speed tends to a limiting value, which is called instantaneous speed v:

The instantaneous velocity v, therefore, is a vector quantity equal to the first derivative of the radius-vector of the moving point with respect to time. Since the secant coincides with the tangent in the limit, the velocity vector v is directed tangentially to the trajectory in the direction of motion (Fig. 3). As D decreases t path D s will approach |Dr| more and more, so the module of instantaneous velocity

Thus, the module of instantaneous speed is equal to the first derivative of the path with respect to time:

At not uniform motion - the instantaneous velocity modulus changes over time. In this case, we use the scalar quantity á vñ - average speed uneven movement:

From fig. 3 it follows that vñ> |ávñ|, because D s> |Dr|, and only in the case of rectilinear motion

If the expression d s = v d t(see formula (2.2)) integrate over time within the range of t before t+ D t, then we find the length of the path traveled by the point in time D t:

When uniform motion the numerical value of the instantaneous speed is constant; then expression (2.3) takes the form

The length of the path traveled by a point in the time interval from t 1 to t 2 is given by the integral

Acceleration and its components

In the case of uneven motion, it is important to know how quickly the speed changes over time. The physical quantity characterizing the rate of change of speed in absolute value and direction is acceleration.

Consider flat Movement, those. movement in which all parts of the trajectory of a point lie in the same plane. Let the vector v define the speed of the point BUT at the time t. For time D t moving point moved to position IN and acquired a speed different from v both in modulus and direction and equal to v 1 = v + Dv. Move the vector v 1 to the point BUT and find Dv (Fig. 4).

Average acceleration uneven movement in the interval from t before t+ D t is called a vector quantity equal to the ratio of the change in speed Dv to the time interval D t

Instant acceleration a (acceleration) of a material point at time t there will be a limit of average acceleration:

Thus, the acceleration a is a vector quantity equal to the first derivative of the velocity with respect to time.

Let us decompose the vector Dv into two components. For this, from the point BUT(Fig. 4) in the direction of the velocity v, we set aside the vector , modulo equal to v 1 . It is obvious that the vector , equal to , determines the change in speed over time D t modulo: . The second component of the vector Dv characterizes the change in speed over time D t in direction.

Tangential and normal acceleration.

Tangential acceleration- acceleration component directed tangentially to the motion trajectory. Coincides with the direction of the velocity vector during accelerated motion and oppositely directed during slow motion. Characterizes the change in the speed module. It is usually denoted or (, etc., in accordance with which letter is chosen to denote acceleration in general in this text).

Sometimes, tangential acceleration is understood as the projection of the tangential acceleration vector - as defined above - onto the unit vector of the tangent to the trajectory, which coincides with the projection of the (total) acceleration vector onto the unit tangent vector, that is, the corresponding expansion factor in the accompanying basis. In this case, not a vector notation is used, but a “scalar” one - as usual for a projection or vector coordinate - .

The magnitude of the tangential acceleration - in the sense of the projection of the acceleration vector onto the unit tangent vector of the trajectory - can be expressed as follows:

where - ground speed along the trajectory, coinciding with the absolute value of the instantaneous speed at a given moment.

If we use the notation for the unit tangent vector, then we can write tangential acceleration in vector form:

Output

The expression for tangential acceleration can be found by differentiating the velocity vector with respect to time, represented as a unit tangent vector:

where the first term is the tangential acceleration and the second is the normal acceleration.

Here we use the notation for the unit normal vector to the trajectory and - for the current length of the trajectory (); the last transition also uses the obvious

and, from geometrical considerations,

Centripetal acceleration (normal)- part of the total acceleration of the point, due to the curvature of the trajectory and the speed of the material point along it. Such an acceleration is directed towards the center of curvature of the trajectory, which is the reason for the term. Formally and essentially, the term centripetal acceleration generally coincides with the term normal acceleration, differing rather only stylistically (sometimes historically).

Especially often people talk about centripetal acceleration when we are talking about uniform motion in a circle or motion more or less close to this particular case.

elementary formula

where - normal (centripetal) acceleration, - (instantaneous) linear speed of movement along the trajectory, - (instantaneous) angular velocity of this movement relative to the center of curvature of the trajectory, is the radius of curvature of the trajectory at a given point. (The connection between the first formula and the second is obvious, considering).

The expressions above include absolute values. They can be easily written in vector form by multiplying by - a unit vector from the center of curvature of the trajectory to its given point:

These formulas are equally applicable to the case of motion with a constant (in absolute value) speed, and to an arbitrary case. However, in the second one, it must be borne in mind that the centripetal acceleration is not the full acceleration vector, but only its component perpendicular to the trajectory (or, which is the same, perpendicular to the instantaneous velocity vector); the full acceleration vector then also includes a tangential component (tangential acceleration), which coincides in direction with the tangent to the trajectory (or, which is the same, with instantaneous speed).

Output

That the decomposition of the acceleration vector into components - one along the vector tangent to the trajectory (tangential acceleration) and another orthogonal to it (normal acceleration) - can be convenient and useful is pretty obvious in itself. This is aggravated by the fact that when moving at a constant velocity, the tangential component will be equal to zero, that is, in this important particular case, only the normal component remains. In addition, as can be seen below, each of these components has pronounced properties and structure of its own, and the normal acceleration contains a rather important and non-trivial geometric content in the structure of its formula. Not to mention the important particular case of motion in a circle (which, moreover, can be generalized to the general case almost without change).

Displacement (in kinematics) is a change in the location of a physical body in space relative to the selected frame of reference. Also, displacement is a vector that characterizes this change. It has the additivity property.

Speed ​​(often denoted from English velocity or French vitesse) is a vector physical quantity that characterizes the speed of movement and direction of movement of a material point in space relative to the selected reference system (for example, angular velocity).

Acceleration (usually denoted in theoretical mechanics) - the time derivative of speed, a vector quantity showing how much the velocity vector of a point (body) changes as it moves per unit time (i.e., acceleration takes into account not only the change in speed, but also its directions).

Tangential (tangential) acceleration is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the 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 the linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis as the tangent circle, which is the trajectory of the body.

Normal acceleration

Normal acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. 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 the 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 in curvilinear motion, it is composed of tangential and normal accelerations according to the vector addition rule and is determined by the formula:

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

The direction of full acceleration is also determined by the vector addition rule:

Strength. Weight. Newton's laws.

Force is a vector physical quantity, which is a measure of the intensity of the impact on a given body of other bodies, as well as fields. The force applied to a massive body is the cause of a change in its speed or the occurrence of deformations in it.

Mass (from the Greek μάζα) is a scalar physical quantity, one of the most important quantities in physics. Initially (XVII-XIX centuries), it characterized the “amount of matter” in a physical object, on which, according to the ideas of that time, both the ability of the object to resist the applied force (inertia) and gravitational properties - weight depended. Closely related to the concepts of "energy" and "momentum" (according to modern ideas- mass is equivalent to rest energy).

Newton's first law

There are such frames of reference, called inertial ones, relative to which a material point, in the absence of external influences, retains the magnitude and direction of its velocity indefinitely.

Newton's second law

In an inertial frame of reference, the acceleration that a material point receives is directly proportional to the resultant of all forces applied to it and inversely proportional to its mass.

Newton's third law

Material points act on each other in pairs with forces of the same nature, directed along the straight line connecting these points, equal in magnitude and opposite in direction:

Pulse. Law of conservation of momentum. Elastic and inelastic shocks.

Impulse (Number of motion) is a vector physical quantity that characterizes the measure of the mechanical motion of a body. In classical mechanics, the momentum of a body is equal to the product the mass m of this body to its speed v, the direction of the momentum coincides with the direction of the velocity vector:

The law of conservation of momentum (Law of conservation of momentum) states that the vector sum of the momenta of all bodies (or particles) of a closed system is a constant value.

In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. From Newton's laws, it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces.

Like any of the fundamental conservation laws, the momentum conservation law describes one of the fundamental symmetries - the homogeneity of space.

Absolutely inelastic impact Such a shock interaction is called, in which the bodies are connected (stick together) with each other and move on as one body.

In a perfectly inelastic impact, mechanical energy is not conserved. It partially or completely passes into the internal energy of bodies (heating).

Absolutely elastic impact is called a collision in which the mechanical energy of a system of bodies is conserved.

In many cases, collisions of atoms, molecules and elementary particles obey the laws of absolutely elastic impact.

With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is fulfilled.

4. Types of mechanical energy. Job. Power. Law of energy conservation.

In mechanics, there are two types of energy: kinetic and potential.

Kinetic energy is the mechanical energy of any freely moving body and is measured by the work that the body could do when it slows down to a complete stop.

So, the kinetic energy of a translationally moving body is equal to half the product of the mass of this body and the square of its speed:

Potential energy is the mechanical energy of a system of bodies, determined by their mutual arrangement and the nature of the forces of interaction between them. Numerically, the potential energy of the system in its given position is equal to the work that the forces acting on the system will produce when the system moves from this position to where the potential energy is conventionally assumed to be zero (E n \u003d 0). The concept of "potential energy" takes place only for conservative systems, i.e. systems in which the work of the acting forces depends only on the initial and final position of the system.

So, for a load of weight P, raised to a height h, the potential energy will be equal to E n = Ph (E n = 0 at h = 0); for a load attached to a spring, E n = kΔl 2 / 2, where Δl is the extension (compression) of the spring, k is its stiffness coefficient (E n = 0 at l = 0); for two particles with masses m 1 and m 2 attracted according to the law of universal gravitation, , where γ is the gravitational constant, r is the distance between particles (E n = 0 as r → ∞).

The term "work" in mechanics has two meanings: work as a process in which a force moves a body acting at an angle other than 90°; work is a physical quantity equal to the product of force, displacement and the cosine of the angle between the direction of the force and displacement:

Work is zero when the body is moving by inertia (F = 0), when there is no movement (s = 0), or when the angle between the movement and the force is 90° (cos a = 0). The SI unit of work is the joule (J).

1 joule is the work done by a force of 1 N when a body moves 1 m along the line of action of the force. To determine the speed of work, enter the value of "power".

Power is a physical quantity equal to the ratio of the work performed over a certain period of time to this period of time.

Distinguish the average power over a period of time:

and instantaneous power at a given time:

Since work is a measure of energy change, power can also be defined as the rate of change in the energy of a system.

The SI unit for power is the watt, which is equal to one joule per second.

The law of conservation of energy is a fundamental law of nature, established empirically and consisting in the fact that for an isolated physical system a scalar physical quantity can be introduced, which is a function of the parameters of the system and called energy, which is conserved over time. Since the law of conservation of energy does not refer to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it can be called not a law, but the principle of conservation of energy.

The basic formulas of the kinematics of a material point, their derivation and presentation of the theory are given.

Content

See also: An example of solving the problem (coordinate method of specifying the movement of a point)

Basic formulas for the kinematics of a material point

We present the basic formulas for the kinematics of a material point. After that, we give their derivation and presentation of the theory.

Radius vector of a material point M in a rectangular coordinate system Oxyz :
,
where are unit vectors (orths) in the direction of the x, y, z axes.

Point speed:
;
.
.
Unit vector in the direction of the tangent to the point path:
.

Point Acceleration:
;
;
;
; ;

Tangential (tangential) acceleration:
;
;
.

Normal acceleration:
;
;
.

Unit vector directed towards the center of curvature of the point trajectory (along the principal normal):
.

.

Radius vector and point trajectory

Consider the motion of a material point M . We choose a fixed rectangular coordinate system Oxyz centered at some fixed point O . Then the position of the point M is uniquely determined by its coordinates (x, y, z). These coordinates are components of the radius vector of the material point.

The radius vector of the point M is the vector drawn from the origin of the fixed coordinate system O to the point M .
,
where are the unit vectors in the direction of the x, y, z axes.

As the point moves, the coordinates change with time. That is, they are functions of time. Then the system of equations
(1)
can be viewed as an equation of a curve given by parametric equations. Such a curve is the trajectory of a point.

The trajectory of a material point is the line along which the point moves.

If the point moves in a plane, then you can choose the axes and coordinate systems so that they lie in this plane. Then the trajectory is determined by two equations

In some cases, time can be excluded from these equations. Then the trajectory equation will have a dependence of the form:
,
where is some function. This dependency contains only variables and . It does not contain a parameter.

Material point speed

The speed of a material point is the time derivative of its radius vector.

According to the definition of speed and the definition of derivative:

Time derivatives, in mechanics, are denoted by a dot above the symbol. Substitute here the expression for the radius vector:
,
where we have explicitly indicated the dependence of coordinates on time. We get:

,
where
,
,

- velocity projections on the coordinate axes. They are obtained by differentiating with respect to time the components of the radius vector
.

In this way
.
Speed ​​module:
.

Tangent to path

From a mathematical point of view, the system of equations (1) can be considered as the equation of a line (curve) given by parametric equations. Time, in this consideration, plays the role of a parameter. From the course mathematical analysis it is known that the direction vector for the tangent to this curve has components:
.
But these are the components of the point velocity vector. I.e the velocity of the material point is directed tangentially to the trajectory.

All this can be demonstrated directly. Let at the moment of time the point be in position with the radius vector (see figure). And at the moment of time - in a position with a radius vector . Draw a straight line through the points. By definition, a tangent is a line that the line tends to when .
Let us introduce the notation:
;
;
.
Then the vector is directed along the straight line.

When tending, the straight line tends to the tangent, and the vector tends to the speed of the point at the moment of time:
.
Since the vector is directed along the straight line, and the straight line is at , then the velocity vector is directed along the tangent.
That is, the velocity vector of the material point is directed along the tangent to the trajectory.

Let's introduce unit length tangent direction vector:
.
Let us show that the length of this vector is equal to one. Indeed, because
, then:
.

Then the point velocity vector can be represented as:
.

Material point acceleration

The acceleration of a material point is the derivative of its velocity with respect to time.

Similarly to the previous one, we obtain the acceleration components (acceleration projections on the coordinate axes):
;
;
;
.
Acceleration module:
.

Tangential (tangential) and normal accelerations

Now consider the question of the direction of the acceleration vector with respect to the trajectory. To do this, apply the formula:
.
Differentiate it with respect to time using the product differentiation rule:
.

The vector is directed tangentially to the trajectory. In what direction is its time derivative directed?

To answer this question, we use the fact that the length of the vector is constant and equal to one. Then the square of its length is also equal to one:
.
Here and below, two vectors in parentheses denote the scalar product of vectors. Differentiate the last equation with respect to time:
;
;
.
Since the scalar product of the vectors and is equal to zero, these vectors are perpendicular to each other. Since the vector is tangent to the path, the vector is perpendicular to the tangent.

The first component is called tangential or tangential acceleration:
.
The second component is called normal acceleration:
.
Then the total acceleration is:
(2) .
This formula is a decomposition of acceleration into two mutually perpendicular components - tangent to the trajectory and perpendicular to the tangent.

Because , then
(3) .

Tangential (tangential) acceleration

Multiply both sides of the equation (2) scalar to :
.
Because , then . Then
;
.
Here we put:
.
From this it can be seen that the tangential acceleration is equal to the projection of the total acceleration on the direction of the tangent to the trajectory or, which is the same, on the direction of the point's velocity.

The tangential (tangential) acceleration of a material point is the projection of its full acceleration on the direction of the tangent to the trajectory (or on the direction of velocity).

The symbol denotes the tangential acceleration vector directed along the tangent to the trajectory. Then is a scalar value equal to the projection of the total acceleration on the direction of the tangent. It can be both positive and negative.

Substituting , we have:
.

Substitute in the formula:
.
Then:
.
That is, the tangential acceleration is equal to the time derivative of the modulus of the point's velocity. In this way, tangential acceleration leads to a change in the absolute value of the speed of the point. As the speed increases, the tangential acceleration is positive (or directed along the speed). As the speed decreases, the tangential acceleration is negative (or opposite to the speed).

Now let's examine the vector.

Consider the unit vector of the tangent to the trajectory . We place its origin at the origin of the coordinate system. Then the end of the vector will be on a sphere of unit radius. When moving a material point, the end of the vector will move along this sphere. That is, it will revolve around its origin. Let be the instantaneous angular velocity of rotation of the vector at time . Then its derivative is the speed of movement of the end of the vector. It is directed perpendicular to the vector. Let's apply the formula for the rotating motion. Vector modulus:
.

Now consider the position of the point for two close times. Let at the moment of time the point is in the position , and at the moment of time - in the position . Let and be unit vectors directed tangentially to the trajectory at these points. Through the points and draw planes perpendicular to the vectors and . Let be a straight line formed by the intersection of these planes. Drop a perpendicular from a point to a line. If the positions of the points and are close enough, then the movement of the point can be considered as a rotation along a circle of radius around the axis, which will be the instantaneous axis of rotation of the material point. Since the vectors and are perpendicular to the planes and , the angle between these planes is equal to the angle between the vectors and . Then the instantaneous speed of rotation of the point around the axis is equal to the instantaneous speed of rotation of the vector:
.
Here, is the distance between the points and .

Thus, we found the modulus of the time derivative of the vector:
.
As we pointed out earlier, the vector is perpendicular to the vector. It can be seen from the above reasoning that it is directed towards the instantaneous center of curvature of the trajectory. This direction is called the principal normal.

Normal acceleration

Normal acceleration

directed along the vector . As we found out, this vector is directed perpendicular to the tangent, towards the instantaneous center of curvature of the trajectory.
Let be a unit vector directed from a material point to the instantaneous center of curvature of the trajectory (along the principal normal). Then
;
.
Since both vectors and have the same direction - towards the center of curvature of the trajectory, then
.

From the formula (2) we have:
(4) .
From the formula (3) find the modulus of normal acceleration:
.

Multiply both sides of the equation (2) scalar to :
(2) .
.
Because , then . Then
;
.
This shows that the modulus of normal acceleration is equal to the projection of the total acceleration on the direction of the principal normal.

The normal acceleration of a material point is the projection of its full acceleration onto the direction perpendicular to the tangent to the trajectory.

Let's substitute . Then
.
That is, normal acceleration causes a change in the direction of the point's velocity, and it is related to the radius of curvature of the trajectory.

From here you can find the radius of curvature of the trajectory:
.

Finally, we note that the formula (4) can be rewritten in the following form:
.
Here we have applied the formula for vector product three vectors:
,
into which they framed
.

So we got:
;
.
Let's equate the modules of the left and right parts:
.
But the vectors and are mutually perpendicular. That's why
.
Then
.
This is a well-known formula from differential geometry for the curvature of a curve.

See also:

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

,

where is the derivative of the speed modulus,  unit vector of the tangent, coinciding in direction with the speed.

Normal acceleration characterizes the change in velocity in the 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 modulus of the acceleration vector can be found by 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 ratios are used:

;
;
;
;

.

Special cases of rectilinear motion:

1) uniform rectilinear motion: ;

2) uniform rectilinear motion:
.

1.4. Rotary motion and its kinematic characteristics

During rotational motion, all points of the body move along 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 body rotation
during
and directed along the axis of rotation so that, looking along it, the rotation of the body is observed to occur in a clockwise direction.

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 an angular displacement.

P For rotational motion, the following formulas are valid:

;
;
.

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

;
;
.

Addiction
expresses the law of rotation of the body.

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

With equally variable rotation:  = const,
,
.

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

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

Rotation frequency - the number of revolutions made by the body per unit of time.

The angular velocity can be expressed as follows:

.

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

2. Dynamics of translational and rotational motions

1. Newton's Laws Newton's first law: every body is in a state of rest or uniform rectilinear motion, until the impact of other bodies will bring it out of this state.

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

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

By force called a vector physical quantity, which is a measure of the mechanical action of one body on another. Mechanical action occurs both in 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 application point.

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

=++...+=.

Mass body is called a scalar quantity, which is a measure inertia body. Under inertia refers to the property of material bodies to keep their speed unchanged in the absence of external influences and to change it gradually (ie with finite acceleration) under the action of a force.

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

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

Newton's second law: the rate of change of the momentum of the 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:

.