Speed. Path.

Let the material point move in the selected CO. The vector drawn from the initial position of a point to the final one is called moving(). Then the vector quantity is called average moving speed. The length of the trajectory section traversed by a point during the interval is called by S(). Average speed characterizes the speed and direction of particle movement. The average speed of a body’s movement along a trajectory is characterized by average ground speed. How fast and in what direction the body is moving at the moment t characterizes instantaneous speed . Instantaneous ground speed. When the modulus of instantaneous speed is equal to the instantaneous ground speed, the instantaneous speed is always directed tangentially to the trajectory. For infinitesimal displacement. For small intervals this is done approximately.

Speed ​​is a vector quantity, which means it can be written in the form . On the other side . Therefore, the projection of speed... Magnitude (module) of speed.

Expression for speed in polar coordinates (): , . The direction is given by an angle or a unit vector. Radius vector of a point, , is a unit vector perpendicular to . .

The distance traveled by the particle from to .

Acceleration. Normal and tangential acceleration.

When a material point moves, its speed changes both in magnitude and direction. How quickly this happens at an arbitrary moment in time is characterized by the vector quantity acceleration. . Acceleration vector projection …

Let us consider the motion of a particle in a plane. The velocity is directed along a tangent trajectory, so we can write . Here the unit vector specifies the direction of the tangent, .

Acceleration directed tangentially to the trajectory, determined by the speed of change in the magnitude of the speed, or module, is called tangential acceleration.

– normal acceleration(characterizes the speed of change in the direction of speed), is a unit vector, perpendicular and directed inside the curve, R is the radius of curvature of the line.

Newton's third law. Galileo's principle of relativity.

Newton's 3rd law: the forces with which 2 bodies act on each other are equal in magnitude, opposite in direction, lie on the same straight line passing through the bodies and have the same physical nature.

Newton's three laws allow us to solve main task of dynamics: Based on given forces, initial positions and initial velocities of bodies, the further movement of the mechanical system can be determined. 1st law gives a criterion for finding ISO; 2nd law gives the dynamic equation of motion; 3rd law allows us to introduce into consideration all the forces acting in the system. When moving from one ISO to another ISO, the velocities are converted according to the law, and the acceleration -, i.e. the acceleration of the bodies does not change, as well as the forces, therefore, the equation of the 2nd law remains unchanged. Consequently, under the same initial conditions (coordinates and speeds), we will obtain the same solution in both cases. This means that ISOs are equivalent.

Galileo's principle of relativity: all mechanical phenomena in different ISOs proceed in the same way under the same initial conditions, as a result of which it is impossible to single out any ISO as absolutely at rest.

Law of conservation of momentum.

In mechanics there are 3 fundamental conservation law(this is a certain function of the coordinates of particle velocities and time, which remains constant during movement). Conservation laws allow you to solve problems using 1st order differential equations. The vector quantity is called impulse material point (momentum - momentum). From Newton's 2nd law it follows that the rate of change of momentum of a mechanical system is equal to the sum of external forces acting on the system. N – number of material points. A system that is not acted upon by external forces is called closed, or isolated. For a closed system, the right side of the equation is equal to 0. This means . We get law of conservation of momentum: The momentum of a closed-loop system is conserved (does not change) over time.

The law of conservation of momentum is a consequence of the homogeneity of space. Notes: 1) The momentum of an open-loop system will be conserved if external forces compensate each other, and their resultant = 0; 2) if the resultant of external forces is , but = 0 its projection onto a certain direction (project OX), then the projection of the momentum onto this direction will be preserved; 3) if external forces are present, but a short-term process is considered (impact, explosion), then the acting external forces can be neglected and the law of conservation of momentum can be used, since dt is small, then the impulse of external forces is small and can be neglected.

Let a system of material points be given, with masses whose radius vectors are relative to some origin O. Point C, the radius vector of which is determined by the expression , is called center of mass, or the center of inertia of the system. Its position relative to the bodies does not depend on O’s choice. Center of mass speed . The ISO associated with the center of mass is called center of mass system.

Conservative forces.

Interaction between bodies located at some distance from each other is carried out through force fields created throughout the surrounding space. If the field does not change, then such a field is called stationary. Let there exist a point O (the center of the force field) such that at any point in space the force acting on the particle lies on a straight line passing through the given point in space and the force center. If the magnitude of the forces depends only on the distance between these points, then we have central force field(ex. Coulomb field). If at all points in space the force is the same in magnitude and direction, then we speak of uniform force field. If the work done on a particle by the forces of a stationary field does not depend on the choice of motion trajectory and is determined only by the initial and final positions of the bodies, then such a field is called conservative.

1) the gravity field is called stationary homogeneous. . This means that the gravity field is conservative.

2) elastic force field. . This means that the elastic force field is conservative.

3) Let us show that any central force field is conservative. , . . Here the work is determined by the start and end positions of the points, and not by the type of trajectory. Therefore, the central force field is conservative. The central forces are:

1) Coulomb interaction force , .

2) gravitational interaction force, .

An equivalent definition of conservative forces is: a force is called conservative, if its work on an arbitrary closed trajectory = 0.

Problem of 2 bodies.

The two-body problem involves the movement of an isolated system of two material points interacting with each other. Due to the isolation of the system, its momentum is conserved, and the center of mass moves at a constant speed relative to the reference frame K'. This allows you to go to the center of mass system (it will be inertial, like K’). – radius vector relative to . - radius vectors and relative to C. We compose the system: . Solving the system, we get: , . The movement of bodies is determined by forces,. We took into account Newton's 3rd law and isotropy of space(if rotating the CO by an arbitrary angle does not lead to a change in the measurement results). We get the equations: , . We solve, and as a result we get: .

The center of mass of a rigid body moves in the same way as a material point of mass m would move under the influence of all external forces acting on the rigid body.

Gyroscopes.

Gyroscope(or top) is a massive solid body, symmetrical to a certain axis, rotating around it at a high angular velocity. Due to the symmetry of the gyroscope, . When trying to rotate a rotating gyroscope around a certain axis, gyroscopic effect– under the influence of forces that, it would seem, should cause a rotation of the axis of the gyroscope OO around the straight line O'O', the axis of the gyroscope rotates around the straight line O''O'' (the axis OO and the straight line O'O' are assumed to lie in the plane of the drawing, and the straight line O''O'' and the forces f1 and f2 are perpendicular to this plane). The explanation of the effect is based on the use of the moment equation. The angular momentum rotates around the OX axis due to the relationship. Together with the OX, the gyroscope also rotates. Due to the gyroscopic effect, the bearing on which the gyroscope rotates begins to act gyroscopic forces. Under the influence of gyroscopic forces, the gyroscope axis tends to take a position parallel to the angular velocity of the Earth's rotation.

The described behavior of the gyroscope is the basis gyroscopic compass. Advantages of the gyroscope: indicates the exact direction to the geographic north pole, its operation is not affected by metal objects.

Gyroscope precession– a special type of gyroscope motion occurs if the moment of external forces acting on the gyroscope, while remaining constant in magnitude, rotates simultaneously with the gyroscope axis, forming a right angle with it all the time. Let us consider the movement of a gyroscope with one fixed point on the axis under the influence of gravity, is the distance from the fixed point to the center of inertia of the gyroscope, and is the angle between the gyroscope and the vertical. the moment is directed perpendicular to the vertical plane passing through the axis of the gyroscope. Equation of motion: momentum increment = Consequently, changes its position in space in such a way that its end describes a circle in the horizontal plane. Over a period of time, the gyroscope rotates through an angle The gyroscope axis describes a cone around a vertical axis with angular velocity – angular velocity of precession.

Harmonic vibrations.

Oscillations– processes characterized by varying degrees of repeatability over time. Depending on the physical nature of the repeating process, vibrations are distinguished: mechanical, electromagnetic, electromechanical and others. All these processes, despite their different physical nature, are described by the same mathematical equations and have a number of common properties. Consider a small ball of mass m suspended on a light elastic spring of stiffness k. In the equilibrium position (x=0), the sum of the forces acting on the ball is equal to 0, i.e. . When the ball deviates from its equilibrium position, its movement will be described by the equation: . Let's write the equation in the following form: . The position of the body is described through the cosine (or sine) function, which is called harmonic, therefore such oscillations are called harmonic. – vibration amplitude– gives the maximum deviation from the equilibrium position. – oscillation phase – determined by the displacement of the body at a given moment in time. – initial phase. The cosine function has a period. This means that the state of the oscillating body is repeated when the phase changes by . The period of time during which the phase changes by is called period of oscillation . Period– the time it takes to complete one complete oscillation. Oscillation frequency– number of oscillations per unit time, . – circular (cyclic) frequency, i.e. number of vibrations per second. Knowing the initial position and speed of the body, we can determine the amplitude and initial phase: .The movement of a body during harmonic vibration occurs under the influence quasi-elastic force: , which is conservative, and, therefore, the law of conservation of energy is satisfied, . Average value of kinetic and potential energies by time: .

Damped oscillations.

In real physical systems, resistance forces always act, as a result of which the amplitude of oscillations decreases over time. Let us consider the movement of a body in a viscous medium when the drag forces are opposite to the speed of the body: , is the drag coefficient. . Let us substitute - the 2nd order differential equation is reduced to a quadratic algebraic equation. The oscillatory process is possible if the resistance forces are small enough. This means that the condition must be met. In this case . Therefore, the general solution to our equation will be the function - kinematic law of damped oscillations. We can say that harmonic oscillations are observed with a frequency, while the amplitude of the oscillations decreases according to an exponential law. The rate of decay is determined by the quantity attenuation coefficient. Attenuation is also characterized damping decrement, which shows how many times the amplitude of oscillations has decreased in a time equal to the period: . The logarithm of this expression is called logarithmic damping decrement: . In damped systems, the following quantity is also used: quality factor: .

Wave equation.

The equation of any wave is a solution to some differential equation called wave. Based on the physical properties of the medium and the basic laws of mechanics, we obtain the wave equation from an explicit expression for the plane wave equation.

You can write: – wave equation. The wave equation will be satisfied by any wave of arbitrary frequency propagating with speed. determined by the physical properties of the environment. In the case of a plane wave propagating in the x direction, the wave equation is written as: .

Elastic wave energy.

Let a plane longitudinal wave propagate in the OX direction in some elastic medium. Her equation: . Particles of the medium, deviating from the equilibrium position, move at certain speeds. Therefore, they have kinetic and potential energies. Let us select in the medium a cylindrical volume V with base area S and height x. Its magnitude is such that we can consider particle speed and about relative offset identical. Energy, contained in this volume. Thus, elastic wave energy density . Let's substitute the equation of a plane wave into it, transform and use the fact that: . Then we find with period-average energy density: . From the expression for energy density it is clear that its value changes over time from 0 to a certain maximum value, which means that energy from vibration sources is transferred by a wave from one place in space to another at a speed. The wave carries out the process of transferring energy, but not matter. Energy transfer is carried out through the forces of elastic interaction between particles of the medium. The amount of energy transferred through a certain surface per unit time is called flow of energy through this surface: . For a more detailed characterization of the energy transfer process, the vector energy flux density. In magnitude, it is equal to the energy flow transferred through the area, perpendicular to the direction of wave propagation, divided by the area of ​​this area: - last thing - vector Umov. In direction it coincides with the direction of wave propagation. Average . The modulus of this expression is called wave intensity.

Addition of speeds in the service station.

In the 19th century, classical mechanics was faced with the problem of extending this rule for adding velocities to optical (electromagnetic) processes. Essentially, there was a conflict between two ideas of classical mechanics, transferred to the new field of electromagnetic processes. For example, if we consider the example with waves on the surface of water from the previous section and try to generalize it to electromagnetic waves, we will get a contradiction with observations (see, for example, Michelson’s experiment). The classic rule for adding velocities corresponds to the transformation of coordinates from one system of axes to another system moving relative to the first without acceleration. If with such a transformation we retain the concept of simultaneity, that is, we can consider two events simultaneous not only when they are registered in one coordinate system, but also in any other inertial system, then the transformations are called Galilean. In addition, with Galilean transformations, the spatial distance between two points - the difference between their coordinates in one ISO - is always equal to their distance in another inertial frame. The second idea is the principle of relativity. Being on a ship moving uniformly and rectilinearly, its movement cannot be detected by any internal mechanical effects. Does this principle apply to optical effects? Is it not possible to detect the absolute movement of a system by the optical or, what is the same thing, electrodynamic effects caused by this movement? Intuition (related quite clearly to the classical principle of relativity) says that absolute motion cannot be detected by any kind of observation. But if light propagates at a certain speed relative to each of the moving inertial systems, then this speed will change when moving from one system to another. This follows from the classical rule of adding velocities. In mathematical terms, the speed of light will not be invariant under Galilean transformations. This violates the principle of relativity, or rather, does not allow the principle of relativity to be extended to optical processes. Thus, electrodynamics destroyed the connection between two seemingly obvious provisions of classical physics - the rule of adding velocities and the principle of relativity. Moreover, these two provisions in relation to electrodynamics turned out to be incompatible. The theory of relativity provides the answer to this question. It expands the concept of the principle of relativity, extending it to optical processes. The rule for adding velocities is not canceled completely, but is only refined for high velocities using the Lorentz transformation.

If some object has velocity components relative to the system S and - relative to S", then the following relationship exists between them:

In these relations, the relative speed of movement of the reference frames v is directed along the x axis. Relativistic addition of velocities, like the Lorentz transformation, at low velocities () transforms into the classical law of addition of velocities.

If an object moves at the speed of light along the x axis relative to the system S, then it will have the same speed relative to S": This means that the speed is invariant (the same) in all ISOs.

Barometric formula.

The barometric formula gives the dependence of atmospheric pressure on altitude measured from the Earth's surface. It is assumed that the temperature of the atmosphere does not change with altitude. To derive the formula, we select a vertical cylinder: cross section S. A small cylindrical volume of height dh is identified in it. It is in equilibrium: it is acted upon by the force of gravity mg, the vertically upward force of gas pressure F1, and the vertically directed downward pressure force F2. Their sum = 0. In projection: -mg+ F1-. F2=0 . From the Clapeyron-Mendeleev equation . We integrate over the range from 0 to and get: – barometric formula, used to determine height. The change in temperature can be neglected.

Gas pressure on the wall.

Maxwell distribution.

Let there be n identical molecules in a state of random thermal motion at a certain temperature. After each act of collision between molecules, their speeds change randomly. As a result of an unimaginably large number of collisions, a stationary equilibrium state is established, when the number of molecules in a given speed range remains constant.

As a result of each collision, the velocity projections of the molecules undergo a random change by , , , and the changes in each velocity projection are independent of each other. We will assume that force fields do not act on particles. Let us find under these conditions what number of particles dn out of the total number n has a speed in the range from υ to υ+Δυ. At the same time, we cannot say anything definite about the exact value of the speed of a particular particle υi, since the collisions and movements of each of the molecules cannot be traced either experimentally or in theory. Such detailed information would hardly be of practical value.

Speed ​​is a vector quantity. For the projection of velocity onto the x-axis (x-th component of velocity), we then have where A1 is a constant equal to

A graphical representation of the function is shown in the figure. It can be seen that the fraction of molecules with speed is not zero. At , (this is the physical meaning of constant A1).

The given expression and graph are valid for the distribution of gas molecules over the x-components of velocity. Obviously, from the y- and z-components of velocity one can also obtain:

The probability that the speed of a molecule simultaneously satisfies three conditions: the x-component of the speed lies in the range from , to + ,; y-component, in the range from to + ; The z-component, in the interval from to +d, will be equal to the product of the probabilities of each of the conditions (events) separately: where, or ) is the number of molecules in a parallelepiped with sides , , d, that is, in a volume dV= d located at a distance from the origin of coordinates in velocity space. This quantity () cannot depend on the direction of the velocity vector. Therefore, it is necessary to obtain the distribution function of molecules by speed, regardless of their direction, that is, by the absolute value of the speed. If you collect together all the molecules in a unit volume, the velocities of which are in the range from υ to υ+dυ in all directions, and release them, then in one second they will find themselves in a spherical layer of thickness dυ and radius υ. This spherical layer consists of those parallelepipeds about which mentioned above.

The volume of this spherical layer is . Total number of molecules in the layer: this implies Maxwell's law of distribution of molecules according to absolute values ​​of velocities: where is the fraction of all particles in a spherical layer of volume dV whose velocities lie in the range from υ to υ+dυ. For dυ = 1 we get probability density, or molecular velocity distribution function: This function denotes the fraction of molecules in a unit volume of gas whose absolute velocities are contained in a unit speed interval that includes a given speed. Let's denote: and we get: The graph of this function is shown in the figure. That's what it is Maxwell distribution. Or in another way

.

Entropy.

Thermodynamic entropy S, often simply called entropy, in chemistry and thermodynamics is a function of the state of a thermodynamic system. The concept of entropy was first introduced by Rudolf Clausius, who defined change in entropy of a thermodynamic system during a reversible process as the ratio of the change in the total amount of heat ΔQ to the absolute temperature T (that is, the change in heat at a constant temperature): . For example, at a temperature of 0 °C, water can be in a liquid state and, with little external influence, begins to quickly turn into ice, releasing a certain amount of heat. In this case, the temperature of the substance remains 0 °C. The state of a substance changes, accompanied by a change in heat, due to a change in structure.

This formula is only applicable for an isothermal process (occurring at a constant temperature). Its generalization to the case of an arbitrary quasi-static process looks like this: , where dS is the increment (differential) of entropy, and δQ is an infinitesimal increment in the amount of heat. It is necessary to pay attention to the fact that the thermodynamic definition in question applies only to quasi-static processes(consisting of continuously successive equilibrium states).

Entropy is an additive quantity, i.e. The entropy of a system is equal to the sum of the entropies of its individual parts.

Boltzmann established connection between entropy and the probability of a given state. Later this connection was presented in the form of Planck’s formula: , where the constant k = 1.38×10−23 J/K is called the Boltzmann constant by Planck, and Ω is the (thermodynamic probability) statistical weight of the state, is the number of possible microstates (ways) by which one can go to a given macroscopic state. This postulate, called Boltzmann's principle by Albert Einstein, laid the foundation for statistical mechanics, which describes thermodynamic systems using the statistical behavior of their constituent components. Boltzmann's principle connects the microscopic properties of a system (Ω) with one of its thermodynamic properties (S). According to the definition, entropy is a function of the state, that is, it does not depend on the method of achieving this state, but is determined by the parameters of this state. Since Ω can only be a natural number (1, 2, 3, ...), the Boltzmann entropy must be non-negative - based on the properties of the logarithm.

Entropy in open systems:

Due to the second law of thermodynamics, the entropy Si of a closed system cannot decrease ( law of non-decreasing entropy). Mathematically, this can be written as follows: , index i denotes the so-called internal entropy corresponding to a closed system. In an open system, heat flows are possible both from the system and into it. If there is a heat flow, the amount of heat δQ1 enters the system at temperature T1 and the amount of heat δQ2 leaves at temperature T2. The entropy increment associated with these heat flows is equal to:

In stationary systems, usually δQ1 = δQ2, T1 > T2, so dSo< 0. Поскольку здесь изменение энтропии отрицательно, то часто употребляют выражение «приток негэнтропии», вместо оттока энтропии из системы. Negentropy is thus defined as the reciprocal of entropy.

The total change in entropy of an open system will be equal to: dS = dSi + dSo.

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. The vector characterizing this change is also called displacement. It has the property of additivity. The length of the segment is the displacement module, measured in meters (SI).

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

The displacement module coincides with the distance traveled if and only if the direction of displacement 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 period of time) is called moving.

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

Speed

To characterize the motion of a material point, a vector quantity is introduced - speed, which is defined as rapidity movement and his direction at a given moment in time.

Let a 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 the point will go along the path D s and will receive an elementary (infinitesimal) displacement Dr.

Average speed vector is called the ratio of the increment Dr of the radius vector of a 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 the average speed tends to a limiting value called instantaneous speed v:

Instantaneous speed 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 in the limit coincides with the tangent, the velocity vector v is directed tangent to the trajectory in the direction of motion (Fig. 3). As D decreases t path D s will increasingly approach |Dr|, so the absolute value of the instantaneous velocity

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

At uneven movement - the module of instantaneous speed changes over time. In this case, we use the scalar quantity b vñ - average speed uneven movement:

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

If expression d s = v d t(see formula (2.2)) integrate over time ranging from 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) will take the form

The length of the path traveled by a point during a period of time from t 1 to t 2, given by the integral

Acceleration and its components

In the case of uneven movement, it is important to know how quickly the speed changes over time. A physical quantity characterizing the rate of change in speed in magnitude and direction is acceleration.

Let's consider flat movement, those. a movement in which all parts of a point’s trajectory lie in the same plane. Let the vector v specify the speed of the point A at a point in time t. During time D t the moving point has moved to position IN and acquired a speed different from v both in magnitude and direction and equal to v 1 = v + Dv. Let's move the vector v 1 to the point A and find Dv (Fig. 4).

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

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

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

Let us decompose the vector Dv into two components. To do this from the point A(Fig. 4) in the direction of velocity v we plot the vector equal in absolute value to v 1 . Obviously, 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 in the opposite direction during slow motion. Characterizes the change in speed module. It is usually designated 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 coefficient in the accompanying basis. In this case, not a vector notation is used, but a “scalar” one - as usual for the projection or coordinates of a vector - .

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

where is the 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 the tangential acceleration in vector form:

Conclusion

The expression for tangential acceleration can be found by differentiating with respect to time the velocity vector, represented in terms of the 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 geometric considerations,

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

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

Elementary formula

where is the normal (centripetal) acceleration, is the (instantaneous) linear speed of movement along the trajectory, is the (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, given).

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 a 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 must keep in mind that centripetal acceleration is not the full acceleration vector, but only its component perpendicular to the trajectory (or, what is the same, perpendicular to the instantaneous velocity vector); the full acceleration vector then also includes a tangential component (tangential acceleration), the direction coinciding with the tangent to the trajectory (or, what is the same, with the instantaneous speed).

Conclusion

The fact that the decomposition of the acceleration vector into components - one along the tangent to the vector trajectory (tangential acceleration) and the other orthogonal to it (normal acceleration) - can be convenient and useful is quite obvious in itself. This is aggravated by the fact that when moving at a constant speed, 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 clearly defined properties and structure, and normal acceleration contains quite 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 practically without changes).

In order to be able to solve various problems on the movement of bodies in physics, you need to know the definitions of physical quantities, as well as the formulas by which they are related. This article will cover the questions of what tangential velocity is, what total acceleration is, and what components make it up.

Concept of speed

The two main quantities of the kinematics of movement of bodies in space are speed and acceleration. Speed ​​describes the speed of movement, so the mathematical form for it is as follows:

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Here l¯ is the displacement vector. In other words, speed is the time derivative of the distance traveled.

As you know, every body moves along an imaginary line, which is called a trajectory. The velocity vector is always directed tangentially to this trajectory, no matter what point the moving body is located at.

There are several names for the quantity v¯, if we consider it together with the trajectory. So, since it is directed tangentially, it is called tangential velocity. It can also be spoken of as a linear physical quantity as opposed to angular velocity.

Speed ​​is calculated in meters per second in SI, but in practice kilometers per hour are often used.

The concept of acceleration

In contrast to speed, which characterizes the speed at which a body passes a trajectory, acceleration is a quantity that describes the rate of change in speed, which is written mathematically as follows:

Like speed, acceleration is a vector characteristic. However, its direction is not related to the velocity vector. It is determined by the change in direction v¯. If during the movement the speed does not change its vector, then the acceleration a¯ will be directed along the same line as the speed. This acceleration is called tangential. If the speed changes direction while maintaining its absolute value, then the acceleration will be directed towards the center of curvature of the trajectory. It's called normal.

Acceleration is measured in m/s2. For example, the well-known acceleration of gravity is tangential when an object rises or falls vertically. Its value near the surface of our planet is 9.81 m/s2, that is, for every second of fall, the speed of the body increases by 9.81 m/s.

The cause of acceleration is not speed, but force. If a force F acts on a body of mass m, then it will inevitably create an acceleration a, which can be calculated as follows:

This formula is a direct consequence of Newton's second law.

Full, normal and tangential acceleration

Speed ​​and acceleration as physical quantities were discussed in previous paragraphs. Now we will study in more detail which components make up the total acceleration a¯.

Let us assume that the body moves with speed v¯ along a curved path. Then the equality will be true:

The vector u¯ has unit length and is directed along the tangent line to the trajectory. Using this representation of the velocity v¯, we obtain the equality for the total acceleration:

a¯ = dv¯/dt = d(v*u¯)/dt = dv/dt*u¯ + v*du¯/dt.

The first term obtained in the right equality is called tangential acceleration. Velocity is related to it by the fact that it quantifies the change in the absolute value of v¯ without taking into account its direction.

The second term is the normal acceleration. It quantitatively describes the change in the velocity vector, without taking into account the change in its modulus.

If we denote the tangential and normal components of the total acceleration a as at and an, then the modulus of the latter can be calculated using the formula:

a = √(at2 + an2).

Relationship between tangential acceleration and speed

The corresponding connection is described by kinematic expressions. For example, in the case of motion in a straight line with constant acceleration, which is tangential (the normal component is zero), the following expressions are valid:

In the case of circular motion with constant acceleration, these formulas are also valid.

Thus, whatever the trajectory of the body’s movement, the tangential acceleration through the tangential velocity is calculated as the time derivative of its module, that is:

For example, if the speed changes according to the law v = 3*t3 + 4*t, then at will be equal to:

at = dv/dt = 9*t2 + 4.

Speed ​​and normal acceleration

We write explicitly the formula for the normal component an, we have:

an¯ = v*du¯/dt = v*du¯/dl*dl/dt = v2/r*re¯

Where re¯ is a vector of unit length, which is directed towards the center of curvature of the trajectory. This expression establishes the relationship between tangential velocity and normal acceleration. We see that the latter depends on the modulus v at a given time and on the radius of curvature r.

Normal acceleration appears whenever the velocity vector changes, but it is zero if this vector remains in the same direction. It makes sense to talk about the quantity an¯ only when the curvature of the trajectory is a finite quantity.

We noted above that when moving in a straight line, there is no normal acceleration. However, in nature there is a type of trajectory along which an has a finite value, and at = 0 for |v¯| = const. This trajectory is a circle. For example, rotation at a constant frequency of a metal shaft, carousel or planet around its own axis occurs with constant normal acceleration an and zero tangential acceleration at.

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 a problem (coordinate method of specifying the movement of a point)

Basic formulas for the kinematics of a material point

Let us present the basic formulas of the kinematics of a material point. After which we will give their conclusion and presentation of the theory.

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

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

Acceleration point:
;
;
;
; ;

Tangential (tangential) acceleration:
;
;
.

Normal acceleration:
;
;
.

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

.

Radius vector and point trajectory

Let's consider the movement of the material point M. Let us choose a fixed rectangular coordinate system Oxyz with a center at some fixed point O. Then the position of point M is uniquely determined by its coordinates

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

When a point moves, the coordinates change over time.
(1)
That is, they are functions of time.

Then the system of equations

can be thought of as the equation of a curve defined 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 the axes and coordinate systems can be selected so that they lie in this plane. Then the trajectory is determined by two equations

In some cases, time can be eliminated from these equations.

Then the trajectory equation will have the form:

where is some function. This dependency contains only the variables and . It does not contain the parameter.

Velocity of a material point
,
The speed of a material point is the derivative of its radius vector with respect to time.

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

In mechanics, derivatives with respect to time are denoted by a dot above the symbol. Let's substitute here the expression for the radius vector:
.

where we have clearly indicated the dependence of coordinates on time. We get:
.
Where
.

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

Thus
.
Speed ​​Module: Tangent to the path.

From a mathematical point of view, the system of equations (1) can be considered as an equation of a line (curve) defined by parametric equations. Time, in this consideration, plays the role of a parameter. From the course of mathematical analysis it is known that the direction vector for the tangent to this curve has the components:
But these are the components of the point’s velocity vector. That is
;
;
.
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 a position with the radius vector (see figure). And at the moment of time - in a position with the radius vector.
.
Let's draw a straight line through the points.
By definition, a tangent is a straight line to which the straight line tends as .

Let us introduce the following notation: tangent direction vector of unit length:
.
Let us show that the length of this vector is equal to one. Indeed, since
, That:
.

Then the speed vector of the point can be represented as:
.

Acceleration of a material point

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

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

Tangential (tangent) and normal acceleration

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

The vector is directed tangentially to the trajectory. In which 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 unity. Then the square of its length is also equal to one:
.
Here and below, two vectors in parentheses denote the scalar product of vectors. Let's differentiate the last equation with respect to time:
;
;
.
Since the scalar product of vectors and is equal to zero, these vectors are perpendicular to each other. Since the vector is directed tangent to the trajectory, 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 represents the decomposition of acceleration into two mutually perpendicular components - tangent to the trajectory and perpendicular to the tangent.

Since then
(3) .

Tangential (tangential) acceleration

Let's multiply both sides of the equation (2) scalar to :
.
Because , then .
;
.
Then
.
Here we put:

From this we can see that the tangential acceleration is equal to the projection of the total acceleration onto the direction of the tangent to the trajectory or, which is the same, onto the direction of the point’s velocity.

Tangential (tangential) acceleration of a material point is the projection of its total acceleration onto the direction of the tangent to the trajectory (or to the direction of velocity).

We use the symbol to denote the tangential acceleration vector directed along the tangent to the trajectory. Then is a scalar quantity equal to the projection of the total acceleration onto the direction of the tangent. It can be both positive and negative.
.

Substituting , we have:
.
Let's put it into the formula:
.
Then: That is, the tangential acceleration is equal to the time derivative of the absolute velocity of the point. Thus,. As the speed increases, the tangential acceleration is positive (or directed along the speed). As the speed decreases, the tangential acceleration is negative (or in the opposite direction to the speed).

Now let's examine the vector.

Consider a unit vector tangent to the trajectory.
.

Let's 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 a material point moves, the end of the vector will move along this sphere. That is, it will rotate around its origin. Let be the instantaneous angular velocity of rotation of the vector at the moment of 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 rotating motion. Vector module:
.
Now consider the position of the point for two close moments in time. Let the point be in position at the moment of time, and in position at the moment of time.

Let and be unit vectors directed tangentially to the trajectory at these points. Through the points and we draw planes perpendicular to the vectors and .

Let and be unit vectors directed tangentially to the trajectory at these points. Through the points and we draw planes perpendicular to the vectors and .

Let be a straight line formed by the intersection of these planes. From a point we lower a perpendicular to a straight line.
If the positions of the points are close enough, then the movement of the point can be considered as 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 points and . (2) Thus, we found the modulus of the time derivative of the vector:
(4) .
As we indicated earlier, the vector is perpendicular to the vector. (3) From the above reasoning it is clear that it is directed towards the instantaneous center of curvature of the trajectory. This direction is called the principal normal.
.

Let's multiply both sides of the equation (2) scalar to :
(2) .
.
Because , then .
;
.
Normal acceleration

The normal acceleration of a material point is the projection of its total 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 speed of a point, and it is related to the radius of curvature of the trajectory.
.

From here you can find the radius of curvature of the trajectory: (4) And in conclusion, we note that the formula
.
can be rewritten as follows:
,
Here we have applied the formula for the cross product of three vectors:
.

which they framed
;
.
So we got:
.
Let's equate the modules of the left and right parts:
.
But the vectors are also mutually perpendicular. That's why
.
Then

This is a well-known formula from differential geometry for the curvature of a curve.

See also: 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.

Figure 1 – Tangential acceleration

The direction of the tangential acceleration vector coincides with the direction of the linear velocity or is opposite to it, from Fig. 1. That is, the tangential acceleration vector lies on the same axis with the tangent circle, which is the trajectory of the body. 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, shown in Fig. 1. Normal acceleration characterizes the change in speed in direction and is denoted by n. The normal acceleration vector is directed along the radius of curvature of the trajectory. Full acceleration

(9)

(10)

in curvilinear motion, it consists of tangential and normal accelerations according to the rule of vector addition and is determined by the formula:

(11)

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

1.1.5 Translational and rotational motion of an absolutely rigid body The movement of the body is considered progressive

, if any straight line segment rigidly connected to the body constantly moves parallel to itself. During translational motion, all points of the body make the same movements, travel the same paths, have equal speeds and accelerations, and describe the same trajectories. Rotation of a rigid body around a fixed axis

When a body rotates, the radius of the circle described by a point of this body will rotate through a certain angle over a time interval. Due to the invariance of the relative position of the points of the body, the radii of the circles described by any other points of the body rotate through the same angle during the same time. This angle is a value that characterizes the rotational motion of the entire body as a whole. From this we can conclude that to describe the rotational motion of an absolutely rigid body around a fixed axis, you need to know only one variable - the angle through which the body will rotate in a certain time.

The relationship between linear and angular velocities for each point of a rigid body is given by the formula:

(12)