Point acceleration for all 3 ways to accelerate movement

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

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

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

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

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

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

Natural axes and natural trihedron

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

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

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

Normal and tangential acceleration of a point

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

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

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

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

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

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

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

Special cases.

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

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

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

Conclusion: normal acceleration exists only at curvilinear

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

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

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

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

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

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

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

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

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

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

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

Equation of uniform motion of a point

Acceleration decomposition a (t) (\displaystyle \mathbf (a) (t)\ \ ) to tangential and normal a n (\displaystyle \mathbf (a)_(n)); (τ (\displaystyle \mathbf (\tau ) )- unit tangent vector).

Tangential acceleration- acceleration component directed tangentially to the trajectory of motion. Characterizes the change in the velocity module in contrast to the normal component, which characterizes the change in the direction of velocity. Tangential acceleration is equal to the product of the unit vector directed along the velocity of motion and the derivative of the velocity modulus with respect to time. Thus, it is directed in the same direction as the velocity vector during accelerated motion (positive derivative) and in the opposite direction during slow motion (negative derivative).

Usually indicated by the symbol chosen for acceleration, with the addition of a subscript indicating the tangential component: a τ (\displaystyle \mathbf (a) _(\tau )\ \ ) or a t (\displaystyle \mathbf (a)_(t)\ \ ), w τ (\displaystyle \mathbf (w) _(\tau )\ \ ),u τ (\displaystyle \mathbf (u)_(\tau )\ \ ) etc.

Sometimes it is not a vector form that is used, but a scalar one - a τ (\displaystyle a_(\tau )\ \ ), denoting the projection of the total acceleration vector onto the unit vector of the tangent to the trajectory, which corresponds to the expansion coefficient along the accompanying basis.

Encyclopedic YouTube

• 1 / 5

  The magnitude of tangential acceleration as a projection of the acceleration vector onto the tangent to the trajectory can be expressed as follows:

  a τ = d v d t , (\displaystyle a_(\tau )=(\frac (dv)(dt)),)

  Where v = d l / d t (\displaystyle v\ =dl/dt)- 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 e τ (\displaystyle \mathbf (e)_(\tau )\ ), then we can write the tangential acceleration in vector form:

  a τ = d v d t e τ . (\displaystyle \mathbf (a) _(\tau )=(\frac (dv)(dt))\mathbf (e) _(\tau ).)

  Conclusion

  Conclusion 1

  The expression for tangential acceleration can be found by differentiating with respect to time the velocity vector, presented in the form v = v e τ (\displaystyle \mathbf (v) =v\,\mathbf (e) _(\tau )) through the unit tangent vector e τ (\displaystyle \mathbf (e)_(\tau )):

  a = d v d t = d (v e τ) d t = d v d t e τ + v d e τ d t = d v d t e τ + v d e τ d l d l d t = d v d t e τ + v 2 R e n , (\displaystyle \mathbf (a) =(\frac (d\mathbf ( v) )(dt))=(\frac (d(v\,\mathbf (e) _(\tau )))(dt))=(\frac (\mathrm (d) v)(\mathrm (d ) t))\mathbf (e) _(\tau )+v(\frac (d\mathbf (e) _(\tau ))(dt))=(\frac (\mathrm (d) v)(\ mathrm (d) t))\mathbf (e) _(\tau )+v(\frac (d\mathbf (e) _(\tau ))(dl))(\frac (dl)(dt))= (\frac (\mathrm (d) v)(\mathrm (d) t))\mathbf (e) _(\tau )+(\frac (v^(2))(R))\mathbf (e) _(n)\ ,)

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

  The notation used here is e n (\displaystyle e_(n)\ ) for a unit vector normal to the trajectory and l (\displaystyle l\ )- for the current trajectory length ( l = l (t) (\displaystyle l=l(t)\ )); the last transition also uses the obvious

  d l / d t = v (\displaystyle dl/dt=v\ )

  and, from geometric considerations,

  d e τ d l = e n R . (\displaystyle (\frac (d\mathbf (e) _(\tau ))(dl))=(\frac (\mathbf (e) _(n))(R)).)

  Conclusion 2

  If the trajectory is smooth (which is assumed), then:

  Both follow from the fact that the angle of the vector to the tangent will not be lower than the first order in . From here the desired formula immediately follows.

  Less strictly speaking, projection v (\displaystyle \mathbf (v)\ ) to the tangent at small d t (\displaystyle dt\ ) will practically coincide with the length of the vector v (\displaystyle \mathbf (v)\ ), since the angle of deviation of this vector from the tangent at small d t (\displaystyle dt\ ) is always small, which means the cosine of this angle can be considered equal to unity.

  Notes

  The absolute value of tangential acceleration depends only on the ground acceleration, coinciding with its absolute value, in contrast to the absolute value of normal acceleration, which does not depend on the ground acceleration, but depends on the ground speed.

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

  The second component of acceleration, equal to

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

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

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

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

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

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

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

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

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

  · 5) , - uniform curvilinear movement;

  · 6) , - curvilinear uniform motion;

  · 7) , - curvilinear movement with variable acceleration.

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

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

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

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

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

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

  Tangential component of acceleration

  Normal component of acceleration

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

  tangent to the curve and modulo equal to the product

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

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

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

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

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

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

  Vector quantity

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

  Substituting (6.6) into (6.5), we get

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

  Newton's third law

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

  F 12 = – F 21, (7.1)

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

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

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

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

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

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

  Hooke's law

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

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

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

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

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

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

  F tr = f N ,

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

  f = tga 0.

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

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

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

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

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

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

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

  Liquid (viscous) is the friction between a solid and a liquid or gaseous medium or its layers.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  Using expression (17.1), we obtain

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  a full job

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

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

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

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

  d ( T+P) = 0,

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

  The movement of a material point along a curved path is always accelerated, since even if the speed does not change in numerical value, it always changes in direction.

  In general, acceleration during curvilinear motion can be represented as a vector sum of tangential (or tangential) acceleration t and normal acceleration n: =t+n- rice. 1.4.

  Tangential acceleration characterizes the rate of change in velocity modulo. The value of this acceleration will be:

  Normal acceleration characterizes the rate of change in speed in direction. The numerical value of this acceleration, where r- radius of the contacting circle, i.e. a circle drawn through three infinitely close points B¢ , A, B, lying on the curve (Fig. 1.5). Vector n directed along the normal to the trajectory to the center of curvature (the center of the osculating circle).

  Numerical value of total acceleration

  where is the angular velocity.

  where is the angular acceleration.

  Angular acceleration is numerically equal to the change in angular velocity per unit time.

  In conclusion, we present a table that establishes an analogy between the linear and angular kinematic parameters of motion.

  End of work -

  This topic belongs to the section:

  Short course in physics

  Ministry of Education and Science of Ukraine.. Odessa National Maritime Academy..

  If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:

  What will we do with the received material:

  If this material was useful to you, you can save it to your page on social networks:

  Tweet

  All topics in this section:

  Basic SI units
  Currently, the International System of Units - SI - is generally accepted. This system contains seven basic units: meter, kilogram, second, mole, ampere, kelvin, candela and two additional ones -

  Mechanics
  Mechanics is the science of the mechanical movement of material bodies and the interactions between them that occur during this process. Mechanical movement is understood as a change in mutual sex over time.

  Newton's laws
  Dynamics is a branch of mechanics that studies the movement of material bodies under the influence of forces applied to them. Mechanics is based on Newton's laws. Newton's first law

  Law of conservation of momentum
  Let's consider the derivation of the law of conservation of momentum based on Newton's second and third laws.

  Relationship between work and change in kinetic energy
  Rice. 3.3 Let a body of mass m move along the x axis under

  Relationship between work and change in potential energy
  Rice. 3.4 We will establish this connection using the example of the work of gravity

  Law of conservation of mechanical energy
  Let us consider a closed conservative system of bodies. This means that the bodies of the system are not affected by external forces, and internal forces are conservative in nature. Full mechanical

  Collisions
  Let us consider an important case of interaction of solid bodies - collisions. Collision (impact) is the phenomenon of a finite change in the velocities of solid bodies over very short periods of time when they are not

  Basic law of the dynamics of rotational motion
  Rice. 4.3 To derive this law, consider the simplest case

  Law of conservation of angular momentum
  Let us consider an isolated body, i.e. a body that is not acted upon by an external moment of force. Then Mdt = 0 and from (4.5) it follows d(Iw)=0, i.e. Iw=const. If an isolated system consists

  Gyroscope
  A gyroscope is a symmetrical solid body that rotates around an axis that coincides with the axis of symmetry of the body, passing through the center of mass, and corresponding to the greatest moment of inertia.

  General characteristics of oscillatory processes. Harmonic vibrations
  Oscillations are movements or processes that have varying degrees of repeatability over time. In technology, devices using oscillatory processes can perform op.

  Oscillations of a spring pendulum
  Rice. 6.1 Let us attach a body of mass m to the end of the spring, which can

  Energy of harmonic vibration
  Let us now consider, using the example of a spring pendulum, the processes of energy change in a harmonic oscillation. It is obvious that the total energy of the spring pendulum is W=Wk+Wp, where the kinetic

  Addition of harmonic vibrations of the same direction
  The solution to a number of issues, in particular, the addition of several oscillations of the same direction, is greatly facilitated if the oscillations are depicted graphically, in the form of vectors on a plane. The resulting

  Damped oscillations
  In real conditions, resistance forces are always present in systems that oscillate. As a result, the system gradually expends its energy to perform work against resistance forces and

  Forced vibrations
  In real conditions, an oscillating system gradually loses energy to overcome frictional forces, so the oscillations are damped. For the oscillations to be undamped, it is necessary somehow

  Elastic (mechanical) waves
  The process of propagation of disturbances in a substance or field, accompanied by the transfer of energy, is called a wave. Elastic waves - the process of mechanical propagation in an elastic medium

  Wave interference
  Interference is the phenomenon of superposition of waves from two coherent sources, as a result of which a redistribution of wave intensity occurs in space, i.e. interference occurs

  Standing waves
  A special case of interference is the formation of standing waves. Standing waves arise from the interference of two counterpropagating coherent waves with the same amplitude. This situation may cause trouble

  Doppler effect in acoustics
  Sound waves are elastic waves with frequencies from 16 to 20,000 Hz, perceived by the human hearing organs. Sound waves in liquid and gaseous media are longitudinal. Into hard

  Basic equation of the molecular kinetic theory of gases
  Let us consider an ideal gas as the simplest physical model. An ideal gas is one for which the following conditions are met: 1) the dimensions of the molecules are so small that

  Distribution of molecules by speed
  Fig. 16.1 Let us assume that we were able to measure the speeds of all

  Barometric formula
  Let us consider the behavior of an ideal gas in a gravity field. As you know, as you rise from the surface of the Earth, the pressure of the atmosphere decreases. Let's find the dependence of atmospheric pressure on altitude

  Boltzmann distribution
  Let us express the gas pressure at heights h and h0 through the corresponding number of molecules per unit volume and u0, assuming that at different heights T = const: P =

  The first law of thermodynamics and its application to isoprocesses
  The first law of thermodynamics is a generalization of the law of conservation of energy taking into account thermal processes. Its formulation: the amount of heat imparted to the system is spent on doing work

  Number of degrees of freedom. Internal energy of an ideal gas
  The number of degrees of freedom is the number of independent coordinates that describe the movement of a body in space. A material point has three degrees of freedom, since when it moves in p

  Adiabatic process
  Adiabatic is a process that occurs without heat exchange with the environment. In an adiabatic process, dQ = 0, therefore the first law of thermodynamics in relation to this process is

  Reversible and irreversible processes. Circular processes (cycles). Operating principle of a heat engine
  Reversible processes are those that satisfy the following conditions. 1. After passing through these processes and returning the thermodynamic system to its original state in

  Ideal Carnot heat engine
  Rice. 25.1 In 1827, the French military engineer S. Carnot, re

  Second law of thermodynamics
  The first law of thermodynamics, which is a generalization of the law of conservation of energy taking into account thermal processes, does not indicate the direction of the occurrence of various processes in nature. Yes, first

  A process is impossible, the only result of which would be the transfer of heat from a cold body to a hot one
  In a refrigeration machine, heat is transferred from a cold body (the freezer) to a warmer environment. This would seem to contradict the second law of thermodynamics. Really against it

  Entropy
  Let us now introduce a new parameter of the state of a thermodynamic system - entropy, which fundamentally differs from other state parameters in the direction of its change. Elementary treason

  Discreteness of electric charge. Law of conservation of electric charge
  The source of the electrostatic field is an electric charge - an internal characteristic of an elementary particle that determines its ability to enter into electromagnetic interactions.

  Electrostatic field energy
  Let's first find the energy of a charged flat capacitor. Obviously, this energy is numerically equal to the work that needs to be done to discharge the capacitor.

  Main characteristics of current
  Electric current is the ordered (directed) movement of charged particles. The current strength is numerically equal to the charge passed through the cross section of the conductor per unit

  Ohm's law for a homogeneous section of a chain
  A section of the circuit that does not contain an EMF source is called homogeneous. Ohm experimentally established that the current strength in a homogeneous section of the circuit is proportional to the voltage and inversely proportional

  Joule-Lenz law
  Joule and, independently of him, Lenz experimentally established that the amount of heat released in a conductor with resistance R during time dt is proportional to the square of the current, resistive

  Kirchhoff's rules
  Rice. 39.1 To calculate complex DC circuits using

  Contact potential difference
  If two dissimilar metal conductors are brought into contact, then electrons are able to move from one conductor to another and back. The equilibrium state of such a system

  Seebeck effect
  Rice. 41.1 In a closed circuit of two dissimilar metals per g

  Peltier effect
  The second thermoelectric phenomenon - the Peltier effect - is that when an electric current is passed through the contact of two dissimilar conductors, a release or absorption occurs in it.

  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 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 the 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 with virtually no changes).