Speed ​​is a physical quantity that characterizes the speed of movement and direction of movement of a material point relative to the chosen reference system; by definition, equal to the derivative of the radius vector of a point with respect to time.

Speed ​​in a broad sense is the speed of change of any quantity (not necessarily the radius vector) depending on another (more often it means changes in time, but also in space or any other). So, for example, they talk about angular velocity, the rate of temperature change, the rate of a chemical reaction, the group velocity, the rate of connection, etc. Mathematically, the “rate of change” is characterized by the derivative of the quantity under consideration.

Acceleration is denoted by the rate of change of speed, that is, the first derivative of speed with respect to time, a vector quantity showing how much the velocity vector of a body changes as it moves per unit time:

acceleration is a vector, that is, it takes into account not only the change in the magnitude of the speed (the magnitude of the vector quantity), but also the change in its direction. In particular, the acceleration of a body moving in a circle with a constant absolute velocity is not zero; the body experiences a constant magnitude (and variable in direction) acceleration directed towards the center of the circle (centripetal acceleration).

The unit of acceleration in the International System of Units (SI) is meters per second per second (m/s2, m/s2),

The derivative of acceleration with respect to time, that is, the quantity characterizing the rate of change of acceleration, is called jerk:

Where is the jerk vector.

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

Average acceleration

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

where is the acceleration vector.

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

At time t1 (see Figure 1.8) the body has speed 0. At time t2 the body has speed . According to the rule of vector subtraction, we find the vector of change in speed Δ = - 0. Then the acceleration can be determined as follows:

The SI unit of acceleration is 1 meter per second per second (or meter per second squared), that is

A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.

Instant acceleration

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

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

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

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

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

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

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

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

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

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

Where - acceleration vector.

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

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

Rice. 1.8. Average acceleration.

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

A meter per second squared is equal to the acceleration of a point moving in a straight line, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.

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

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

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

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

V 2 then the direction of the acceleration vector is opposite to the direction of the velocity vector 2. In other words, in this case what happens is slowing down, in this case the acceleration will be negative (and

Rice. 1.9. Instant acceleration.

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

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

Rice. 1.10. Tangential acceleration.

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

Normal acceleration

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

Full acceleration

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

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

= τ + n

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

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

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

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

Rice. 1.10. Tangential acceleration.

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

Normal acceleration

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

Full acceleration

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

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

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

Force. Weight. Newton's laws.

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

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

Newton's first law

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

Newton's second law

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

Newton's third law

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

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

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

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

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

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

Absolutely inelastic impact They call this impact interaction in which bodies connect (stick together) with each other and move on as one body.

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

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

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

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

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

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

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

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

Potential energy is the mechanical energy of a system of bodies, determined by their relative position and the nature of the interaction forces between them. Numerically, the potential energy of a system in its given position is equal to the work that will be done by the forces acting on the system when moving the system from this position to the one where the potential energy is conventionally assumed to be zero (E n = 0). The concept of “potential energy” applies only to conservative systems, i.e. systems in which the work of the acting forces depends only on the initial and final positions of the system.

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

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

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

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

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

The average power over a period of time is distinguished:

and instantaneous power at a given time:

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

The SI unit of power is the watt, equal to one joule divided by a second.

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

Physics exam questions(Part I, 2011).

Kinematics of translational motion. Frames of reference. Trajectory, path length, movement. Speed ​​and acceleration. Average, average ground, instantaneous speed. Normal, tangential and full acceleration.

Kinematic characteristics of rotational motion around a fixed axis: angular velocity, angular acceleration.

Dynamics of translational motion. Newton's laws. (Savelyev I.V. T.1 § 7, 9, 11). Basic physical quantities and their dimensions. (Savelyev I.V. T.1 § 10). Types of forces in mechanics. (Savelyev I.V. T.1 § 13–16).

Kinetic and potential energy. Mechanical work and power. Conservative and non-conservative forces. Work in the field of these forces. Law of energy conservation.

Impulse of a mechanical system. Law of conservation of momentum.

The moment of force relative to a point and relative to the axis of rotation.

The angular momentum of a material point relative to the point and relative to the axis of rotation. The moment of momentum of a body relative to an axis. Law of conservation of angular momentum.

The basic law of the dynamics of rotational motion. Moments of inertia of homogeneous bodies of regular geometric shape. Steiner's theorem on parallel axes.

Kinetic energy, work and power during rotational motion. Comparison of basic formulas and laws of translational and rotational motion.

Kinematics of harmonic oscillations. Quantities characterizing harmonic oscillations: period, frequency, amplitude, phase. Relationship between oscillation period and cyclic frequency. Dependences of displacement, speed and acceleration on time. Relevant graphs.

Equation of harmonic vibrations in differential form. Dependence of displacement on time. Relationship between cyclic frequency and mass of an oscillating point. Energy of harmonic vibrations (kinetic, potential and total). Relevant graphs.

Mathematical and physical pendulums. Formulas for the period of small oscillations. (Savelyev I.V. T.1 § 54).

Addition of harmonic vibrations of the same direction and the same frequency. Vector diagram. (Savelyev T.1 § 55).

Damped oscillations. Equation of damped oscillations in differential form. Dependence of the displacement and amplitude of damped oscillations on time. Attenuation coefficient. Logarithmic decrement of oscillations. (Savelyev I.V. T.1 § 58).

Forced vibrations. Equation of forced oscillations in differential form. Displacement, amplitude and frequency of forced oscillations. The phenomenon of resonance. Graph of amplitude versus frequency.

Waves. Wave propagation in an elastic medium. Transverse and longitudinal waves. Wave front and wave surfaces. Wavelength. Traveling wave equation. (Savelyev T.2 § 93-95).

Formation of standing waves. Standing wave equation. Standing wave amplitude. (Savelyev I.V. T.2 § 99)

Two approaches to the study of macrosystems: molecular kinetic and thermodynamic. Basic parameters of macrosystems. Equation of state of an ideal gas (Clapeyron-Mendeleev equation). (Savelyev I.V. T.1 § 79–81, 86).

Equation of state of a real gas (van der Waals equation). Theoretical van der Waals isotherm and experimental isotherm of real gas. Critical state of matter. (Savelyev I.V. T.1 § 91, § 123–124).

Internal energy of the system. Internal energy of an ideal gas. Two ways to change internal energy. Quantity of heat. Heat capacity. Relationship between specific and molar heat capacities.

Work with volume changes. The first law of thermodynamics. Mayer's formula. Application of the first law of thermodynamics to isoprocesses of an ideal gas.

Classical theory of heat capacity of an ideal gas. Boltzmann's theorem on the uniform distribution of energy over the degrees of freedom of a molecule. Calculation of the internal energy of an ideal gas and its heat capacities through the number of degrees of freedom. (Savelyev I.V. T.1 § 97).

Application of the first law of thermodynamics to an adiabatic process. Poisson's equation. (Savelyev I.V. T.1 § 88).

1. Kinematics of translational motion. Frames of reference. Trajectory, path length, movement. Speed ​​and acceleration. Average, average ground, instantaneous speed. Normal, tangential and full acceleration.

Kinematics of translational motion

During the translational motion of a body, all points of the body move equally, and, instead of considering the movement of each point of the body, we can consider the movement of only one point of it.

The main characteristics of the movement of a material point: the trajectory of movement, the movement of the point, the path it has traveled, coordinates, speed and acceleration.

The line along which a material point moves in space is called trajectory.

By moving material point over a certain period of time is called the displacement vector ∆r=r-r 0 , directed from the position of a point at the initial moment of time to its position at the final moment.

Speed material point is a vector characterizing the direction and speed of movement of the material point relative to the reference body. Acceleration vector characterizes the speed and direction of change in the speed of a material point relative to the reference body.

average speed- vector physical quantity equal to the ratio of the displacement vector to the period of time during which this displacement occurs:

Instantspeed - vector physical quantity equal to the first derivative from radius vector in time:

Instantaneous speedv is a vector quantity equal to the first derivative of the radius - the vector of a moving point with respect to time. Since the secant in the limit coincides with the tangent, then velocity vectorvdirected tangentially to the trajectory in the direction of movement (Figure 1.2).

As ∆t decreases, the path ∆S will increasingly approach |∆r|, therefore instantaneous speed module:

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

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

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

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

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

a= a τ + a n

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

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

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

where a – acceleration vector.

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

At time t1 (see Fig. 1.8) the body has a speed V 0. At time t2 the body has a speed V. According to the rule of subtracting vectors, we find the vector of change in speed ΔV = V - V 0 Then the acceleration can be determined as follows:

Rice. 1.8. Average acceleration.

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

A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.

You can also enter average moving speed, which will be vector, equal to the ratio movements by the time during which it was completed:

The average speed determined in this way can be equal to zero even if the point (body) actually moved (but at the end of the time interval returned to its original position).

If the movement occurred in a straight line (and in one direction), then the average ground speed is equal to the module of the average speed along the movement.

The movement of bodies occurs in space and time. Therefore, to describe the movement of a material point, it is necessary to know in which places in space this point was located and at what moments in time it passed this or that position.

Reference body - an arbitrarily selected body, relative to which the position of other bodies is determined.

Reference system - a set of coordinate systems and clocks associated with a reference body.

The most commonly used coordinate system is Cartesian - the orthonormal basis of which is formed by three modulo unit and mutually orthogonal vectors i j k r r r drawn from the origin.

Arbitrary point position M characterized radius vector R r connecting the origin O with a dot M . r x i y j z k r r r r = + + , r = r = x 2 + y 2+ z 2 r

The motion of a material point is completely determined if the Cartesian coordinates of the material point are given depending on time: x = x(t) y = y(t) z =z(t)

These equations are called kinematic equations of motion of a point . They are equivalent to one vector equation of motion of a point.

A line described by a moving material point (or body) relative to a chosen reference system is called trajectory . The trajectory equation can be obtained by eliminating the parameter t from kinematic equations. Depending on the shape of the trajectory, the movement can be straightforward or curvilinear .

Length of the path point is the sum of the lengths of all sections of the trajectory traversed by this point during the period of time under consideration s = s(t) . Path length - scalar function of time.

Move vector r r r 0 r r r = - vector 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).

The line along which a material point moves in space is called the trajectory of its movement. In other words, trajectory of movement call the set of all successive positions occupied by a material point as it moves in space.

One of the basic concepts of mechanics is the concept of a material point, which means a body with mass whose dimensions can be neglected when considering its motion. The movement of a material point is the simplest problem of mechanics, which will allow us to consider more complex types of movements.

The movement of a material point occurs in space and changes over time. Real space is three-dimensional, and the position of a material point at any moment in time is completely determined by three numbers - its coordinates in the chosen reference system. The number of independent quantities whose specification is necessary to unambiguously determine the position of a body is called the number of its degrees of freedom. As a coordinate system, we will choose a rectangular, or Cartesian, coordinate system. To describe the movement of a point, in addition to a coordinate system, you also need to have a device with which you can measure various periods of time. Let's call such a device a clock. The chosen coordinate system and the clock associated with it form a reference system.

D
Ecartesian coordinates X,Y,Z determine the radius vector in space z, the tip of which describes the trajectory of a material point as it changes over time. The length of a point's trajectory represents the distance traveled S(t). Path S(t) is a scalar quantity. Along with the distance traveled, the movement of a point is characterized by the direction in which it moves. The difference between two radius vectors taken at different times forms the point displacement vector (Fig.).

In order to characterize how quickly the position of a point in space changes, the concept of speed is used. Under the average speed of movement along the trajectory for a finite time  t understand the ratio of the final path traveled during this time  S In time:

. (1.1)

The speed of a point moving along a trajectory is a scalar quantity. Along with it, we can talk about the average speed of movement of a point. This speed is a quantity directed along the displacement vector,

. (1.2)

If moments in time t 1 , And t 2 infinitely close, then time  t infinitesimal and in this case denoted by dt. During dt a point travels an infinitesimal distance dS. Their ratio forms the instantaneous speed of the point

. (1.3)

Derivative of the radius vector r in time determines the instantaneous speed of movement of the point.

. (1.4)

Since the displacement coincides with an infinitesimal element of the trajectory dr= dS, then the velocity vector is directed tangentially to the trajectory, and its magnitude:

. (1.5)

N
and fig. shows the dependence of the distance traveled S from time t. Speed ​​vector v(t) is directed tangent to the curve S(t) at the moment of time t. From Fig. it can be seen that the angle of inclination of the tangent to the axis t equals

.

Integrating expression (1.5) in the time interval from t 0 before t, we obtain a formula that allows us to calculate the path traveled by the body in time t-t 0 if the time dependence of its speed is known v(t)

. (1.6)

G
The geometric meaning of this formula is clear from Fig. By definition of the integral, the distance traveled is the area enclosed by the curve v=v(t) in the range from t 0 before t.In the case of uniform motion, when the speed maintains its constant value throughout the entire movement, v=const; hence the expression

, (1.7)

Where S 0 - the path traveled to the initial time t 0 .

The derivative of velocity with respect to time, which is the second derivative with respect to time of the radius vector, is called the acceleration of the point:

. (1.8)

The acceleration vector a is directed along the velocity increment vector dv. Let a = const. This important and frequently encountered case is called uniformly accelerated or uniformly decelerated (depending on the sign of the quantity a) motion. Let's integrate expression (1.8) in the range from t= 0 to t:

(1.9)

(1.10)

and use the following initial conditions:
.

Thus, with uniformly accelerated motion

. (1.11)

In particular, with one-dimensional motion, for example along the axis X,
. The case of rectilinear motion is shown in Fig. At large times, the dependence of the coordinate on time is a parabola.

IN In general, the movement of a point can be curvilinear. Let's consider this type of movement. If the trajectory of a point is an arbitrary curve, then the speed and acceleration of the point as it moves along this curve change in magnitude and direction.

Let's choose an arbitrary point on the trajectory. Like any vector, the acceleration vector can be represented as the sum of its components along two mutually perpendicular axes. As one of the axes, we take the direction of the tangent at the considered point of the trajectory, then the other axis will be the direction of the normal to the curve at the same point. The acceleration component directed tangentially to the trajectory is called tangential acceleration a t, and directed perpendicular to it - normal acceleration a n .

We obtain formulas expressing the quantities a t, And a n through motion characteristics. For simplicity, let us consider a flat curve instead of an arbitrary curvilinear trajectory. The final formulas remain valid in the general case of a non-planar trajectory.

B
Thanks to acceleration, the point acquires speed over time dt small change dv. In this case, the tangential acceleration, directed tangentially to the trajectory, depends only on the magnitude of the speed, but not on its direction. This change in velocity is equal to dv. Therefore, tangential acceleration can be written as the time derivative of the velocity:

. (1.12)

On the other hand, change dv n, directed perpendicular to v, characterizes only the change in the direction of the velocity vector, but not its magnitude. In Fig. shows the change in the velocity vector caused by the action of normal acceleration. As can be seen from Fig.
, and thus, up to a value of the second order of smallness, the velocity remains unchanged v=v".

Let's find the value a n. The easiest way to do this is to take the simplest case of curvilinear motion - uniform motion in a circle. Wherein a t=0. Consider the movement of a point over time dt along an arc dS circle radius R.

WITH
crust v And v", as noted, remain equal in magnitude. Shown in Fig. The triangles thus turn out to be similar (as isosceles with equal angles at the vertices). From the similarity of triangles it follows
, from where we find the expression for normal acceleration:

. (1.13)

The formula for total acceleration during curved motion is:

. (1.14)

We emphasize that relations (1.12), (1.13) and (1.14) are valid for any curvilinear motion, and not just for circular motion. This is due to the fact that any section of a curvilinear trajectory in a sufficiently small neighborhood of a point can be approximately replaced by an arc of a circle. The radius of this circle, called the radius of curvature of the trajectory, will vary from point to point and requires special calculation. Thus, formula (1.14) remains valid in the general case of a spatial curve.

2. Kinematic characteristics of rotational motion around a fixed axis: angular velocity, angular acceleration.

The motion of a rigid body in which two of its points ABOUT And ABOUT"remain motionless is called rotational movement around a fixed axis, and a fixed straight line OO" call axis of rotation. Let an absolutely rigid body rotate around a fixed axis OO" (Fig. 2.12).

Rice. 2.12

Let's follow some point M this solid body. During dt dot M makes an elementary movement dr . At the same rotation angle dφ, another point, separated from the axis by a greater or lesser distance, makes a different movement. Consequently, neither the movement of a certain point of a rigid body itself, nor the first derivative, nor the second derivative can serve as a characteristic of the motion of the entire rigid body. During the same time dt radius vector R drawn from the point 0 " exactly M, will turn at an angle dφ. The radius vector of any other point will rotate by the same angle (since the body is absolutely solid, otherwise the distance between the points must change). Angle of rotation dφ characterizes the movement of the entire body over time dt. It is convenient to introduce the vector of elementary rotation of the body, numerically equal to dφ and directed along the axis of rotation OO" so that, looking along the vector, we see a clockwise rotation (the direction of the vector and the direction of rotation are related by the “gimlet rule”). Elementary rotations satisfy the usual rule of vector addition:

Angular velocity body rotation

Angular velocity body at a given moment t is the value to which the average angular velocity tends if it tends to zero.

The angular velocity of a rigid body is the first derivative of the rotation angle with respect to time.

Dimension: [radian/time]; ; .

Angular velocity can be represented as a vector. The angular velocity vector is directed along the axis of rotation in the direction from which the rotation is visible counterclockwise.

If the angular velocity is not a constant value, then another characteristic of rotation is introduced - angular acceleration.

Angular acceleration characterizes the change in the angular velocity of a body over time.

If over a period of time the angular velocity receives an increment, then the average angular acceleration is equal to

rotation, - one of the simplest types of rigid body motion. V. movement around a fixed axis is a movement in which all points of the body, moving in parallel planes, describe circles with centers lying on one fixed line perpendicular to the planes of these circles and called. axis of rotation. Speed ​​of an arbitrary point of the body v = , where w - angular velocity body, r - radius vector drawn to a point from the center of the circle it describes. Angular acceleration body e = M/I, where M is the external moment. forces relative to the axis of rotation, I is the moment of inertia of the body relative to the same axis.

V. d. around a fixed point is movement, in which all points of the body move along concentric surfaces. spheres with centers at a fixed point. At each moment of time, this movement can be considered as rotation around an instantaneous axis of rotation passing through a fixed point. The speed of an arbitrary point of the body is v =, here r is the radius vector drawn to the point from a fixed point of the body. Basic law of dynamics: dL/dt = M, where L - angular momentum body relative to a fixed point, M is the moment relative to the same point of all external. forces applied to a body are called the main point of external forces. This law is also valid for the rotation of a rigid body around its center of inertia, regardless of whether the latter is at rest or moves arbitrarily. The theory of V. d. has many. applications in celestial mechanics, ext. ballistics, gyroscope theory, theory of machines and mechanisms.

Distance traveledS , moving dr, speed v, tangential and normal acceleration a t, And a n, are linear quantities. To describe curvilinear motion, along with them, you can use angular quantities.

Let us consider in more detail the important and frequently encountered case of motion in a circle. In this case, along with the length of the circular arc, the movement can be characterized by the angle of rotation φ around the axis of rotation. Size

(1.15)

called angular speed. Angular velocity is a vector, the direction of which is associated with the direction of the axis of rotation of the body (Fig.).

Let us pay attention to the fact that, while the angle of rotation itself φ is a scalar, infinitesimal rotation dφ - a vector quantity whose direction is determined by the right-hand rule, or gimlet, and is associated with the axis of rotation. If the rotation is uniform, then ω =const and the point on the circle rotates through equal angles around the axis of rotation in equal times. The time during which it makes a full revolution, i.e. turns at an angle 2π, called period of movement T. Expression (1.15) can be integrated over the range from zero to T and get angular frequency

. (1.16)

The number of revolutions per unit time is the reciprocal of the period - the cyclic rotation frequency

ν =1/ T. (1.17)

It is not difficult to obtain a connection between the angular and linear velocity of a point. When moving in a circle, the element of the arc is related to an infinitesimal rotation by the relation dS = R dφ. Substituting it into (1.15), we find

v = ω r. (1.18)

Formula (1.18) relates the values ​​of angular and linear velocities. Relation connecting vectors ω And v, follows from Fig. Namely, the linear velocity vector is the vector product of the angular velocity vector and the radius vector of the point r:

. (1.19)

Thus, the angular velocity vector is directed along the axis of rotation of the point and is determined by the rule of the right hand or gimlet.

Angular acceleration- time derivative of the angular velocity vector ω (respectively, the second time derivative of the rotation angle)

Let us express tangential and normal acceleration in terms of angular velocities and acceleration. Using connection (1.18), (1.12) and (1.13), we obtain

a t = β · R, a = ω 2 · R. (1.20)

Thus, for total acceleration we have

. (1.21)

Magnitude β plays the role of tangential acceleration: if β = 0.total acceleration when rotating a point is not zero, a =R·ω 2 ≠ 0.

3. Dynamics of translational motion. Newton's laws. (Savelyev I.V. T.1 § 7, 9, 11). Basic physical quantities and their dimensions. (Savelyev I.V. T.1 § 10). Types of forces in mechanics. (Savelyev I.V. T.1 § 13–16).

When bodies move, their speeds usually change either in magnitude or in direction, or simultaneously in both magnitude and direction.

If you throw a stone at an angle to the horizon, then its speed will change both in magnitude and direction.

A change in the speed of a body can occur either very quickly (the movement of a bullet in the barrel when fired from a rifle) or relatively slowly (the movement of a train when it departs). In order to be able to find the speed at any time, it is necessary to enter a value characterizing the rate of change of speed. This quantity is calledacceleration.

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

Where - acceleration vector .

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

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

Rice. 1.8. Average acceleration.

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

A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.