The movement of a body is called translational if. Vivid examples of such a movement

Forward movement

Fig 1. Translational movement of a body on a plane from left to right, with a segment arbitrarily selected in it AB. At first rectilinear, then curvilinear, turning into rotation of each point around its center with equal for a given moment angular velocities and equal turning radius values. Points O- instantaneous turning centers to the right. R- they are equal for each end of the segment, but the instantaneous radii of rotation are different for different moments of time.

Forward movement- this is the mechanical movement of a system of points (body), in which any straight line segment associated with a moving body, the shape and dimensions of which do not change during movement, remains parallel to its position at any previous moment in time.

The above illustration shows that, in contrast to the common statement. forward motion is not the opposite of rotational motion, but in general case can be considered as a set of turns - not completed rotations. This implies that rectilinear motion is a rotation around a center of rotation infinitely distant from the body.

In general, translational motion occurs in three-dimensional space, but its main feature - maintaining the parallelism of any segment to itself, remains in force.

Mathematically, translational motion in its own way the final result is equivalent to parallel carry. However, considered as physical process it represents in three-dimensional space a variant screw motion(See Fig. 2)

Examples of translational motion

For example, an elevator car moves forward. Also, to a first approximation, the cabin of the Ferris wheel performs translational motion. However, strictly speaking, the movement of the Ferris wheel cabin cannot be considered progressive.

One of the most important characteristics The movement of a point is its trajectory, which in general is a spatial curve that can be represented as conjugate arcs of different radii, each emanating from its own center, the position of which can change over time. In the limit, a straight line can be considered as an arc whose radius is equal to infinity.

Fig.2 Example of 3D translational motion of a body

In this case, it turns out that with translational motion in each at the moment time, any point of the body rotates around its instantaneous center of rotation, and the length of the radius at a given moment is the same for all points of the body. The velocity vectors of the points of the body, as well as the accelerations they experience, are identical in magnitude and direction.

When solving problems theoretical mechanics It can be convenient to consider the motion of a body as the addition of the motion of the center of mass of the body and the rotational motion of the body itself around the center of mass (this circumstance was taken into account when formulating Koenig’s theorem).

Device examples

Commercial scales, the cups of which move progressively, but not rectilinearly

The principle of translational motion is implemented in a drawing device - a pantograph, the leading and driven arms of which always remain parallel, that is, they move forward. In this case, any point on the moving parts makes specified movements in the plane, each around its instantaneous center of rotation with the same angular velocity for all moving points of the device.

It is important that the leading and driven arms of the device, although moving in harmony, represent two different bodies. Therefore, the radii of curvature along which given points on the leading and driven arms move can be made unequal, and this is precisely the point of using a device that allows you to reproduce any curve on a plane on a scale determined by the ratio of the lengths of the arms.

In fact, the pantograph provides synchronous translational movement of a system of two bodies: the “reader” and the “writer”, the movement of each of which is illustrated in the above drawing.

See also

• Rectilinear movement of a point
• Centripetal and centrifugal forces

Notes

Literature

• Newton I. Mathematical principles of natural philosophy. Per. and approx. A. N. Krylova. M.: Nauka, 1989
• S. E. Khaikin. Inertial forces and weightlessness. M.: “Science”, 1967. Newton I. Mathematical principles of natural philosophy. Per. and approx. A. N. Krylova.
• Frisch S. A. and Timoreva A. V. Well general physics, Textbook for physics, mathematics and physics and technology faculties state universities, Volume I. M.: GITTLE, 1957

Links

• V. I. Gervids Translational and rotational movements(flash). NRNU MEPhI (10.03.2011). - Physical demonstrations. Retrieved December 19, 2011.

Wikimedia Foundation. 2010.

• Miranda, Edison
• Zubkov, Valentin Ivanovich

See what “Forward movement” is in other dictionaries:

Forward movement- Forward movement. The movement of a straight segment AB occurs parallel to itself. FORWARD MOTION, movement of a body in which any straight line drawn in the body moves parallel to itself. During forward movement... ... Illustrated Encyclopedic Dictionary

FORWARD MOTION- TV movement body, while a straight line connecting any two points of the body moves, remaining parallel to its initial direction. With P. d., all points of the body describe the same trajectories and have the same ... ... Physical encyclopedia

forward motion- advancement, progress, step forward, the ice has broken, improvement, growth, shift, step, movement forward, progress, development Dictionary of Russian synonyms. forward movement noun, number of synonyms: 11 forward movement... Dictionary of synonyms

forward motion- solid body; translational motion The movement of a body in which a straight line connecting any two points of this body moves while remaining parallel to its initial direction... Polytechnic terminological explanatory dictionary

FORWARD MOTION- moving forward. Dictionary foreign words, included in the Russian language. Pavlenkov F., 1907 ... Dictionary of foreign words of the Russian language

FORWARD MOTION- movement of a body in which any straight line drawn in the body moves parallel to itself. During translational motion, all points of the body describe the same trajectories and have the same speeds and accelerations at each moment of time... Big Encyclopedic Dictionary

forward motion- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN advancetransiational advanceheadwayforward motion ... Technical Translator's Guide

forward motion- movement of a body in which any straight line (for example, AB in the figure) drawn in the body moves parallel to itself. During translational motion, all points of the body describe the same trajectories and have the same... ... Encyclopedic Dictionary

FORWARD MOTION- movement of a body, in which any straight line (for example, AB in the figure) drawn in the body moves parallel to itself. With P.D., all points of the body describe the same trajectories and have the same velocities and accelerations at each moment of time... Natural science. Encyclopedic Dictionary

forward motion- slenkamasis judesys statusas T sritis automatika atitikmenys: engl. translational motion; translational movement vok. fortschreitende Bewegung, f; Schiebung, f rus. forward movement, n pranc. mouvement de translation, m … Automatikos terminų žodynas

Books

• Progressive movement to Central Asia in trade and diplomatic-military relations. Additional material for the history of the Khiva campaign of 1873, Lobysevich F.I.. The book is a reprint edition of 1900. Despite the fact that it was carried out serious work to restore the original quality of the publication, some pages may...

In kinematics, as in statics, we will consider all rigid bodies as absolutely rigid. Kinematics problems solid breaks down into two parts:

1) motion assignment and definition kinematic characteristics general body movements; 2) determination of the kinematic characteristics of the movement of individual points of the body.

Let's start by considering the translational motion of a rigid body.

Translational motion is such a motion of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.

Translational motion should not be confused with rectilinear motion. When a body moves forward, the trajectories of its points can be any curved lines. Let's give examples.

1. The car body on a straight horizontal section of the road moves forward. In this case, the trajectories of its points will be straight lines.

2. The pair AB (Fig. 131), when the cranks (VI and ) rotate, also moves translationally (any straight line drawn in it remains parallel to its initial direction). The points of the partner move in circles.

The properties of translational motion are determined by the following theorem: during translational motion, all points of the body describe identical (overlapping, coinciding) trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.

To prove this, let us consider a rigid body undergoing translational motion relative to the reference frame Oxyz. Let us take two arbitrary points A and B in the body, the positions of which at time t are determined by radius vectors (Fig. 132); Let's draw a vector A B connecting these points. Then

(35)

In this case, the length AB is constant, like the distance between points of a rigid body, and the direction AB remains unchanged, since the body moves translationally. Thus, the vector AB remains constant throughout the movement of the body (). As a result, as can be seen from equality (35) (and directly from the drawing), the trajectory of point B is obtained from the trajectory of the point by parallel displacement of all its points by a constant vector AB. Consequently, the trajectories of points A and B will really be the same (when superimposed, coinciding) curves.

To find the velocities of points A and B, we differentiate both sides of equality (35) with respect to time. We get

But the derivative of constant vector A B is equal to zero. Derivatives of vectors with respect to time give the velocities of points A and B. As a result, we find that

i.e. that the velocities of points A and B of the body at any moment of time are the same both in magnitude and direction. Taking the derivatives with respect to time from both sides of the resulting equality, we find:

Consequently, the accelerations of points A and B of the body at any moment of time are also identical in magnitude and direction.

Since points A and B were chosen arbitrarily, it follows from the results found that for all points of the body their trajectories, as well as velocities and accelerations at any time, will be the same. Thus, the theorem is proven.

The velocities and accelerations of the points of a moving body form vector fields- velocity field and acceleration field of body points.

From what has been proven it follows that the fields of velocities and accelerations of points of a body moving translationally will be homogeneous (Fig. 133), but not stationary at all, i.e., changing with time (see § 32).

It also follows from the theorem that the translational motion of a rigid body is completely determined by the motion of any one of its points. Consequently, the study of the translational motion of a body comes down to the problem of the kinematics of a point, which we have already considered.

In translational motion, the velocity v common to all points of the body is called the speed of translational motion of the body, and the acceleration a is called the acceleration of translational motion of the body. Vectors can be depicted as applied to any point on the body.

Note that the concepts of speed and acceleration of a body make sense only in translational motion. In all other cases, the points of the body, as we will see, move with at different speeds and accelerations, and the terms “body speed” or “body acceleration” for these movements lose their meaning.

Translational and rotational motion

The simplest movement of a body is one in which all points of the body move equally, describing the same trajectories. This movement is called progressive . We obtain this type of motion by moving the splinter so that it remains parallel to itself at all times. trajectories can be either straight or curved lines.
The needle of a sewing machine, the piston in the cylinder of a steam engine or engine moves progressively internal combustion, car body (but not wheels!) when driving on a straight road, etc.

Another simple type of movement is rotational body movement, or rotation. During rotational motion, all points of the body move in circles whose centers lie on a straight line. This straight line is called the axis of rotation. The circles lie in parallel planes, perpendicular to the axis of rotation. Points of the body lying on the axis of rotation remain motionless. Rotation is not a translational movement: when the axis rotates.

Trajectory path movement speed acceleration definition

The line along which a material point moves is called trajectory . The length of the trajectory is called the path. The unit of path is meter.
Path = speed * time. S=v*t.
Directed line segment drawn from initial position moving point to its final position is called moving (s). Displacement is a vector quantity. The unit of movement is meter.
Speed - vector physical quantity, characterizing the speed of movement of the body, numerically equal to the ratio movements over a short period of time to the value of this period of time.
The speed formula is v = s/t. Unit of speed - m/s
Acceleration - vector physical quantity characterizing the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred. Formula for calculating acceleration: a=(v-v0)/t; The unit of acceleration is meter/(squared second).

Acceleration components tangential and normal acceleration

Tangential acceleration directed tangentially to the trajectory

Normal acceleration is directed normal to the trajectory

Tangential acceleration characterizes the change in speed in magnitude. If the speed does not change in magnitude, then the tangential component is equal to zero, and the normal component of acceleration is equal to the full acceleration.

Normal acceleration characterizes the change in speed in direction. If the direction of speed does not change, the movement occurs along a straight path.

In general full acceleration:

So, the normal component of the acceleration vector

The rate of change over time in the direction of the tangent to the trajectory. It is larger (), the more the trajectory is curved and the faster the particle moves along the trajectory.

4)Corner path

Corner path – this is the elementary rotation angle:

Radian is an angle that cuts an arc on a circle equal to the radius.

The direction of the angular path is determined by the rule right screw: if the head of the screw is rotated in the direction of movement of the point along the circle, then the translational movement of the tip of the screw will indicate the direction .

Angular velocity (average and instantaneous)

Average angular velocity – this is a physical quantity numerically equal to the ratio of the angular path to the period of time:

Instantaneous angular velocity – this is a physical quantity that is numerically equal to the change in the limit of the ratio of the angular path to the time interval as this interval tends to zero, or is the first derivative of the angular path with respect to time:

, .

Newton's laws

Newton's first law

• Inertial is called that frame of reference relative to which any material point, isolated from external influences, is either at rest or maintains a state of uniform rectilinear movement.
• Newton's first law reads:

In essence, this law postulates the inertia of bodies, which seems obvious today. But this was far from the case at the dawn of natural exploration. Aristotle argued that the cause of all movement is force, that is, movement by inertia did not exist for him. [ source?]

Newton's second law

Newton's second law - differential law motion, describing the relationship between the force applied to a material point and its acceleration.

Newton's second law states that

With a suitable choice of units of measurement, this law can be written as a formula:

where is the acceleration of the body;

Force applied to a body;

m- body weight.

Or more known form:

If several forces act on a body, then Newton’s second law is written:

In the case when the mass of a material point changes with time, Newton's second law is formulated in general view: the rate of change of momentum of a point is equal to the force acting on it.

where is the impulse (amount of movement) of the point;

t- time;

Derivative with respect to time.

Newton's second law is valid only for speeds much less than the speed of light and in inertial systems countdown.

Newton's third law

This law explains what happens to two interacting bodies. Let us take for example a closed system consisting of two bodies. The first body can act on the second with some force, and the second - on the first with force. How do the forces compare? Newton's third law states: the action force is equal in magnitude and opposite in direction to the reaction force. We emphasize that these forces are applied to different bodies, and therefore are not compensated at all.

The law itself:

Conclusions

Some interesting conclusions immediately follow from Newton's laws. Thus, Newton’s third law says that, no matter how bodies interact, they cannot change their total momentum: law of conservation of momentum. Next, we must require that the interaction potential of two bodies depends only on the modulus of the difference in the coordinates of these bodies U(| r 1 − r 2 |). Then there arises law of conservation of total mechanical energy interacting bodies:

Newton's laws are the basic laws of mechanics. All other laws of mechanics can be derived from them.

Steiner's theorem

Steiner's theorem - formulation

According to Steiner's theorem, it is established that the moment of inertia of a body when calculating relative to an arbitrary axis corresponds to the sum of the moment of inertia of the body relative to an axis that passes through the center of mass and is parallel to this axis, as well as plus the product of the square of the distance between the axes and the mass of the body, according to the following formula (1):

Where in the formula we take the following values, respectively: d – the distance between the axes ОО1║О’O1’;
J0 is the moment of inertia of the body, calculated relative to the axis that passes through the center of mass and will be determined by relation (2):

J0 = Jd = mR2/2 (2)

For example, for the hoop in the figure the moment of inertia about the axis O'O', equals

The moment of inertia of a straight rod of length , the axis is perpendicular to the rod and passes through its end.

10) angular momentum law of conservation of angular momentum

The angular momentum (momentum of motion) of a material point A relative to a fixed point O is a physical quantity defined by the vector product:

Where r- radius vector drawn from point O to point A, p=m v- momentum of the material point (Fig. 1); L- pseudovector,

Fig.1

Momentum relative to fixed axis z is called a scalar quantity L z, equal to projection to this axis of the angular momentum vector defined relative to arbitrary point About this axis. The angular momentum L z does not depend on the position of point O on the z axis.

When an absolutely rigid body rotates around a fixed axis z, each point of the body moves along a circle of constant radius r i with speed v i. The speed v i and momentum m i v i are perpendicular to this radius, i.e. the radius is the arm of the vector m i v i . This means we can write that the angular momentum individual particle equals

and is directed along the axis in the direction determined by the right screw rule.

Law of conservation of angular momentum Mathematically expressed through the vector sum of all angular momentum relative to the selected axis for closed system bodies, which remains constant until the system is acted upon external forces. In accordance with this, the angular momentum of a closed system in any coordinate system does not change with time.

The law of conservation of angular momentum is a manifestation of the isotropy of space with respect to rotation.

In simplified form: , if the system is in equilibrium.

Rigid body dynamics

Rotation around a fixed axis. The angular momentum of a rigid body relative to a fixed axis of rotation is equal to

The direction of projection coincides with the direction i.e. determined by the gimlet rule. Magnitude

is called the moment of inertia of a rigid body with respect to Differentiating , we get

This equation is called the basic equation of the dynamics of the rotational motion of a rigid body around a fixed axis. Let us also calculate the kinetic energy of a rotating rigid body:

and the work of an external force when turning a body:

Plane motion of a rigid body. Plane motion is a superposition of translational motion of the center of mass and rotational motion in the center of mass system (see Section 1.2). The motion of the center of mass is described by Newton’s second law and is determined by the resulting external force(equation (11)). Rotational motion in the center of mass system obeys equation (39), in which only real external forces must be taken into account, since the moment of inertia forces relative to the center of mass equal to zero(similar to the moment of gravity, example 1 from Section 1.6). Kinetic energy plane motion is equal to the equation Momentum momentum relative to a fixed axis, perpendicular to the plane movement, is calculated by the formula (see equation where is the arm of the velocity of the center of mass relative to the axis, and the signs are determined by the choice of the positive direction of rotation.

Movement with fixed point. The angular velocity of rotation, directed along the axis of rotation, changes its direction both in space and in relation to the solid body itself. Equation of motion

which is called the basic equation of motion of a rigid body with a fixed point, allows you to find out how the angular momentum changes. Since the vector in the general case is not parallel to the vector, then for

To close the equations of motion, we must learn to relate these quantities to each other.

Gyroscopes. A gyroscope is a rigid body that rotates rapidly about its axis of symmetry. The problem of the movement of the gyroscope axis can be solved in the gyroscopic approximation: both vectors are directed along the axis of symmetry. A balanced gyroscope (fixed at the center of mass) has the property of being inertialess; its axis stops moving as soon as it disappears external influence(goes to zero). This allows you to use a gyroscope to maintain orientation in space.

A heavy gyroscope (Fig. 12), in which the center of mass is displaced at a distance from the point of attachment, is subject to a moment of gravity directed perpendicularly since the axis of the gyroscope performs regular rotation around vertical axis(precession of the gyroscope).

The end of the vector rotates along a horizontal circle of radius a with angular velocity

The angular velocity of precession does not depend on the angle of inclination of the a-axis.

Conservation laws- fundamental physical laws, according to which, under certain conditions, some measurable physical quantities characterizing a closed physical system do not change over time.

· Law of conservation of energy

Law of conservation of momentum

Law of conservation of angular momentum

Law of conservation of mass

Law of Conservation electric charge

· Lepton number conservation law

Law of conservation of baryon number

· Law of conservation of parity

moment of force

The moment of force relative to the axis of rotation is a physical quantity equal to the product of the force by its arm.

The moment of force is determined by the formula:

M - FI, where F is force, I is force arm.

The shoulder of force is called shortest distance from the line of action of the force to the axis of rotation of the body.

The moment of force characterizes the rotating effect of a force. This action is dependent on both strength and leverage. The larger the shoulder, the less force must be applied,

The SI unit of moment of force is a moment of force of 1 N, the arm of which is equal to 1 m - newton meter (N m).

Rule of Moments

A rigid body capable of rotating around a fixed axis is in equilibrium if the moment of force M rotating it clockwise is equal to the moment of force M2 rotating it counterclockwise:

M1 = -M2 or F 1 ll = - F 2 l 2.

The moment of a pair of forces is the same about any axis perpendicular to the plane of the pair. The total moment M of a pair is always equal to the product of one of the forces F and the distance I between the forces, which is called the shoulder of the pair, regardless of what segments and /2 the position of the axis of the pair’s shoulder is divided into:

M = Fll + Fl2=F(l1 + l2) = Fl.

If a body rotates around a fixed axis z with angular velocity, then linear speed i th point , R i– distance to the axis of rotation. Hence,

Here Ic– moment of inertia about the instantaneous axis of rotation passing through the center of inertia.

Work of moment of forces.

Work of force.
Job constant force, acting on a rectilinearly moving body
, where is the displacement of the body, is the force acting on the body.

In general, work variable force, acting on a body moving along curvilinear trajectory . Work is measured in Joules [J].

The work of a moment of force acting on a body rotating around a fixed axis, where is the moment of force and is the angle of rotation.
In general.
The work done by the body turns into its kinetic energy.

Mechanical vibrations.

Oscillations- a process of changing the states of the system that is repeated to one degree or another over time.

Oscillations are almost always associated with the alternating transformation of the energy of one form of manifestation into another form.

The difference between an oscillation and a wave.

Various fluctuations physical nature have a lot general patterns and are closely interconnected by waves. Therefore, the study of these patterns is carried out by the generalized theory of wave oscillations. Fundamental difference from waves: during vibrations there is no transfer of energy; these are, so to speak, “local” energy transformations.

Oscillation Characteristics

Amplitude (m) - maximum deviation fluctuating value from some averaged value for the system.

Time lapse (sec), through which any indicators of the state of the system are repeated (the system makes one complete oscillation), is called the period of oscillation.

The number of oscillations per unit time is called the oscillation frequency ( Hz, sec -1).

Oscillation period and frequency – reciprocals;

In circular or cyclic processes, instead of the “frequency” characteristic, the concept is used circular or cyclic frequency (Hz, sec -1, rev/sec), showing the number of oscillations in time 2π:

Oscillation phase -- determines the displacement at any time, i.e. determines the state of the oscillatory system.

Pendulum mat physical spring

. Spring pendulum- this is a load of mass m, which is suspended on an absolutely elastic spring and performs harmonic oscillations under the action elastic force F = –kx, where k is the spring stiffness. The equation of motion of a pendulum has the form

From formula (1) it follows that the spring pendulum performs harmonic oscillations according to the law x = Асos(ω 0 t+φ) with a cyclic frequency

and period

Formula (3) is true for elastic vibrations within the limits within which Hooke's law is satisfied, i.e. if the mass of the spring is small compared to the mass of the body. Potential energy spring pendulum, using (2) and formula potential energy the previous section is equal to

2. Physical pendulum- a rigid body that oscillates under the influence of gravity around a stationary horizontal axis, which passes through point O, which does not coincide with the center of mass C of the body (Fig. 1).

Fig.1

If the pendulum is deflected from the equilibrium position by a certain angle α, then, using the equation of dynamics of the rotational motion of a rigid body, the moment M of the restoring force

where J is the moment of inertia of the pendulum relative to the axis that passes through the suspension point O, l is the distance between the axis and the center of mass of the pendulum, F τ ≈ –mgsinα ≈ –mgα is the restoring force (the minus sign indicates that the directions of F τ and α are always opposite; sinα ≈ α since the oscillations of the pendulum are considered small, i.e. the pendulum is deflected by small angles from the equilibrium position). We write equation (4) as

Taking

we get the equation

identical to (1), the solution of which (1) will be found and written as:

From formula (6) it follows that with small oscillations the physical pendulum performs harmonic oscillations with a cyclic frequency ω 0 and a period

where the value L=J/(m l) - .

Point O" on the continuation of straight line OS, which is located at a distance of given length L from the point O of the pendulum suspension, is called swing center physical pendulum(Fig. 1). Applying Steiner's theorem for the moment of inertia of the axis, we find

i.e. OO" is always greater than OS. The suspension point O of the pendulum and the center of swing O" have interchangeability property: if the suspension point is moved to the center of swing, then the previous suspension point O will be the new center of swing, and the period of oscillation of the physical pendulum will not change.

3. Math pendulum is an idealized system consisting of a material point of mass m, which is suspended on an inextensible weightless thread, and which oscillates under the influence of gravity. Good approximation mathematical pendulum there is a small heavy ball that is suspended on a long thin thread. Moment of inertia of a mathematical pendulum

Where l- length of the pendulum.

Since the mathematical pendulum is special case physical pendulum, if we assume that all its mass is concentrated at one point - the center of mass, then, by substituting (8) into (7), we find an expression for the period of small oscillations of a mathematical pendulum

Comparing formulas (7) and (9), we see that if the reduced length L of the physical pendulum is equal to the length l mathematical pendulum, then the periods of oscillation of these pendulums are the same. Means, reduced length of a physical pendulum- this is the length of a mathematical pendulum whose period of oscillation coincides with the period of oscillation of a given physical pendulum.

Gar. fluctuations and character.

Oscillations movements or processes characterized by a certain repeatability over time are called. Oscillatory processes are widespread in nature and technology, for example, the swing of a clock pendulum, alternating electric current etc

The simplest type of oscillations are harmonic vibrations- oscillations in which the fluctuating quantity changes over time according to the law of sine (cosine). Harmonic oscillations of a certain value s are described by an equation of the form

where ω 0 - circular (cyclic) frequency, A - maximum value fluctuating value, called vibration amplitude, φ - initial phase of oscillation at time t=0, (ω 0 t+φ) - oscillation phase at time t. The oscillation phase is the value of the oscillating quantity at a given moment in time. Since the cosine has a value ranging from +1 to –1, s can take values ​​from +A to –A.

Certain states of a system that performs harmonic oscillations are repeated after a period of time T, called period of oscillation, during which the oscillation phase receives an increment (change) of 2π, i.e.

Magnitude, inverse period hesitation,

i.e. the number of complete oscillations that occur per unit time is called vibration frequency. Comparing (2) and (3), we find

Frequency unit - hertz(Hz): 1 Hz is the frequency of a periodic process, during which one process cycle is completed in 1 s.

Oscillation amplitude

The amplitude of a harmonic oscillation is called highest value displacement of a body from its equilibrium position. The amplitude can take different meanings. It will depend on how much we shift the body in starting moment time from the equilibrium position.

The amplitude is determined initial conditions, that is, the energy imparted to the body at the initial moment of time. Since sine and cosine can take values ​​in the range from -1 to 1, the equation must contain a factor Xm, expressing the amplitude of the oscillations. Equation of motion at harmonic vibrations:

x = Xm*cos(ω0*t).

Faded. kolb and their har

Damped oscillations

Damping of oscillations is called gradual decrease amplitude of oscillations over time, due to the loss of energy by the oscillatory system.

Natural oscillations without damping are an idealization. The reasons for attenuation may be different. IN mechanical system The presence of friction leads to damping of oscillations. IN electromagnetic circuit Heat losses in the conductors forming the system lead to a decrease in vibration energy. When all the energy stored in the oscillatory system is used up, the oscillations will stop. Therefore the amplitude damped oscillations decreases until it becomes equal to zero.

where β – attenuation coefficient

In new notations differential equation damped oscillations has the form:

. where β – attenuation coefficient, where ω 0 is the frequency of undamped free vibrations in the absence of energy losses in the oscillatory system.

This is a second order linear differential equation.

Damped frequency:

In any oscillatory system, damping leads to a decrease in frequency and, accordingly, an increase in the oscillation period.

(physical meaning has only a real root, therefore ).

Period of damped oscillations:

.

The meaning that was put into the concept of the period for continuous oscillations, is not suitable for damped oscillations, since oscillatory system never returns to its original state due to losses vibrational energy. In the presence of friction, vibrations are slower: .

Period of damped oscillations is the minimum period of time during which the system passes the equilibrium position twice in one direction.

Amplitude of damped oscillations:

For a spring pendulum.

The amplitude of damped oscillations is not a constant value, but changes over time, the faster the higher coefficientβ. Therefore, the definition for amplitude, given earlier for undamped free oscillations, must be changed for damped oscillations.

For small attenuations amplitude of damped oscillations is called the largest deviation from the equilibrium position over a period.

The amplitude of damped oscillations changes according to exponential law:

Let the oscillation amplitude decrease by “e” times during time τ (“e” is the base natural logarithm, e ≈ 2.718). Then, on the one hand, , and on the other hand, having described the amplitudes A zat. (t) and A zat. (t+τ), we have . From these relations it follows βτ = 1, hence

Forced vibrations.

Waves and their characteristics

Wave - excitation of the medium, propagating in space and time or in phase space with energy transfer and without mass transfer

By their nature, waves are divided into:

Based on distribution in space: standing, running.

By the nature of the waves: oscillatory, solitary (solitons).

By type of waves: transverse, longitudinal, mixed type.

According to the laws describing the wave process: linear, nonlinear.

According to the properties of the substance: waves in discrete structures, waves in continuous substances.

By geometry: spherical (spatial), one-dimensional (flat), spiral.

Wave Characteristics

Temporal and spatial periodicity

temporal periodicity - the rate of phase change over time at some given point, called the wave frequency;
spatial periodicity - the rate of phase change (time lag of the process) at a certain point in time with a change in coordinate - wavelength λ.

Temporal and spatial periodicities are interrelated. In a simplified form for linear waves, this dependence has the following form:

where c is the speed of wave propagation in a given medium.

Wave intensity

To characterize the intensity wave process three parameters are used: the amplitude of the wave process, the energy density of the wave process and the energy flux density.

Thermodynamic systems

In thermodynamics they study physical systems, consisting of large number particles and are in a state of thermodynamic equilibrium or close to it. Such systems are called thermodynamic systems.

The unit of measurement for the number of particles in thermodynamic system Usually the Avogadro number is used (approximately 6·10^23 particles per mole of substance), which gives an idea of ​​what order of magnitude we are talking about.

Thermodynamic equilibrium is a state of a system in which the macroscopic quantities of this system (temperature, pressure, volume, entropy) remain unchanged over time under conditions of isolation from the environment.

Thermodynamic parameters

There are extensive state parameters proportional to the mass of the system:

volume, internal energy, entropy, enthalpy, Gibbs energy, Helmholtz energy (free energy),

and intensive state parameters independent of the mass of the system:

pressure, temperature, concentration, magnetic induction, etc.

Laws ideal gas

Boyle's Law - Mariotte. Let the gas be in conditions where its temperature is maintained constant (such conditions are called isothermal ).Then for a given mass of gas, the product of pressure and volume is a constant:

This formula is called isotherm equation. Graphically the dependence of p on V for different temperatures shown in the figure.

Gay-Lussac's law. Let the gas be in conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be achieved by placing gas in a cylinder closed by a movable piston. Then a change in gas temperature will lead to movement of the piston and a change in volume. The gas pressure will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:

Graphically, the dependence of V on T for different pressures shown in the figure.

The motion of a rigid body is divided into types:

• progressive;
• rotational along a fixed axis;
• flat;
• rotational around a fixed point;
• free.

The first two of them are the simplest, and the rest are represented as a combination of basic movements.

Progressive call the motion of a rigid body in which any straight line drawn in it moves while remaining parallel to its initial direction.

Rectilinear motion is translational, but not every translational motion will be rectilinear. In the presence of translational motion, the path of the body is represented in the form of curved lines.

Figure 1. Progressive curvilinear movement cab view wheel

Theorem 1

The properties of translational motion are determined by the theorem: during translational motion, all points of the body describe identical trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.

Consequently, the translational motion of a rigid body is determined by the movement of any of its points. This comes down to the point kinematics problem.

Definition 2

If there is translational motion, then the total speed for all points of the body υ → is called forward motion speed, and acceleration a → - acceleration of forward motion. The image of the vectors υ → and a → is usually indicated as applied at any point of the body.

The concept of speed and acceleration of a body makes sense only in the presence of translational motion. In other cases, points of the body are characterized by different velocities and accelerations.

Definition 3

Rotational motion of an absolutely rigid body around a fixed axis- this is the movement of all points of the body located in planes perpendicular to a fixed straight line, called the axis of rotation, and the description of circles whose centers are located on this axis.

To determine the position of a rotating body, it is necessary to draw an axis of rotation along which the A z axis is directed, a stationary half-plane passing through the body and moving with it, as shown in Figure 2.

Figure 2. Body rotation angle

The position of the body at any moment of time will be characterized by the corresponding sign in front of the angle φ between the half-planes, which is called the angle of rotation of the body. When it is laid aside, starting from a stationary plane (counterclockwise direction), the angle takes positive value, against the plane – negative. Angle measurements are made in radians. To determine the position of the body at any time, one should take into account the dependence of the angle φ on t, that is, φ = f (t). The equation is the law of rotational motion of a rigid body around a fixed axis.

In the presence of such rotation, the values ​​of the rotation angles of the radius vector of various points of the body will be similar.

The rotational motion of a rigid body is characterized by angular velocity ω and angular acceleration ε.

The equations of rotational motion are obtained from the equations of translational motion, using the replacement of displacement S by angular displacement φ, speed υ by angular velocity ω, and acceleration a by angular ε.

Rotational and translational movement. Formulas

Rotational motion problems

Given a material point that moves rectilinearly according to the equation s = t 4 + 2 t 2 + 5. Calculate instantaneous speed and the acceleration of the point at the end of the second second after the start of movement, average speed and the distance traveled during this period of time.

Given: s = t 4 + 2 t 2 + 5, t = 2 s.

Find: s ; υ; υ; α.

Solution

s = 2 4 + 2 2 2 + 5 = 29 m.

υ = d s d t = 4 t 3 + 4 t = 4 2 3 + 4 2 = 37 m/s.

υ = ∆ s ∆ t = 29 2 = 14.5 m/s.

a = d υ d t = 12 t 2 + 4 = 12 · 2 2 + 4 = 52 m/s 2.

Answer: s = 29 m; υ = 37 m/s; υ = 14.5 m/s; α = 52 m/s 2

Example 2

A body is given that rotates around a fixed axis according to the equation φ = t 4 + 2 t 2 + 5. Calculate instantaneous angular velocity, angular acceleration body at the end of 2 seconds after the start of movement, the average angular velocity and angle of rotation for a given period of time.

Given:φ = t 4 + 2 t 2 + 5, t = 2 s.

Find: φ ; ω ; ω ; ε.

Solution

φ = 2 4 + 2 2 2 + 5 = 29 r a d.

ω = d φ d t = 4 t 3 + 4 t = 4 2 3 + 4 2 = 37 r a d / s.

ω = ∆ φ ∆ t = 29 2 = 14.5 r a d / s.

ε = d ω d t = 12 2 + 4 = 12 · 2 2 + 4 = 52 r a d / s 2 .

Answer: φ = 29 r a d; ω = 37 r a d / s; ω = 14.5 r a d / s; ε = 52 r a d / s 2.

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