If the body moves forward. Movement of bodies Forward movement – ​​Knowledge Hypermarket

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 internal combustion engine, the body of a car (but not the wheels!) move forward when driving on a straight road, etc.

Another simple type of movement is rotational body movement, or rotation. At rotational movement all points of the body move in circles whose centers lie on a straight line. This straight line is called the axis of rotation. 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.
A directed line segment drawn from the initial position of a 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 a body, numerically equal to the ratio of movement 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 is 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, the total acceleration is:

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 motion.
• 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 is a differential law of motion that describes the relationship between the force applied to a material point and its acceleration.

Newton's second law states that

At suitable choice 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 mass.

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 velocities much lower than the speed of light and in inertial frames of reference.

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 the fixed axis z is called a scalar quantity L z equal to the projection onto this axis of the angular momentum vector defined relative to an arbitrary point O of 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 velocity v i and the 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 that we can write that the angular momentum of an individual particle is equal to

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

Law of conservation of angular momentum It is expressed mathematically through the vector sum of all angular momentum about a selected axis for a closed system of bodies, which remains constant until the system is acted upon by 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 is zero ( similar to the moment of gravity, example 1 from Section 1.6). The kinetic energy of plane motion is equal to the equation The angular momentum relative to a fixed axis perpendicular to the plane of motion 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 a 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 the external influence disappears (turns 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 force directed perpendicularly since the axis of the gyroscope performs regular rotation around the 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 energy conservation

Law of conservation of momentum

Law of conservation of angular momentum

Law of conservation of mass

Law of conservation of electric charge

Law of conservation of lepton number

Law of conservation of baryon number

· Law of conservation of parity

Moment of power

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 arm of a force is the 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 velocity i th point , R i– distance to the axis of rotation. Hence,

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

Work of moment of forces.

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

In general, the work done by a variable force acting on a body moving along a curved path . 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 many common patterns and are closely interconnected by waves. Therefore, the study of these patterns is carried out by the generalized theory of wave oscillations. The fundamental difference from waves: during oscillations there is no transfer of energy; these are, so to speak, “local” energy transformations.

Oscillation Characteristics

Amplitude (m)- the maximum deviation of a fluctuating quantity from some average value for the system.

Time interval (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).

The oscillation period and frequency are reciprocal quantities;

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 of an 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 in which Hooke’s law is satisfied, that is, if the mass of the spring is small compared to the mass of the body. The potential energy of a spring pendulum, using (2) and the potential energy formula of the previous section, is equal to

2. Physical pendulum is a solid body that oscillates under the influence of gravity around a fixed horizontal axis that 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 from the equilibrium position by small angles). 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 reduced 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. A good approximation of a mathematical pendulum 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 a mathematical pendulum is a special case of a 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 electricity 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 - the maximum value of the fluctuating quantity, 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.

The reciprocal of the oscillation period is

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 displace the body at the initial moment of time from the equilibrium position.

The amplitude is determined by the 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 for harmonic vibrations:

x = Xm*cos(ω0*t).

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Damped oscillations

Damping of oscillations is a gradual decrease in the 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 a mechanical system, vibrations are damped by the presence of friction. In an electromagnetic circuit, heat losses in the conductors forming the system lead to a decrease in oscillation 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 oscillations 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.

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

Period of damped oscillations:

.

The meaning that was put into the concept of a period for undamped oscillations is not suitable for damped oscillations, since the oscillatory system never returns to its original state due to losses of oscillatory 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 greater the coefficient β. 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 an exponential law:

Let the oscillation amplitude decrease by “e” times during time τ (“e” is the base of the 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 a 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 of the 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

Thermodynamics is the study of physical systems consisting of a large number of particles and being 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 a thermodynamic system is usually the Avogadro number (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 that do not depend on the mass of the system:

pressure, temperature, concentration, magnetic induction, etc.

Ideal gas laws

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 various temperatures is 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 various pressures is shown in the figure.

Translational motion is such a motion of a rigid body when every straight line mentally drawn in the body moves parallel to itself.

Theorem. During translational motion, all points of the body describe identical (congruent) trajectories and have geometrically equal velocities and accelerations at each moment of time.

Proof. Let the body move forward (Fig. 91). Let us arbitrarily select two points in the body and . The vector of these points, during the translational motion of the body, is a constant vector - its direction remains constant in accordance with the definition of translational motion, its module - due to the constant distances between the points of an absolutely rigid body. Therefore, for the radius vectors of the selected points at any time, the following relation holds:

This equality means that if the position of a point at some point in time becomes known, then the position of the point at this moment is found by shifting the point by a vector value that is the same at all times. Therefore, if the geometric locus of the position (trajectory) of the point is known, then the geometric locus of the position (trajectory) of the point is obtained by shifting the trajectory of the point in the direction and by the magnitude of the vector. Which proves the congruence of the trajectories of points and . Since the points are chosen arbitrarily, the trajectories of all points of the body are congruent.

Differentiating the written equality successively twice in time, we are convinced of the validity of the second part of the theorem:

The speed common to all points of the body is called the speed of the body; the acceleration common to all points is the acceleration of the body. Let us immediately note that these terms make sense only in forward motion; in all other cases of body movement, individual points of the body have different speeds and acceleration.

From all that has been said, it follows that the study of the translational motion of a body comes down to the problem of the kinematics of a point. Namely, a point in the body is selected whose movement is determined most simply, and its trajectory, speed, and acceleration are determined by the methods of the point’s kinematics. The trajectories, velocities and accelerations of the remaining points are determined by simply transferring the kinematic characteristics of the selected point.

Determine the trajectory, speed and acceleration of point M, rigidly connected to link AB of the twin-wheel mechanism (Fig. 92), if , and angle .

We notice that the link AB of the mechanism moves forward. The movement of its point A, which also serves as the end of the crank, is easily determined. Let's select this point and find its kinematic characteristics.

It is immediately clear that the trajectory of point A is a circle with a center at the point and radius . By shifting this circle so that its center is at point O, and , we obtain the trajectory of point M.

>>Physics: Movement of bodies. Forward movement

A description of the movement of a body is considered complete only when it is known how each point moves.
We paid a lot of attention to describing the movement of the point. It is for a point that the concepts of coordinates, speed, acceleration, trajectory are introduced. In general, the task of describing the motion of bodies is complex. It is especially difficult if the bodies are noticeably deformed during movement. It is easier to describe the movement of a body whose relative positions do not change. Such a body is called absolutely solid. In fact, there are no absolutely solid bodies. But in cases where real bodies deform little when moving, they can be considered as absolutely solid. (Another abstract model introduced when considering motion.) However, the motion of an absolutely rigid body in the general case turns out to be very complex. Any complex motion of an absolutely rigid body can be represented as the sum of two independent motions: translational and rotational.
Forward movement. The simplest movement of rigid bodies is progressive.
Progressive is the motion of a rigid body in which any segment connecting any two points of the body remains parallel to itself.
During translational motion, all points of the body make the same movements, describe the same trajectories, travel the same paths, and have equal velocities and accelerations at each moment of time. Let's show it.
Let the body move forward ( Fig.2.1). Let's connect two of its arbitrary points B And A segment. The distance does not change, since the body is absolutely rigid. During translational motion, the magnitude and direction of the vector remain constant. As a result, the trajectories of points B And A are the same, since they can be completely combined by parallel transfer to the vector.

According to Figure 2.1, moving points A And B are the same and take place at the same time. Therefore, the points A And B have the same speeds and accelerations.
It is quite obvious that to describe the translational motion of a rigid body it is enough to describe the movement of any one of its points. Only with translational motion can we talk about the speed and acceleration of the body. With any other movement of a body, its points have different speeds and accelerations, and the terms “body speed” and “body acceleration” for non-translational motion lose their meaning.
A desk drawer, the pistons of a car engine relative to the cylinders, and carriages on a straight section move approximately progressively railway, cutter lathe relative to the bed. Movement of a bicycle pedal or Ferris wheel cabin in parks ( Fig.2.2, 2.3) are also examples of translational motion.

For description forward motion For a rigid body, it is enough to write the equation of motion of one of its points.

G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade

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Progressive is the movement of a rigid body in which any straight line invariably associated with this body remains parallel to its initial position.

Theorem. During the translational motion of a rigid body, all its points describe identical trajectories and at each given moment have equal velocity and acceleration in magnitude and direction.

Proof. Let's draw through two points and , a linearly moving body segment
and consider the movement of this segment in position
. At the same time, the point describes the trajectory
, and point – trajectory
(Fig. 56).

Considering that the segment
moves parallel to itself, and its length does not change, it can be established that the trajectories of points And will be the same. This means that the first part of the theorem is proven. We will determine the position of the points And vector method relative to a fixed origin . Moreover, these radii - vectors are dependent
. Because. neither the length nor the direction of the segment
does not change when the body moves, then the vector

. Let's move on to determining the velocities using dependence (24):

, we get
.

Let's move on to determining accelerations using dependence (26):

, we get
.

From the proven theorem it follows that the translational motion of a body will be completely determined if the motion of only one point is known. Therefore, the study of the translational motion of a rigid body comes down to the study of the movement of one of its points, i.e. to the point kinematics problem.

Topic 11. Rotational motion of a rigid body

Rotational This is the movement of a rigid body in which two of its points remain motionless throughout the entire movement. In this case, the straight line passing through these two fixed points is called axis of rotation.

During this movement, each point of the body that does not lie on the axis of rotation describes a circle, the plane of which is perpendicular to the axis of rotation, and its center lies on this axis.

We draw through the axis of rotation a fixed plane I and a movable plane II, invariably connected to the body and rotating with it (Fig. 57). The position of plane II, and accordingly the entire body, in relation to plane I in space, is completely determined by the angle . When a body rotates around an axis this angle is a continuous and unambiguous function of time. Therefore, knowing the law of change of this angle over time, we can determine the position of the body in space:

- law of rotational motion of a body. (43)

In this case, we will assume that the angle measured from a fixed plane in the direction opposite to the clockwise movement, when viewed from the positive end of the axis . Since the position of a body rotating around a fixed axis is determined by one parameter, such a body is said to have one degree of freedom.

Angular velocity

The change in the angle of rotation of a body over time is called angular body speed and is designated
(omega):

.(44)

Angular velocity, just like linear velocity, is a vector quantity, and this vector built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the rotation of the body counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (44). Application point on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action. If we denote the orth-vector of the rotation axis by , then we obtain the vector expression for angular velocity:

. (45)

Angular acceleration

The rate of change in the angular velocity of a body over time is called angular acceleration body and is designated (epsilon):

. (46)

Angular acceleration is a vector quantity, and this vector built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the direction of rotation of the epsilon counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (46). Application point on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action.

If we denote the orth-vector of the rotation axis by , then we obtain the vector expression for angular acceleration:

. (47)

If the angular velocity and acceleration are of the same sign, then the body rotates expedited, and if different - slowly. An example of slow rotation is shown in Fig. 58.

Let us consider special cases of rotational motion.

1. Uniform rotation:

,
.

,
,
,

,
. (48)

2. Equal rotation:

.

,
,
,
,
,
,
,
,

,
,
.(49)

Relationship between linear and angular parameters

Consider the movement of an arbitrary point
rotating body. In this case, the trajectory of the point will be a circle with radius
, located in a plane perpendicular to the axis of rotation (Fig. 59, A).

Let us assume that at the moment of time the point is in position
. Let us assume that the body rotates in a positive direction, i.e. in the direction of increasing angle . At a moment in time
the point will take position
. Let's denote the arc
. Therefore, over a period of time
the point has passed the way
. Her average speed , and when
,
. But, from Fig. 59, b, it's clear that
. Then. Finally we get

. (50)

Here - linear speed of the point
. As was obtained earlier, this speed is directed tangentially to the trajectory at a given point, i.e. tangent to the circle.

Thus, the module of the linear (circumferential) velocity of a point of a rotating body is equal to the product of the absolute value of the angular velocity and the distance from this point to the axis of rotation.

Now let's connect the linear components of the acceleration of a point with the angular parameters.

,
. (51)

The modulus of the tangential acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the angular acceleration of the body and the distance from this point to the axis of rotation.

,
. (52)

The modulus of normal acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the square of the angular velocity of the body and the distance from this point to the axis of rotation.

Then the expression for the total acceleration of the point takes the form

. (53)

Vector directions ,,shown in Figure 59, V.

Flat motion of a rigid body is a movement in which all points of the body move parallel to some fixed plane. Examples of such movement:

The motion of any body whose base slides along a given fixed plane;

Rolling of a wheel along a straight section of track (rail).

We obtain the equations of plane motion. To do this, consider a flat figure moving in the plane of the sheet (Fig. 60). Let us relate this movement to a fixed coordinate system
, and with the figure itself we connect the moving coordinate system
, which moves with it.

Obviously, the position of a moving figure on a stationary plane is determined by the position of the moving axes
relative to fixed axes
. This position is determined by the position of the moving origin , i.e. coordinates ,and rotation angle , a moving coordinate system, relatively fixed, which we will count from the axis in the direction opposite to the clockwise movement.

Consequently, the movement of a flat figure in its plane will be completely determined if the values ​​of ,,, i.e. equations of the form:

,
,
. (54)

Equations (54) are equations of plane motion of a rigid body, since if these functions are known, then for each moment of time it is possible to find from these equations, respectively ,,, i.e. determine the position of a moving figure at a given moment in time.

Let's consider special cases:

1.

, then the movement of the body will be translational, since the moving axes move while remaining parallel to their initial position.

2.

,

. With this movement, only the angle of rotation changes , i.e. the body will rotate about an axis passing perpendicular to the drawing plane through the point .

Decomposition of the motion of a flat figure into translational and rotational

Consider two consecutive positions And
occupied by the body at moments of time And
(Fig. 61). Body from position to position
can be transferred as follows. Let's move the body first progressively. In this case, the segment
will move parallel to itself to position
, and then let's turn body around a point (pole) at an angle
until the points coincide And .

Hence, any plane motion can be represented as the sum of translational motion together with the selected pole and rotational motion, relative to this pole.

Let's consider methods that can be used to determine the velocities of points of a body performing plane motion.

1. Pole method. This method is based on the resulting decomposition of plane motion into translational and rotational. The speed of any point of a flat figure can be represented in the form of two components: translational, with a speed equal to the speed of an arbitrarily chosen point -poles , and rotational around this pole.

Let's consider a flat body (Fig. 62). The equations of motion are:
,
,
.

From these equations we determine the speed of the point (as with the coordinate method of specifying)

,
,
.

Thus, the speed of the point - the quantity is known. We take this point as a pole and determine the speed of an arbitrary point
bodies.

Speed
will consist of a translational component , when moving along with the point , and rotational
, when rotating the point
relative to the point . Point speed move to point
parallel to itself, since during translational motion the velocities of all points are equal both in magnitude and direction. Speed
will be determined by dependence (50)
, and this vector is directed perpendicular to the radius
in the direction of rotation
. Vector
will be directed along the diagonal of a parallelogram built on vectors And
, and its module is determined by the dependency:

, .(55)

2. Theorem on the projections of velocities of two points of a body.

The projections of the velocities of two points of a rigid body onto a straight line connecting these points are equal to each other.

Consider two points of the body And (Fig. 63). Taking a point beyond the pole, we determine the direction depending on (55):
. We project this vector equality onto the line
and considering that
perpendicular
, we get

3. Instantaneous velocity center.

Instantaneous velocity center(MCS) is a point whose speed at a given time is zero.

Let us show that if a body does not move translationally, then such a point exists at every moment of time and, moreover, is unique. Let at a moment in time points And bodies lying in section , have speeds And , not parallel to each other (Fig. 64). Then point
, lying at the intersection of perpendiculars to the vectors And , and there will be an MCS, since
.

Indeed, if we assume that
, then according to Theorem (56), the vector
must be perpendicular at the same time
And
, which is impossible. From the same theorem it is clear that no other section point at this moment in time cannot have a speed equal to zero.

Using the pole method
- pole, determine the speed of the point (55): because
,
. (57)

A similar result can be obtained for any other point of the body. Therefore, the speed of any point on the body is equal to its rotational speed relative to the MCS:

,
,
, i.e. the velocities of body points are proportional to their distances to the MCS.

From the three considered methods for determining the velocities of points of a flat figure, it is clear that the MCS is preferable, since here the speed is immediately determined both in magnitude and in the direction of one component. However, this method can be used if we know or can determine the position of the MCS for the body.

Determining the position of the MCS

1. If we know for this provision body, the direction of the velocities of two points of the body, then the MCS will be the point of intersection of perpendiculars to these velocity vectors.

2. The velocities of two points of the body are antiparallel (Fig. 65, A). In this case, the perpendicular to the velocities will be common, i.e. The MCS is located somewhere on this perpendicular. To determine the position of the MCS, it is necessary to connect the ends of the velocity vectors. The point of intersection of this line with the perpendicular will be the desired MCS. In this case, the MCS is located between these two points.

3. The velocities of two points of the body are parallel, but not equal in magnitude (Fig. 65, b). The procedure for obtaining the MDS is similar to that described in paragraph 2.

d) The velocities of two points are equal in both magnitude and direction (Fig. 65, V). We obtain the case of instantaneous translational motion, in which the velocities of all points of the body are equal. Consequently, the angular velocity of the body in this position is zero:

4. Let us determine the MCS for a wheel rolling without sliding on a stationary surface (Fig. 65, G). Since the movement occurs without sliding, at the point of contact of the wheel with the surface the speed will be the same and equal to zero, since the surface is stationary. Consequently, the point of contact of the wheel with a stationary surface will be the MCS.

Determination of accelerations of points of a plane figure

When determining the accelerations of points of a flat figure, there is an analogy with methods for determining velocities.

1. Pole method. Just as when determining velocities, we take as a pole an arbitrary point of the body whose acceleration we know or we can determine. Then the acceleration of any point of a flat figure is equal to the sum of the accelerations of the pole and the acceleration in rotational motion around this pole:

In this case, the component
determines the acceleration of a point as it rotates around the pole . When rotating, the trajectory of the point will be curvilinear, which means
(Fig. 66).

Then dependence (58) takes the form
. (59)

Taking into account dependencies (51) and (52), we obtain
,
.

2. Instant acceleration center.

Instant acceleration center(MCU) is a point whose acceleration at a given time is zero.

Let us show that at any given moment of time such a point exists. We take a point as a pole , whose acceleration
we know. Finding the angle , lying within
, and satisfying the condition
. If
, That
and vice versa, i.e. corner delayed in direction . Let's postpone from the point at an angle to vector
line segment
(Fig. 67). The point obtained by such constructions
there will be an MCU.

Indeed, the acceleration of the point
equal to the sum of accelerations
poles and acceleration
in rotational motion around a pole :
.

,
. Then
. On the other hand, acceleration
forms with the direction of the segment
corner
, which satisfies the condition
. A minus sign is placed in front of the tangent of the angle , since rotation
relative to the pole counterclockwise, and the angle
is deposited clockwise. Then
.

Hence,
and then
.

Special cases of determining the MCU

1.
. Then
, and, therefore, the MCU does not exist. In this case, the body moves translationally, i.e. the velocities and accelerations of all points of the body are equal.

2.
. Then
,
. This means that the MCU lies at the intersection of the lines of action of the accelerations of the points of the body (Fig. 68, A).

3.
. Then,
,
. This means that the MCU lies at the intersection of perpendiculars to the accelerations of points of the body (Fig. 68, b).

4.
. Then
,

. This means that the MCU lies at the intersection of rays drawn to the accelerations of points of the body at an angle (Fig. 68, V).

From the considered special cases we can conclude: if we accept the point
beyond the pole, then the acceleration of any point of a flat figure is determined by the acceleration in rotational motion around the MCU:

. (60)

Complex point movement a movement in which a point simultaneously participates in two or more movements is called. With such movement, the position of the point is determined relative to the moving and relatively stationary reference systems.

The movement of a point relative to a moving reference frame is called relative motion of a point . We agree to denote the parameters of relative motion
.

The movement of that point of the moving reference system with which the moving point relative to the stationary reference system currently coincides is called portable movement of the point . We agree to denote the parameters of portable motion
.

The movement of a point relative to a fixed frame of reference is called absolute (complex) point movement . We agree to denote the parameters of absolute motion
.

As an example of complex movement, we can consider the movement of a person in a moving vehicle (tram). In this case, the human movement is related to the moving coordinate system - the tram and to the fixed coordinate system - the earth (road). Then, based on the definitions given above, the movement of a person relative to the tram is relative, the movement together with the tram relative to the ground is portable, and the movement of a person relative to the ground is absolute.

We will determine the position of the point
radii - vectors relative to the moving
and motionless
coordinate systems (Fig. 69). Let us introduce the following notation: - radius vector defining the position of the point
relative to the moving coordinate system
,
;- radius vector that determines the position of the beginning of the moving coordinate system (point ) (dots );- radius – a vector that determines the position of a point
relative to a fixed coordinate system
;
,.

Let us obtain conditions (constraints) corresponding to relative, portable and absolute motions.

1. When considering relative motion, we will assume that the point
moves relative to the moving coordinate system
, and the moving coordinate system itself
relative to a fixed coordinate system
doesn't move.

Then the coordinates of the point
will change in relative motion, but the orth-vectors of the moving coordinate system will not change in direction:

,

,

.

2. When considering portable motion, we will assume that the coordinates of the point
relative to the moving coordinate system are fixed, and the point moves along with the moving coordinate system
relatively stationary
:

,

,

,.

3. With absolute motion, the point also moves relatively
and together with the coordinate system
relatively stationary
:

Then the expressions for the velocities, taking into account (27), have the form

,
,

Comparing these dependencies, we obtain the expression for absolute speed:
. (61)

We obtained a theorem on the addition of the velocities of a point in complex motion: the absolute speed of a point is equal to the geometric sum of the relative and portable speed components.

Using dependence (31), we obtain expressions for accelerations:

,

Comparing these dependencies, we obtain an expression for absolute acceleration:
.

We found that the absolute acceleration of a point is not equal to the geometric sum of the relative and portable acceleration components. Let us determine the absolute acceleration component in parentheses for special cases.

1. Portable translational movement of the point
. In this case, the axes of the moving coordinate system
move all the time parallel to themselves, then.

,

,

,
,
,
, Then
. Finally we get

. (62)

If the portable motion of a point is translational, then the absolute acceleration of the point is equal to the geometric sum of the relative and portable components of the acceleration.

2. The portable movement of the point is non-translational. This means that in this case the moving coordinate system
rotates around the instantaneous axis of rotation with angular velocity (Fig. 70). Let us denote the point at the end of the vector through . Then, using the vector method of specifying (15), we obtain the velocity vector of this point
.

On the other side,
. Equating the right-hand sides of these vector equalities, we obtain:
. Proceeding similarly for the remaining unit vectors, we obtain:
,
.

In the general case, the absolute acceleration of a point is equal to the geometric sum of the relative and translational components of the acceleration plus the doubled vector product of the angular velocity vector of the translational motion and the linear velocity vector of the relative motion.

The double vector product of the angular velocity vector of the portable motion and the linear velocity vector of the relative motion is called Coriolis acceleration and is designated

. (64)

Coriolis acceleration characterizes the change in relative speed in translational motion and the change in translational velocity in relative motion.

Headed
according to the vector product rule. The Coriolis acceleration vector is always directed perpendicular to the plane formed by the vectors And , in such a way that, looking from the end of the vector
, see the turn To , through the smallest angle, counterclockwise.

The Coriolis acceleration modulus is equal to.