Parting

Harmonic oscillations dependence x t. Harmonic vibrations

The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.

So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) performs a harmonic oscillatory motion on a vertical screen (Fig. 1).

The displacement from the equilibrium position during harmonic vibrations is described by an equation (it is called the kinematic law of harmonic motion) of the form:

where x - displacement - a value characterizing the position of the oscillating point at time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given time; A - oscillation amplitude - the maximum displacement of the body from the equilibrium position; T - oscillation period - the time of one complete oscillation; those. the shortest amount of time after which values are repeated physical quantities characterizing the oscillation; - initial phase;

The phase of the oscillation at time t. The oscillation phase is an argument of a periodic function, which, for a given oscillation amplitude, determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.

If at the initial moment of time the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law

If the oscillating point at is in the position stable balance, then the displacement of the point from the equilibrium position changes according to the law

The value V, the reciprocal of the period and equal to the number of complete oscillations performed in 1 s, is called the oscillation frequency:

If in time t the body makes N complete oscillations, then

the value , showing how many oscillations the body makes in s, is called cyclic (circular) frequency.

The kinematic law of harmonic motion can be written as:

Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).

Figure 2, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case .

Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:

where is the amplitude of the velocity projection on the x-axis.

This formula shows that during harmonic oscillations, the projection of the body velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by (Fig. 2, b).

To find out the dependence of acceleration, we find the time derivative of the velocity projection:

where is the amplitude of the acceleration projection on the x-axis.

For harmonic oscillations, the acceleration projection leads the phase shift by k (Fig. 2, c).

Similarly, you can build dependency graphs

Considering that , the formula for acceleration can be written

those. for harmonic oscillations, the acceleration projection is directly proportional to the displacement and opposite in sign, i.e. acceleration is directed in the direction opposite to the displacement.

So, the acceleration projection is the second derivative of the displacement, then the resulting ratio can be written as:

The last equality is called equation of harmonic oscillations.

A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic oscillations - harmonic oscillator equation.

This is a periodic oscillation, in which the coordinate, speed, acceleration, characterizing the movement, change according to the sine or cosine law. The harmonic oscillation equation establishes the dependence of the body coordinate on time

The cosine graph has a maximum value at the initial moment, and the sine graph has a zero value at the initial moment. If we begin to investigate the oscillation from the equilibrium position, then the oscillation will repeat the sinusoid. If we begin to consider the oscillation from the position of the maximum deviation, then the oscillation will describe the cosine. Or such an oscillation can be described by the sine formula with an initial phase.

Mathematical pendulum

Oscillations of a mathematical pendulum.
Mathematical pendulum is a material point suspended on a weightless inextensible thread (physical model).
We will consider the movement of the pendulum under the condition that the deflection angle is small, then, if we measure the angle in radians, the statement is true: .
The force of gravity and the tension of the thread act on the body. The resultant of these forces has two components: tangential, which changes the acceleration in magnitude, and normal, which changes the acceleration in direction (centripetal acceleration, the body moves in an arc).
Because the angle is small, then the tangential component is equal to the projection of gravity on the tangent to the trajectory: . Angle in radians is equal to the ratio arc length to the radius (thread length), and the arc length is approximately equal to the offset ( x ≈ s): .
Let's compare the resulting equation with the equation of oscillatory motion. It can be seen that or is a cyclic frequency during oscillations of a mathematical pendulum.
Oscillation period or (Galileo's formula).	Galileo formula
The most important conclusion: the period of oscillation of a mathematical pendulum does not depend on the mass of the body!
Similar calculations can be done using the law of conservation of energy. We take into account that the potential energy of the body in the gravitational field is equal to , and the total mechanical energy is equal to the maximum potential or kinetic:
Let's write down the law of conservation of energy and take the derivative of the left and right parts of the equation: . Because the derivative of a constant value is equal to zero, then . The derivative of the sum is equal to the sum of the derivatives: and.
Therefore: , which means.

Ideal gas equation of state (Mendeleev-Clapeyron equation).
An equation of state is an equation that relates the parameters of a physical system and uniquely determines its state. In 1834 the French physicist B. Clapeyron, who worked for a long time in St. Petersburg, derived the equation of state for an ideal gas for a constant mass of gas. In 1874 D. I. Mendeleev derived an equation for an arbitrary number of molecules.
In MKT and ideal gas thermodynamics macroscopic parameters are: p, V, T, m. We know that . Consequently,. Given that , we get:.
The product of constant values is a constant value, therefore: - universal gas constant (universal, because it is the same for all gases).
Thus, we have: Equation of state (Mendeleev-Clapeyron equation).
Other forms of writing the equation of state of an ideal gas.
1. Equation for 1 mole of a substance. If n \u003d 1 mol, then, denoting the volume of one mole V m, we get:. For normal conditions, we get:
2. Write the equation in terms of density: - Density depends on temperature and pressure!
3. Clapeyron equation. It is often necessary to investigate the situation when the state of the gas changes with its constant amount (m=const) and in the absence of chemical reactions(M=const). This means that the amount of substance n=const. Then:
This entry means that for a given mass of a given gas equality is true:
For a constant mass of an ideal gas, the ratio of the product of pressure and volume to the absolute temperature in a given state is a constant value: .
gas laws.
1. Avogadro's law. In equal volumes of different gases at the same external conditions located the same number molecules (atoms). Condition: V 1 =V 2 =…=V n ; p 1 \u003d p 2 \u003d ... \u003d p n; T 1 \u003d T 2 \u003d ... \u003d T n
Proof: Therefore, under the same conditions (pressure, volume, temperature), the number of molecules does not depend on the nature of the gas and is the same.
2. Dalton's Law. The pressure of a mixture of gases is equal to the sum of the partial (private) pressures of each gas. Prove: p=p 1 +p 2 +…+p n Proof:
3. Pascal's law. The pressure produced on a liquid or gas is transmitted in all directions without change.

The equation of state for an ideal gas. gas laws.

Numbers of degrees of freedom: this is the number of independent variables (coordinates) that completely determine the position of the system in space. In some problems, a monatomic gas molecule (Fig. 1, a) is considered as a material point, which is given three degrees of freedom of translational motion. This does not take into account the energy of rotational motion. In mechanics, a molecule of a diatomic gas, in the first approximation, is considered to be a combination of two material points, which are rigidly connected by a non-deformable bond (Fig. 1, b). This system except for three degrees of freedom forward movement has two more degrees of freedom of rotational motion. Rotation around the third axis passing through both atoms is meaningless. This means that a diatomic gas has five degrees of freedom ( i= 5). A triatomic (Fig. 1, c) and polyatomic nonlinear molecule has six degrees of freedom: three translational and three rotational. It is natural to assume that there is no rigid bond between atoms. Therefore, for real molecules, it is also necessary to take into account the degrees of freedom of vibrational motion.

For any number of degrees of freedom of a given molecule, the three degrees of freedom are always translational. None of the translational degrees of freedom has an advantage over the others, which means that each of them has on average the same energy equal to 1/3 of the value<ε 0 >(energy of translational motion of molecules): In statistical physics, Boltzmann's law on the uniform distribution of energy over the degrees of freedom of molecules: for a statistical system that is in a state of thermodynamic equilibrium, each translational and rotational degree of freedom has an average kinetic energy equal to kT / 2, and each vibrational degree of freedom has an average energy equal to kT. The vibrational degree has twice as much energy, because it accounts for both kinetic energy (as in the case of translational and rotational motions) and potential energy, and the average values of potential and kinetic energy are the same. So the average energy of the molecule where i- the sum of the number of translational, the number of rotational in twice the number of vibrational degrees of freedom of the molecule: i=i post + i rotation +2 i vibrations In the classical theory, molecules are considered with a rigid bond between atoms; for them i coincides with the number of degrees of freedom of the molecule. Since in an ideal gas the mutual potential energy of interaction of molecules is equal to zero (molecules do not interact with each other), then the internal energy for one mole of gas will be equal to the sum of the kinetic energies N A of molecules: (1) Internal energy for an arbitrary mass m of gas. where M - molar mass, ν - amount of substance.

The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.

So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) makes a harmonic oscillatory movement on a vertical screen (Fig. 13.2).

The displacement from the equilibrium position during harmonic vibrations is described by an equation (it is called the kinematic law of harmonic motion) of the form:

\(x = A \cos \Bigr(\frac(2 \pi)(T)t + \varphi_0 \Bigl)\) or \(x = A \sin \Bigr(\frac(2 \pi)(T) t + \varphi"_0 \Bigl)\)

where X- mixing - a value that characterizes the position of the oscillating point at the moment of time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given point in time; BUT- oscillation amplitude - the maximum displacement of the body from the equilibrium position; T- oscillation period - the time of one complete oscillation; those. the smallest period of time after which the values of physical quantities characterizing the oscillation are repeated; \(\varphi_0\) - initial phase; \(\varphi = \frac(2 \pi)(T)t + \varphi"_0\) - phase of oscillation at time t. The oscillation phase is an argument of a periodic function, which, for a given oscillation amplitude, determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.

If at the initial time t0 = 0 the oscillating point is maximally displaced from the equilibrium position, then \(\varphi_0 = 0\), and the displacement of the point from the equilibrium position changes according to the law

\(x = A \cos \frac(2 \pi)(T)t.\)

If the oscillating point at t 0 \u003d 0 is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law

\(x = A \sin \frac(2 \pi)(T)t.\)

the value V, the reciprocal of the period and equal to the number of complete oscillations performed in 1 s, is called oscillation frequency:

\(\nu = \frac(1)(T) \)(in SI the unit of frequency is hertz, 1Hz = 1s -1).

If in time t body commits N full swing, then

\(T = \frac(t)(N) ; \nu = \frac(N)(t).\)

The value \(\omega = 2 \pi \nu = \frac(2 \pi)(T)\) , showing how many oscillations the body makes in 2 \(\pi\) With, called cyclic (circular) frequency.

The kinematic law of harmonic motion can be written as:

\(x = A \cos(2\pi \nu t + \varphi_0), x = A \cos(\omega t + \varphi_0).\)

Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).

Figure 13.3, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case \(\varphi_0=0\), i.e. \(~x=A\cos \omega t.\)

Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:

\(\upsilon_x = x" A \sin \omega t = \omega A \cos \Bigr(\omega t + \frac(\pi)(2) \Bigl) ,\)

where \(~\omega A = |\upsilon_x|_m\) is the amplitude of the velocity projection on the axis X.

This formula shows that during harmonic oscillations, the projection of the body's velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by \(\frac(\pi)(2)\) (Fig. 13.3 , b).

To find out the dependence of the acceleration a x (t) find the time derivative of the velocity projection:

\(~ a_x = \upsilon_x" = -\omega^2 A \cos \omega t = \omega^2 \cos(\omega t + \pi),\)

where \(~\omega^2 A = |a_x|_m\) is the amplitude of the acceleration projection onto the axle X.

For harmonic vibrations, the projection acceleration ahead of the phase shift by k (Fig. 13.3, c).

Similarly, you can plot \(~x(t), \upsilon_x (t)\) and \(~a_x(t),\) if \(~x = A \sin \omega t\) with \(\varphi_0 =0.\)

Considering that \(A \cos \omega t = x\), the formula for acceleration can be written

\(~a_x = - \omega^2 x,\)

So, the acceleration projection is the second derivative of the displacement and x \u003d x "", then the resulting ratio can be written as:

\(~a_x + \omega^2 x = 0\) or \(~x"" + \omega^2 x = 0.\)

The last equality is called equation of harmonic oscillations.

A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic oscillations - harmonic oscillator equation.

Literature

Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - S. 368-370.

Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form

(Any non-trivial solution of this differential equation is a harmonic oscillation with a cyclic frequency)

Types of vibrations

Free oscillations are performed under the action of the internal forces of the system after the system has been taken out of equilibrium. For free vibrations to be harmonic, it is necessary that oscillatory system was linear (described by linear equations of motion), and there was no energy dissipation in it (the latter would cause damping).

Forced oscillations are performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).

Harmonic vibration equation

Equation (1)

gives the dependence of the fluctuating value S on time t; this is the equation of free harmonic oscillations in explicit form. However, the equation of oscillations is usually understood as a different record of this equation, in differential form. For definiteness, we take equation (1) in the form

Differentiate it twice with respect to time:

It can be seen that the following relation holds:

which is called the equation of free harmonic oscillations (in differential form). Equation (1) is a solution to differential equation (2). Since equation (2) is a second-order differential equation, two initial conditions are necessary to obtain a complete solution (that is, to determine the constants A and   included in equation (1); for example, the position and speed of an oscillatory system at t = 0.

A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in homogeneous field gravity forces. The period of small eigenoscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to

and does not depend on the amplitude and mass of the pendulum.

A physical pendulum is an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

We have considered several physically perfect various systems, and made sure that the equations of motion are reduced to the same form

Difference between physical systems appear only in different definitions of the quantity and in a different physical sense of the variable x: it can be a coordinate, angle, charge, current, etc. Note that in this case, as follows from the very structure of equation (1.18), the quantity always has the dimension of inverse time.

Equation (1.18) describes the so-called harmonic vibrations.

The equation of harmonic oscillations (1.18) is a second-order linear differential equation (since it contains the second derivative of the variable x). The linearity of the equation means that

if any function x(t) is a solution to this equation, then the function Cx(t) will also be his solution ( C is an arbitrary constant);

if functions x 1 (t) and x 2 (t) are solutions of this equation, then their sum x 1 (t) + x 2 (t) will also be a solution to the same equation.

A mathematical theorem is also proved, according to which a second-order equation has two independent solutions. All other solutions, according to the properties of linearity, can be obtained as their linear combinations. It is easy to check by direct differentiation that the independent functions and satisfy equation (1.18). Means, common decision this equation has the form:

where C1,C2 are arbitrary constants. This solution can also be presented in another form. We introduce the quantity

and define the angle as:

Then the general solution (1.19) is written as

According to the trigonometry formulas, the expression in brackets is

We finally arrive at general solution of the equation of harmonic oscillations as:

Non-negative value A called oscillation amplitude, - the initial phase of the oscillation. The whole cosine argument - the combination - is called oscillation phase.

Expressions (1.19) and (1.23) are perfectly equivalent, so we can use either of them for reasons of simplicity. Both solutions are periodic functions time. Indeed, the sine and cosine are periodic with a period . Therefore, various states of a system that performs harmonic oscillations are repeated after a period of time t*, for which the oscillation phase receives an increment that is a multiple of :

Hence it follows that

The least of these times

called period of oscillation (Fig. 1.8), a - his circular (cyclic) frequency.

Rice. 1.8.

They also use frequency hesitation

Accordingly, the circular frequency is equal to the number of oscillations per seconds.

So, if the system at time t characterized by the value of the variable x(t), then, the same value, the variable will have after a period of time (Fig. 1.9), that is

The same value, of course, will be repeated after a while. 2T, ZT etc.

Rice. 1.9. Oscillation period

The general solution includes two arbitrary constants ( C 1 , C 2 or A, a), the values of which should be determined by two initial conditions. Usually (though not necessarily) their role is played by initial values variable x(0) and its derivative.

Let's take an example. Let the solution (1.19) of the equation of harmonic oscillations describe the motion of a spring pendulum. The values of arbitrary constants depend on the way in which we brought the pendulum out of equilibrium. For example, we pulled the spring to a distance and released the ball without initial velocity. In this case

Substituting t = 0 in (1.19), we find the value of the constant From 2

The solution thus looks like:

The speed of the load is found by differentiation with respect to time

Substituting here t = 0, find the constant From 1:

Finally

Comparing with (1.23), we find that is the oscillation amplitude, and its initial phase is equal to zero: .

We now bring the pendulum out of equilibrium in another way. Let's hit the load so it gains initial speed, but practically does not shift during the impact. We then have other initial conditions:

our solution looks like

The speed of the load will change according to the law:

Let's put it here: