Definition
Spring pendulum called a system that consists of an elastic spring to which a load is attached.
Let's assume that the weight of the load is $m$, the coefficient of elasticity of the spring is $k$. The mass of the spring in such a pendulum is usually not taken into account. If we consider the vertical movements of the load (Fig. 1), then it moves under the action of gravity and elastic forces, if the system was taken out of equilibrium and left to itself.
Oscillation equations for a spring pendulum
A free-oscillating spring pendulum is an example of a harmonic oscillator. Let us assume that the pendulum oscillates along the X axis. If the oscillations are small, Hooke's law is satisfied, then the equation of motion of the load has the form:
\[\ddot(x)+(\omega )^2_0x=0\left(1\right),\]
where $(шu)^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution to equation (1) is the function:
where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic oscillation frequency of the pendulum, $A$ is the oscillation amplitude; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ - initial phases of oscillations.
In exponential form, the oscillations of a spring pendulum can be written as:
Formulas for the period and frequency of oscillation of a spring pendulum
If Hooke's law is satisfied in elastic oscillations, then the oscillation period of the spring pendulum is calculated using the formula:
Since the oscillation frequency ($\nu $) is the reciprocal of the period, then:
\[\nu =\frac(1)(T)=\frac(1)(2\pi )\sqrt(\frac(k)(m))\left(5\right).\]
Formulas for the amplitude and initial phase of a spring pendulum
Knowing the equation of oscillations of a spring pendulum (1 or 2) and the initial conditions, one can completely describe the harmonic oscillations of a spring pendulum. The initial conditions determine the amplitude ($A$) and the initial phase of oscillations ($\varphi $).
The amplitude can be found as:
the initial phase is:
where $v_0$ is the speed of the load at $t=0\ c$, when the coordinate of the load is equal to $x_0$.
Energy of oscillations of a spring pendulum
With a one-dimensional movement of a spring pendulum, there is only one path between two points of its movement, therefore, the condition of force potentiality is satisfied (any force can be considered potential if it depends only on the coordinates). Since the forces acting on a spring pendulum are potential, we can speak of potential energy.
Let the spring pendulum oscillate in the horizontal plane (Fig. 2). For the zero potential energy of the pendulum, we take the position of its equilibrium, where we place the origin of coordinates. Friction forces are not taken into account. Using the formula relating potential force and potential energy for the one-dimensional case:
considering that for a spring pendulum $F=-kx$,
then the potential energy ($E_p$) of the spring pendulum is:
We write the law of conservation of energy for a spring pendulum as:
\[\frac(m(\dot(x))^2)(2)+\frac(m((\omega )_0)^2x^2)(2)=const\ \left(10\right), \]
where $\dot(x)=v$ - speed of cargo movement; $E_k=\frac(m(\dot(x))^2)(2)$ - kinetic energy of the pendulum.
From formula (10) we can draw the following conclusions:
- The maximum kinetic energy of a pendulum is equal to its maximum potential energy.
- The time-average kinetic energy of the oscillator is equal to its time-average potential energy.
Examples of problems with a solution
Example 1
Exercise. A small ball with a mass of $m=0.36$ kg is attached to a horizontal spring whose elasticity coefficient is $k=1600\ \frac(N)(m)$. What was the initial displacement of the ball from the equilibrium position ($x_0$), if during oscillations it passes it with a speed $v=1\ \frac(m)(c)$?
Solution. Let's make a drawing.
According to the law of conservation of mechanical energy (since we believe that there are no friction forces), we write:
where $E_(pmax)$ is the potential energy of the ball at its maximum displacement from the equilibrium position; $E_(kmax\ )$ - kinetic energy of the ball, at the moment of passing the equilibrium position.
The potential energy is:
In accordance with (1.1), we equate the right sides of (1.2) and (1.3), we have:
\[\frac(mv^2)(2)=\frac(k(x_0)^2)(2)\left(1.4\right).\]
From (1.4) we express the desired value:
Calculate the initial (maximum) displacement of the load from the equilibrium position:
Answer.$x_0=1.5$ mm
Example 2
Exercise. A spring pendulum oscillates according to the law: $x=A(\cos \left(\omega t\right),\ \ )\ $where $A$ and $\omega $ are constants. When the restoring force reaches $F_0 for the first time, the potential energy of the weight is $E_(p0)$. At what point in time will this happen?
Solution. The restoring force for a spring pendulum is the elastic force equal to:
We find the potential energy of the load oscillations as:
At the point in time to be found $F=F_0$; $E_p=E_(p0)$, then:
\[\frac(E_(p0))(F_0)=-\frac(A)(2)(\cos \left(\omega t\right)\ )\to t=\frac(1)(\omega ) \ arc(\cos \left(-\frac(2E_(p0))(AF_0)\right)\ ).\]
Answer.$t=\frac(1)(\omega )\ arc(\cos \left(-\frac(2E_(p0))(AF_0)\right)\ )$
If the ball is displaced from the equilibrium position by a distance x, then the elongation of the spring will become equal to Δl 0 + x. Then the resulting force will take the value:
Taking into account the equilibrium condition (1.7.1), we obtain:
The minus sign indicates that displacement and force are in opposite directions.
The elastic force f has the following properties:
- It is proportional to the displacement of the ball from the equilibrium position;
- It is always directed towards the equilibrium position.
In order to inform the system of the displacement x, it is necessary to perform work against the elastic force:
This work goes to create a reserve of potential energy of the system:
Under the action of an elastic force, the ball will move towards the equilibrium position with an ever-increasing speed. Therefore, the potential energy of the system will decrease, but the kinetic energy will increase (we neglect the mass of the spring). Having come to the equilibrium position, the ball will continue to move by inertia. This is a slow motion and will stop when the kinetic energy is completely converted into potential. Then the same process will proceed when the ball moves in the opposite direction. If there is no friction in the system, the ball will oscillate indefinitely.
The equation of Newton's second law in this case is:
Let's transform the equation like this:
Introducing the notation , we obtain a linear homogeneous differential equation second order:
By direct substitution, it is easy to verify that common decision equation (1.7.8) has the form:
where a is the amplitude and φ is the initial phase of the oscillation - constant values. Therefore, the oscillation of a spring pendulum is harmonic (Fig. 1.7.2).
Rice. 1.7.2. harmonic oscillation
Due to the periodicity of the cosine, various states of the oscillatory system are repeated after a certain period of time (oscillation period) T, during which the phase of the oscillation receives an increment of 2π. You can calculate the period using the equation:
from where follows:
The number of oscillations per unit time is called the frequency:
The unit of frequency is the frequency of such an oscillation, the period of which is 1 s. This unit is called 1 Hz.
From (1.7.11) it follows that:
Therefore, ω 0 is the number of oscillations made in 2π seconds. The value ω 0 is called the circular or cyclic frequency. Using (1.7.12) and (1.7.13), we write:
Differentiating () with respect to time, we obtain an expression for the speed of the ball:
From (1.7.15) it follows that the speed also changes according to the harmonic law and is ahead of the phase shift by ½π. Differentiating (1.7.15), we get the acceleration:
1.7.2. Mathematical pendulum
Mathematical pendulum called an idealized system consisting of an inextensible weightless thread on which a body is suspended, the entire mass of which is concentrated at one point.
The deviation of the pendulum from the equilibrium position is characterized by the angle φ formed by the thread with the vertical (Fig. 1.7.3).
Rice. 1.7.3. Mathematical pendulum
When the pendulum deviates from the equilibrium position, a torque arises that tends to return the pendulum to the equilibrium position:
Let us write the dynamics equation for the pendulum rotary motion, given that its moment of inertia is equal to ml 2:
This equation can be brought to the form:
Restricting ourselves to the case of small fluctuations sinφ ≈ φ and introducing the notation:
equation (1.7.19) can be represented as follows:
which coincides in form with the equation of oscillations of a spring pendulum. Therefore, its solution will be a harmonic oscillation:
From (1.7.20) it follows that the cyclic oscillation frequency of a mathematical pendulum depends on its length and free fall acceleration. Using the formula for the oscillation period () and (1.7.20), we obtain the known relation:
1.7.3. physical pendulum
A physical pendulum is a rigid body capable of oscillating around a fixed point that does not coincide with the center of inertia. In the equilibrium position, the center of inertia of the pendulum C is under the suspension point O on the same vertical (Fig. 1.7.4).
Rice. 1.7.4. physical pendulum
When the pendulum deviates from the equilibrium position by an angle φ, a torque arises that tends to return the pendulum to the equilibrium position:
where m is the mass of the pendulum, l is the distance between the suspension point and the center of inertia of the pendulum.
Let us write the equation for the dynamics of rotational motion for the pendulum, taking into account that the moment of inertia is equal to I:
For small fluctuations sinφ ≈ φ. Then, introducing the notation:
which also coincides in form with the equation of oscillations of a spring pendulum. From equations (1.7.27) and (1.7.26) it follows that with small deviations of the physical pendulum from the equilibrium position, it performs a harmonic oscillation, the frequency of which depends on the mass of the pendulum, the moment of inertia and the distance between the axis of rotation and the center of inertia. Using (1.7.26), you can calculate the oscillation period:
Comparing formulas (1.7.28) and () we get that a mathematical pendulum with a length:
will have the same oscillation period as the considered physical pendulum. The quantity (1.7.29) is called reduced length physical pendulum. Therefore, the reduced length of a physical pendulum is the length of such a mathematical pendulum, the period of oscillation of which is equal to the period of oscillation of a given physical pendulum.
A point on a straight line connecting the point of suspension with the center of inertia, which lies at a distance of the reduced length from the axis of rotation, is called swing center physical pendulum. According to Steiner's theorem, the moment of inertia of a physical pendulum is:
where I 0 is the moment of inertia about the center of inertia. Substituting (1.7.30) into (1.7.29), we get:
Therefore, the reduced length is always greater than the distance between the suspension point and the pendulum's center of inertia, so that the suspension point and the swing center lie on opposite sides of the center of inertia.
1.7.4. Energy of harmonic vibrations
During harmonic oscillation, there is a periodic mutual transformation of the kinetic energy of the oscillating body E k and the potential energy E p, due to the action of a quasi-elastic force. From these energies, the total energy E of the oscillatory system is added:
Let's write the last expression
But k \u003d mω 2, so we get the expression for the total energy of the oscillating body
Thus, the total energy of a harmonic oscillation is constant and proportional to the square of the amplitude and the square of the circular frequency of the oscillation.
1.7.5. damped vibrations .
When studying harmonic oscillations, the forces of friction and resistance that exist in real systems were not taken into account. The action of these forces significantly changes the nature of the motion, the oscillation becomes fading.
If, in addition to the quasi-elastic force, the resistance forces of the medium (friction forces) act in the system, then Newton's second law can be written as follows:
where r is the coefficient of friction, which characterizes the properties of the medium to resist movement. We substitute (1.7.34b) into (1.7.34a):
The graph of this function is shown in Fig. 1.7.5 as a solid curve 1, and a dashed line 2 shows the change in amplitude:
With very little friction, the period of damped oscillation is close to the period of undamped free oscillation (1.7.35.b)
The rate of decrease in the oscillation amplitude is determined by damping factor: the larger β, the stronger the retarding effect of the medium and the faster the amplitude decreases. In practice, the degree of attenuation is often characterized logarithmic damping decrement, meaning by this a value equal to the natural logarithm of the ratio of two successive oscillation amplitudes separated by a time interval equal to the oscillation period:
;
Therefore, the damping coefficient and the logarithmic damping decrement are related by a fairly simple relationship:
With strong damping, it can be seen from formula (1.7.37) that the oscillation period is an imaginary quantity. The movement in this case is already called aperiodic. The aperiodic motion graph is shown in Fig. 1.7.6. Undamped and damped oscillations are called own or free. They arise as a result of the initial displacement or initial velocity and occur in the absence of external influence due to the initially accumulated energy.
1.7.6. Forced vibrations. Resonance .
compelled Oscillations are called those that arise in the system with the participation of an external force that changes according to a periodic law.
Let us assume that, in addition to the quasi-elastic force and the friction force, an external driving force acts on the material point
,
where F 0 - amplitude; ω - circular frequency of oscillations of the driving force. We compose a differential equation (Newton's second law):
,
The amplitude of the forced oscillation (1.7.39) is directly proportional to the amplitude of the driving force and has a complex dependence on the attenuation coefficient of the medium and the circular frequencies of natural and forced oscillations. If ω 0 and β are given for the system, then the amplitude of forced oscillations has a maximum value at a certain specific frequency of the driving force, called resonant.
The phenomenon itself - reaching the maximum amplitude for given ω 0 and β - is called resonance.
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| Rice. 1.7.7. Resonance |
In the absence of resistance, the amplitude of forced oscillations at resonance is infinitely large. In this case, from ω res = ω 0, i.e. resonance in a system without damping occurs when the frequency of the driving force coincides with the frequency of natural oscillations. Graphical dependence of the amplitude of forced oscillations on the circular frequency of the driving force at different meanings attenuation coefficient is shown in fig. 5.
Mechanical resonance can be both beneficial and detrimental. The harmful effect of resonance is mainly due to the destruction it can cause. So, in technology, taking into account different vibrations, it is necessary to foresee the possible occurrence of resonant conditions, otherwise there may be destruction and disasters. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.
If the attenuation coefficient of the internal organs of a person would not be large, then the resonant phenomena that arose in these organs under the influence of external vibrations or sound waves, could lead to tragic consequences: rupture of organs, damage to ligaments, etc. However, such phenomena are practically not observed under moderate external influences, since the attenuation coefficient of biological systems is quite large. Nevertheless, resonant phenomena under the action of external mechanical vibrations occur in the internal organs. This, apparently, is one of the reasons for the negative impact of infrasonic oscillations and vibrations on the human body.
1.7.7. Self-oscillations
There are also such oscillatory systems that themselves regulate the periodic replenishment of wasted energy and therefore can fluctuate for a long time.
The undamped oscillations that exist in any system in the absence of a variable external influence are called self-oscillations, and the systems themselves self-oscillating.
The amplitude and frequency of self-oscillations depend on the properties in the self-oscillating system itself; in contrast to forced oscillations, they are not determined by external influences.
In many cases, self-oscillating systems can be represented by three main elements (Fig. 1.7.8): 1) the actual oscillating system; 2) energy source; 3) a regulator of energy supply to the actual oscillatory system. Oscillating system by channel feedback(Fig. 6) affects the regulator, informing the regulator about the state of this system.
A classic example of a mechanical self-oscillating system is a clock, in which a pendulum or balance is an oscillatory system, a spring or a raised weight is a source of energy, and an anchor is a regulator of the energy supply from the source to the oscillatory system.
Many biological systems (heart, lungs, etc.) are self-oscillatory. A typical example of an electromagnetic self-oscillating system is generators of self-oscillating oscillations.
1.7.8. Addition of vibrations in one direction
Consider the addition of two harmonic oscillations of the same direction and the same frequency:
x 1 \u003d a 1 cos (ω 0 t + α 1), x 2 \u003d a 2 cos (ω 0 t + α 2).
A harmonic oscillation can be specified using a vector, the length of which is equal to the amplitude of the oscillations, and the direction forms an angle with some axis equal to the initial phase of the oscillations. If this vector rotates with an angular velocity ω 0, then its projection on the selected axis will change according to the harmonic law. Based on this, we choose some axis X and represent the oscillations using the vectors a 1 and a 2 (Fig. 1.7.9).
From Figure 1.7.6 it follows that
.
Schemes in which oscillations are depicted graphically as vectors on a plane are called vector diagrams.
It follows from formula 1.7.40. That if the phase difference of both oscillations is equal to zero, the amplitude of the resulting oscillation is equal to the sum of the amplitudes of the added oscillations. If the phase difference of the added oscillations is equal to , then the amplitude of the resulting oscillation is equal to . If the frequencies of the added oscillations are not the same, then the vectors corresponding to these oscillations will rotate at different speeds. In this case, the resulting vector pulsates in magnitude and rotates at a non-constant rate. Consequently, as a result of addition, not a harmonic oscillation is obtained, but a complex oscillatory process.
1.7.9. beats
Consider the addition of two harmonic oscillations of the same direction, slightly different in frequency. Let the frequency of one of them be equal to ω , and the frequency of the second ω + ∆ω, and ∆ω<<ω. Положим, что амплитуды складываемых колебаний одинаковы и начальные фазы обоих колебаний равны нулю. Тогда уравнения колебаний запишутся следующим образом:
x 1 \u003d a cos ωt, x 2 \u003d a cos (ω + ∆ω) t.
Adding these expressions and using the formula for the sum of cosines, we get:
Oscillations (1.7.41) can be considered as a harmonic oscillation with frequency ω, the amplitude of which varies according to the law . This function is periodic with a frequency twice the frequency of the expression under the module sign, i.e. with frequency ∆ω. Thus, the frequency of amplitude pulsations, called the beat frequency, is equal to the difference in the frequencies of the added oscillations.
1.7.10. Addition of mutually perpendicular vibrations (Lissajous figures)
If a material point oscillates both along the x-axis and along the y-axis, then it will move along some curvilinear trajectory. Let the oscillation frequency be the same and the initial phase of the first oscillation be equal to zero, then we write the oscillation equations in the form:
Equation (1.7.43) is the equation of an ellipse, the axes of which are arbitrarily oriented relative to the x and y coordinate axes. The orientation of the ellipse and the size of its semiaxes depend on the amplitudes a and b and the phase difference α. Let's consider some special cases:
(m=0, ±1, ±2, …). In this case, the equation has the formThis is the equation of an ellipse, the axes of which coincide with the coordinate axes, and its semiaxes are equal to the amplitudes (Fig. 1.7.12). If the amplitudes are equal, then the ellipse becomes a circle.
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| Fig.1.7.12 |
If the frequencies of mutually perpendicular oscillations differ by a small amount ∆ω, they can be considered as oscillations of the same frequency, but with a slowly changing phase difference. In this case, the oscillation equations can be written
x=a cos ωt, y=b cos[ωt+(∆ωt+α)]
and the expression ∆ωt+α is considered as a phase difference that slowly changes with time according to a linear law. The resulting movement in this case follows a slowly changing curve, which will successively take the form corresponding to all values of the phase difference from -π to +π.
If the frequencies of mutually perpendicular oscillations are not the same, then the trajectory of the resulting motion has the form of rather complex curves called Lissajous figures. Let, for example, the frequencies of the added oscillations be related as 1 : 2 and phase difference π/2. Then the oscillation equations have the form
x=a cos ωt, y=b cos.
While along the x-axis the point manages to move from one extreme position to another, along the y-axis, having left the zero position, it manages to reach one extreme position, then another and return. The curve view is shown in fig. 1.7.13. The curve with the same frequency ratio, but the phase difference equal to zero is shown in Fig. 1.7.14. The ratio of the frequencies of the added oscillations is inverse to the ratio of the number of points of intersection of the Lissajous figures with straight lines parallel to the coordinate axes. Therefore, by the appearance of the Lissajous figures, one can determine the ratio of the frequencies of the added oscillations or an unknown frequency. If one of the frequencies is known.
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| Fig.1.7.13 |
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| Fig.1.7.14 |
The closer to unity the rational fraction expressing the ratio of vibration frequencies, the more complex the resulting Lissajous figures.
1.7.11. Wave propagation in an elastic medium
If in any place of an elastic (solid liquid or gaseous) medium vibrations of its particles are excited, then due to the interaction between particles this vibration will propagate in the medium from particle to particle with a certain speed υ. the process of propagation of vibrations in space is called wave.
The particles of the medium in which the wave propagates are not involved by the wave in translational motion, they only oscillate around their equilibrium positions.
Depending on the directions of particle oscillations with respect to the direction in which the wave propagates, there are longitudinal and transverse waves. In a longitudinal wave, the particles of the medium oscillate along the propagation of the wave. In a transverse wave, the particles of the medium oscillate in directions perpendicular to the direction of wave propagation. Elastic transverse waves can arise only in a medium with shear resistance. Therefore, in liquid and gaseous media, only longitudinal waves can occur. In a solid medium, the occurrence of both longitudinal and transverse waves is possible.
On fig. 1.7.12 shows the motion of particles during propagation in a medium of a transverse wave. Numbers 1, 2, etc. denote particles lagging behind each other by a distance equal to (¼ υT), i.e. by the distance traveled by the wave in a quarter of the period of oscillations made by the particles. At the time taken as zero, the wave, propagating along the axis from left to right, reached particle 1, as a result of which the particle began to move upward from the equilibrium position, dragging the next particles with it. After a quarter of the period, particle 1 reaches the uppermost equilibrium position of particle 2. After another quarter of the period, the first part will pass the equilibrium position, moving in the direction from top to bottom, the second particle will reach the uppermost position, and the third particle will begin to move upward from the equilibrium position. At the moment of time equal to T, the first particle will complete the complete oscillation cycle and will be in the same state of motion as the starting moment. The wave by the time T, having passed the path (υT), will reach particle 5.
On Fig. 1.7.13 shows the movement of particles during propagation in a medium of a longitudinal wave. All considerations concerning the behavior of particles in a transverse wave can also be applied to this case with the displacements up and down replaced by displacements to the right and left.
It can be seen from the figure that during the propagation of a longitudinal wave in the medium, alternating condensations and rarefaction of particles are created (the places of condensation are circled in the figure by a dotted line), moving in the direction of wave propagation with a speed υ.
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| Rice. 1.7.15 |
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| Rice. 1.7.16 |
On fig. 1.7.15 and 1.7.16 shows oscillations of particles whose positions and equilibria lie on the axis x. In reality, not only particles oscillate along the axis x, but a collection of particles enclosed in a certain volume. Spreading from the sources of oscillations, the wave process covers more and more parts of space, the locus of points, to which oscillations reach by the time t, is called wave front(or wave front). The wave front is the surface that separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen.
The locus of points oscillating in the same phase is called wave surface . The wave surface can be drawn through any point in the space covered by the wave process. Consequently, there are an infinite number of wave surfaces, while there is only one wave front at any time. The wave surfaces remain stationary (they pass through the equilibrium positions of particles oscillating in the same phase ). The wavefront is constantly moving.
Wave surfaces can be of any shape. In the simplest cases, they have the shape of a plane or sphere. Accordingly, the wave in these cases is called plane or spherical. In a plane wave, the wave surfaces are a set of planes parallel to each other, in a spherical wave - a set of concentric spheres.
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| Rice. 1.7.17 |
Let a plane wave propagate along the axis x. Then all points of the sphere, positions, equilibria of which have the same coordinate x(but the difference in coordinate values y and z), oscillate in the same phase.
On Fig. 1.7.17 shows a curve that gives an offset ξ from the equilibrium position of points with different x at some point in time. This drawing should not be taken as a visible image of a wave. The figure shows a graph of functions ξ (x, t) for some fixed point in time t. Such a graph can be built for both longitudinal and transverse waves.
The distance λ, for a short wave propagates in a time equal to the period of oscillation of the particles of the medium, is called wavelength. It's obvious that
where υ is the wave speed, T is the oscillation period. The wavelength can also be defined as the distance between the nearest points of the medium, oscillating with a phase difference equal to 2π (see Fig. 1.7.14)
Replacing in relation (1.7.45) T through 1/ν (ν is the oscillation frequency), we obtain
This formula can also be arrived at from the following considerations. In one second, the wave source performs ν oscillations, generating in the medium during each oscillation one "crest" and one "trough" of the wave. By the time the source completes the ν -th oscillation, the first "ridge" will have time to go through the path υ. Consequently, ν "crests" and "troughs" of the wave must fit into the length υ.
1.7.12. Plane wave equation
The wave equation is an expression that gives the displacement of an oscillating particle as a function of its coordinates x, y, z and time t :
ξ = ξ (x, y, z; t)
(meaning the coordinates of the equilibrium position of the particle). This function must be periodic with respect to time t , and relative to the coordinates x, y, z. . Periodicity in time follows from the fact that points separated from each other at a distance λ , fluctuate in the same way.
Find the type of function ξ in the case of a plane wave, assuming that the oscillations are harmonic. To simplify, we direct the coordinate axes so that the axis x coincides with the direction of wave propagation. Then the wave surfaces will be perpendicular to the axis x and since all points of the wave surface oscillate equally, the displacement ξ will only depend on x and t:
ξ = ξ (x, t) .
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| Fig.1.7.18 |
Let oscillations of points lying in the plane x = 0 (Fig. 1.7.18), have the form
Let us find the type of oscillation of points in the plane corresponding to an arbitrary value x . To go way from the plane x=0 to this plane, the wave takes time ( υ is the speed of wave propagation). Consequently, oscillations of particles lying in the plane x , will be behind in time by τ from vibrations of particles in the plane x = 0 , i.e. will look like
So, plane wave equation(longitudinal, and transverse), propagating in the direction of the axis x , as follows:
This expression defines the relationship between time t and that place x , in which the phase has a fixed value. The resulting dx/dt value gives the speed at which the given phase value moves. Differentiating the expression (1.7.48), we obtain
The equation of a wave propagating in the direction of decreasing x :
When deriving formula (1.7.53), we assumed that the oscillation amplitude does not depend on x . For a plane wave, this is observed when the wave energy is not absorbed by the medium. When propagating in an energy-absorbing medium, the intensity of the wave gradually decreases with distance from the source of oscillations - wave attenuation is observed. Experience shows that in a homogeneous medium such damping occurs according to an exponential law:
Respectively plane wave equation, considering damping, has the following form:
| (1.7.54) |
(a 0 is the amplitude at the points of the plane x = 0).
Definition
Oscillation frequency($\nu$) is one of the parameters that characterize the fluctuations. This is the reciprocal of the fluctuation period ($T$):
\[\nu=\frac(1)(T)\left(1\right).\]
Thus, the frequency of oscillations is called a physical quantity equal to the number of repetitions of oscillations per unit of time.
\[\nu =\frac(N)(\Delta t)\left(2\right),\]
where $N$ is the number of complete oscillatory motions; $\Delta t$ - time during which these fluctuations occurred.
The cyclic oscillation frequency ($(\omega )_0$) is related to the frequency $\nu $ by the formula:
\[\nu =\frac((\omega )_0)(2\pi )\left(3\right).\]
The unit of frequency in the International System of Units (SI) is hertz or reciprocal second:
\[\left[\nu \right]=c^(-1)=Hz.\]
Spring pendulum
Definition
Spring pendulum called a system that consists of an elastic spring to which a load is attached.
Let's assume that the weight of the load is $m$, the coefficient of elasticity of the spring is $k$. The mass of the spring in such a pendulum is usually not taken into account. If we consider the horizontal movements of the load (Fig. 1), then it moves under the action of the elastic force, if the system was taken out of equilibrium and left to itself. In this case, it is often believed that friction forces can be ignored.
Oscillation equations for a spring pendulum
A spring pendulum that oscillates freely is an example of a harmonic oscillator. Let it perform oscillations along the X axis. If the oscillations are small, Hooke's law is satisfied, then we write the equation for the movement of the load as:
\[\ddot(x)+(\omega )^2_0x=0\left(4\right),\]
where $(\omega )^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution of equation (4) is a sine or cosine function of the form:
where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic oscillation frequency of the spring pendulum, $A$ is the oscillation amplitude; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ - initial phases of oscillations.
Oscillation frequency of the spring pendulum
From formula (3) and $(\omega )_0=\sqrt(\frac(k)(m))$, it follows that the oscillation frequency of the spring pendulum is:
\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(6\right).\]
Formula (6) is valid if:
- the spring in the pendulum is assumed to be weightless;
- the weight attached to the spring is a perfectly rigid body;
- there are no torsional vibrations.
Expression (6) shows that the oscillation frequency of a spring pendulum increases with a decrease in the mass of the load and an increase in the coefficient of elasticity of the spring. The oscillation frequency of a spring pendulum does not depend on the amplitude. If the oscillations are not small, the elastic force of the spring does not obey Hooke's law, then the dependence of the oscillation frequency on the amplitude appears.
Examples of problems with a solution
Example 1
Exercise. The oscillation period of the spring pendulum is $T=5\cdot (10)^(-3)c$. What is the oscillation frequency in this case? What is the cyclic frequency of this weight?
Solution. The oscillation frequency is the reciprocal of the oscillation period, therefore, to solve the problem, it is enough to use the formula:
\[\nu=\frac(1)(T)\left(1.1\right).\]
Calculate the desired frequency:
\[\nu =\frac(1)(5\cdot (10)^(-3))=200\ \left(Hz\right).\]
The cyclic frequency is related to the $\nu$ frequency as:
\[(\omega )_0=2\pi \nu \ \left(1.2\right).\]
Let's calculate the cyclic frequency:
\[(\omega )_0=2\pi \cdot 200\approx 1256\ \left(\frac(rad)(c)\right).\]
Answer.$1)\ \nu =200$ Hz. 2) $(\omega )_0=1256\ \frac(rad)(s)$
Example 2
Exercise. The mass of the load hanging on the elastic spring (Fig. 2) is increased by $\Delta m$, while the frequency decreases by $n$ times. What is the mass of the first load?
\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(2.1\right).\]
For the first load, the frequency will be equal to:
\[(\nu )_1=\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(2.2\right).\]
For the second load:
\[(\nu )_2=\frac(1)(2\pi )\sqrt(\frac(k)(m+\Delta m))\ \left(2.2\right).\]
By the condition of the problem $(\nu )_2=\frac((\nu )_1)(n)$, we find the relation $\frac((\nu )_1)((\nu )_2):\frac((\nu )_1)((\nu )_2)=\sqrt(\frac(k)(m)\cdot \frac(m+\Delta m)(k))=\sqrt(1+\frac(\Delta m)( m))=n\ \left(2.3\right).$
We obtain from equation (2.3) the desired mass of the load. To do this, we square both parts of expression (2.3) and express $m$:
Answer.$m=\frac(\Delta m)(n^2-1)$
Bodies under the action of an elastic force, the potential energy of which is proportional to the square of the displacement of the body from the equilibrium position:
where k is the stiffness of the spring.With free mechanical vibrations, the kinetic and potential energies change periodically. At the maximum deviation of the body from the equilibrium position, its velocity, and hence the kinetic energy, vanishes. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a horizontally located spring, the potential energy is the energy of elastic deformations of the spring.
When the body in its motion passes through the equilibrium position, its speed is maximum. At this moment, it has the maximum kinetic and minimum potential energy. An increase in kinetic energy occurs at the expense of a decrease in potential energy. With further movement, the potential energy begins to increase due to the decrease in kinetic energy, etc.
Thus, during harmonic oscillations, a periodic transformation of kinetic energy into potential energy and vice versa occurs.
If there is no friction in the oscillatory system, then the total mechanical energy during free vibrations remains unchanged.
For spring load:
The start of the oscillatory movement of the body is carried out using the Start button. The Stop button allows you to stop the process at any time.
Graphically shows the relationship between potential and kinetic energies during oscillations at any time. Note that in the absence of damping, the total energy of the oscillatory system remains unchanged, the potential energy reaches its maximum at the maximum deviation of the body from the equilibrium position, and the kinetic energy reaches its maximum value when the body passes through the equilibrium position.
A spring pendulum is a material point of mass , attached to an absolutely elastic weightless spring with stiffness 
. There are two simplest cases: horizontal (Fig. 15, a) and vertical (Fig. 15, b) pendulums.
a)
Horizontal pendulum(Fig. 15a). When shifting cargo 
out of equilibrium 
by the amount 
acts on it in a horizontal direction. restoring elastic force
(Hooke's law).
It is assumed that the horizontal support on which the load slides 
during its vibrations, it is absolutely smooth (no friction).
b) vertical pendulum(fig.15, b). The equilibrium position in this case is characterized by the condition:
where 
- the magnitude of the elastic force acting on the load 
when the spring is statically stretched 
under the influence of gravity 
.
|
a |
|
Fig.15. Spring pendulum: a- horizontal and b– vertical
If the spring is stretched and the load is released, it will begin to oscillate vertically. If the offset at some point in time is 
,
then the elastic force will now be written as 
.
In both cases considered, the spring pendulum performs harmonic oscillations with a period
(27)
and cyclic frequency
.
(28)
Using the example of considering a spring pendulum, we can conclude that harmonic oscillations are a movement caused by a force that increases in proportion to the displacement 
. In this way, if the restoring force looks like Hooke's law
(she got the namequasi-elastic force
), then the system must perform harmonic oscillations. At the moment of passing the equilibrium position, the restoring force does not act on the body, however, the body skips the equilibrium position by inertia and the restoring force changes direction to the opposite.
Mathematical pendulum
Fig.16. Mathematical pendulum
, which performs small oscillations under the action of gravity (Fig. 16).
Oscillations of such a pendulum at small deflection angles 
(not exceeding 5º) can be considered harmonic, and the cyclic frequency of the mathematical pendulum:
,
(29)
and the period:
.
(30)
2.3. Body energy during harmonic vibrations
The energy imparted to the oscillating system during the initial push will be periodically transformed: the potential energy of the deformed spring will be converted into the kinetic energy of the moving load and vice versa.
Let the spring pendulum perform harmonic oscillations with the initial phase 
, i.e. 
(fig.17).
Fig.17. Law of conservation of mechanical energy
when the spring pendulum oscillates
At the maximum deviation of the load from the equilibrium position, the total mechanical energy of the pendulum (the energy of a deformed spring with stiffness 
) is equal to 
. When passing through the equilibrium position ( 
) the potential energy of the spring will become equal to zero, and the total mechanical energy of the oscillatory system will be determined as 
.
Figure 18 shows the dependences of the kinetic, potential and total energy in cases where harmonic oscillations are described by trigonometric functions of the sine (dashed line) or cosine (solid line).
Fig.18. Graphs of the time dependence of the kinetic
and potential energy for harmonic oscillations
From the graphs (Fig. 18) it follows that the frequency of change in kinetic and potential energy is twice as high as the natural frequency of harmonic oscillations.
