Circular frequency of oscillation of a load on a spring. Free vibrations

Body vibrations on a spring

Spring pendulum

Equations of oscillations of a spring pendulum

Oscillation frequency of a spring pendulum

Examples of problems with solutions

Free vibrations. Math pendulum

Energy conversions during free mechanical vibrations

Forced vibrations. Resonance. Self-oscillations

I'M IN. ,
Far Eastern State Interregional Industrial and Economic College, Khabarovsk

Free vibrations are carried out under the influence of internal forces of the system after the system has been removed from its equilibrium position.

In order to free vibrations occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position is proportional to the displacement of the body from the equilibrium position and is directed in the direction opposite to the displacement (see §2.1):

Forces of any other physical nature that satisfy this condition are called quasi-elastic .

Thus, a load of some mass m, attached to the stiffening spring k, the second end of which is fixedly fixed (Fig. 2.2.1), constitute a system capable of performing free harmonic oscillations in the absence of friction. A load on a spring is called linear harmonic oscillator.

The circular frequency ω 0 of free oscillations of a load on a spring is found from Newton’s second law:

When the spring-load system is located horizontally, the force of gravity applied to the load is compensated by the support reaction force. If the load is suspended on a spring, then the force of gravity is directed along the line of movement of the load. In the equilibrium position, the spring is stretched by an amount x 0 equal

Therefore, Newton's second law for a load on a spring can be written as

Equation (*) is called equation of free vibrations . It should be noted that the physical properties of the oscillatory system determine only the natural frequency of oscillations ω 0 or the period T . Parameters of the oscillation process such as amplitude x m and the initial phase φ 0 are determined by the way in which the system was brought out of equilibrium at the initial moment of time.

If, for example, the load was displaced from the equilibrium position by a distance Δ l and then at a point in time t= 0 released without initial speed, then x m = Δ l, φ 0 = 0.

If the load, which was in the equilibrium position, was given an initial speed ± υ 0 with the help of a sharp push, then,

Thus, the amplitude x m free oscillations and its initial phase φ 0 are determined initial conditions .

There are many types of mechanical oscillatory systems that use elastic deformation forces. In Fig. Figure 2.2.2 shows the angular analogue of a linear harmonic oscillator. A horizontally located disk hangs on an elastic thread attached to its center of mass. When the disk is rotated through an angle θ, a moment of force occurs M control of elastic torsional deformation:

Where I = I C is the moment of inertia of the disk relative to the axis, passing through the center of mass, ε is the angular acceleration.

By analogy with a load on a spring, you can get:

Mathematical pendulum called a small body suspended on a thin inextensible thread, the mass of which is negligible compared to the mass of the body. In the equilibrium position, when the pendulum hangs plumb, the force of gravity is balanced by the tension force of the thread. When the pendulum deviates from the equilibrium position by a certain angle φ, a tangential component of gravity appears F τ = - mg sin φ (Fig. 2.3.1). The minus sign in this formula means that the tangential component is directed in the direction opposite to the deflection of the pendulum.

If we denote by x linear displacement of the pendulum from the equilibrium position along an arc of a circle of radius l, then its angular displacement will be equal to φ = x / l. Newton's second law, written for the projections of acceleration and force vectors onto the direction of the tangent, gives:

This relationship shows that a mathematical pendulum is a complex nonlinear system, since the force tending to return the pendulum to the equilibrium position is not proportional to the displacement x, A

Only in case small fluctuations, when approximately can be replaced by a mathematical pendulum is a harmonic oscillator, that is, a system capable of performing harmonic oscillations. In practice, this approximation is valid for angles of the order of 15-20°; in this case, the value differs from by no more than 2%. The oscillations of a pendulum at large amplitudes are not harmonic.

For small oscillations of a mathematical pendulum, Newton's second law is written in the form

This formula expresses natural frequency of small oscillations of a mathematical pendulum .

Hence,

Any body mounted on a horizontal axis of rotation is capable of free oscillations in a gravitational field and, therefore, is also a pendulum. Such a pendulum is usually called physical (Fig. 2.3.2). It differs from the mathematical one only in the distribution of masses. In a stable equilibrium position, the center of mass C the physical pendulum is located below the axis of rotation O on the vertical passing through the axis. When the pendulum is deflected by an angle φ, a moment of gravity arises, tending to return the pendulum to the equilibrium position:

and Newton’s second law for a physical pendulum takes the form (see §1.23)

Here ω 0 - natural frequency of small oscillations of a physical pendulum .

Hence,

Therefore, the equation expressing Newton’s second law for a physical pendulum can be written in the form

Finally, for the circular frequency ω 0 of free oscillations of a physical pendulum, the following expression is obtained:

During free mechanical vibrations, kinetic and potential energies change periodically. At the maximum deviation of a body from its equilibrium position, its speed, and therefore its kinetic energy, vanish. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a spring, potential energy is the energy of elastic deformation of the spring. For a mathematical pendulum, this is the energy in the Earth's gravitational field.

When a body in its motion passes through the equilibrium position, its speed is maximum. The body overshoots the equilibrium position according to the law of inertia. At this moment it has maximum kinetic and minimum potential energy. An increase in kinetic energy occurs due to a decrease in potential energy. With further movement, potential energy begins to increase due to a 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 oscillations remains unchanged.

For spring load(see §2.2):

In real conditions, any oscillatory system is under the influence of friction forces (resistance). In this case, part of the mechanical energy is converted into internal energy of thermal motion of atoms and molecules, and vibrations become fading (Fig. 2.4.2).

The rate at which vibrations decay depends on the magnitude of friction forces. Time interval τ during which the amplitude of oscillations decreases in e≈ 2.7 times, called decay time .

The frequency of free oscillations depends on the rate at which the oscillations decay. As friction forces increase, the natural frequency decreases. However, the change in the natural frequency becomes noticeable only with sufficiently large friction forces, when the natural vibrations quickly decay.

An important characteristic of an oscillatory system performing free damped oscillations is quality factor Q. This parameter is defined as a number N total oscillations performed by the system during the damping time τ, multiplied by π:

Thus, the quality factor characterizes the relative loss of energy in the oscillatory system due to the presence of friction over a time interval equal to one oscillation period.

Oscillations occurring under the influence of an external periodic force are called forced.

An external force does positive work and provides an energy flow to the oscillatory system. It does not allow vibrations to die out, despite the action of friction forces.

A periodic external force can change over time according to various laws. Of particular interest is the case when an external force, varying according to a harmonic law with a frequency ω, acts on an oscillatory system capable of performing its own oscillations at a certain frequency ω 0.

If free oscillations occur at a frequency ω 0, which is determined by the parameters of the system, then steady forced oscillations always occur at frequency ω external force.

After the external force begins to act on the oscillatory system, some time Δ t to establish forced oscillations. The establishment time is, in order of magnitude, equal to the damping time τ of free oscillations in the oscillatory system.

At the initial moment, both processes are excited in the oscillatory system - forced oscillations at frequency ω and free oscillations at natural frequency ω 0. But free vibrations are damped due to the inevitable presence of friction forces. Therefore, after some time, only stationary oscillations at the frequency ω of the external driving force remain in the oscillatory system.

Let us consider, as an example, forced oscillations of a body on a spring (Fig. 2.5.1). An external force is applied to the free end of the spring. It forces the free (left in Fig. 2.5.1) end of the spring to move according to the law

If the left end of the spring is displaced by a distance y, and the right one - to the distance x from their original position, when the spring was undeformed, then the elongation of the spring Δ l equals:

In this equation, the force acting on a body is represented as two terms. The first term on the right side is the elastic force tending to return the body to the equilibrium position ( x= 0). The second term is the external periodic effect on the body. This term is called coercive force.

The equation expressing Newton's second law for a body on a spring in the presence of an external periodic influence can be given a strict mathematical form if we take into account the relationship between the acceleration of the body and its coordinate: Then will be written in the form

Equation (**) does not take into account the action of friction forces. Unlike equations of free vibrations(*) (see §2.2) forced oscillation equation(**) contains two frequencies - the frequency ω 0 of free oscillations and the frequency ω of the driving force.

Steady-state forced oscillations of a load on a spring occur at the frequency of external influence according to the law

x(t) = x mcos(ω t + θ).

Amplitude of forced oscillations x m and the initial phase θ depend on the ratio of frequencies ω 0 and ω and on the amplitude y m external force.

At very low frequencies, when ω<< ω 0 , движение тела массой m, attached to the right end of the spring, repeats the movement of the left end of the spring. Wherein x(t) = y(t), and the spring remains practically undeformed. An external force applied to the left end of the spring does not do any work, since the modulus of this force at ω<< ω 0 стремится к нулю.

If the frequency ω of the external force approaches the natural frequency ω 0, a sharp increase in the amplitude of forced oscillations occurs. This phenomenon is called resonance . Amplitude dependence x m forced oscillations from the frequency ω of the driving force is called resonant characteristic or resonance curve(Fig. 2.5.2).

At resonance, the amplitude x m oscillations of the load can be many times greater than the amplitude y m vibrations of the free (left) end of the spring caused by external influence. In the absence of friction, the amplitude of forced oscillations during resonance should increase without limit. In real conditions, the amplitude of steady-state forced oscillations is determined by the condition: the work of an external force during the oscillation period must be equal to the loss of mechanical energy during the same time due to friction. The less friction (i.e. the higher the quality factor Q oscillatory system), the greater the amplitude of forced oscillations at resonance.

In oscillatory systems with not very high quality factor (< 10) резонансная частота несколько смещается в сторону низких частот. Это хорошо заметно на рис. 2.5.2.

The phenomenon of resonance can cause the destruction of bridges, buildings and other structures if the natural frequencies of their oscillations coincide with the frequency of a periodically acting force, which arises, for example, due to the rotation of an unbalanced motor.

Forced vibrations are undamped fluctuations. The inevitable energy losses due to friction are compensated by the supply of energy from an external source of periodically acting force. There are systems in which undamped oscillations arise not due to periodic external influences, but as a result of the ability of such systems to regulate the supply of energy from a constant source. Such systems are called self-oscillating, and the process of undamped oscillations in such systems is self-oscillations . In a self-oscillating system, three characteristic elements can be distinguished - an oscillatory system, an energy source, and a feedback device between the oscillatory system and the source. Any mechanical system capable of performing its own damped oscillations (for example, the pendulum of a wall clock) can be used as an oscillatory system.

The energy source can be the deformation energy of a spring or the potential energy of a load in a gravitational field. A feedback device is a mechanism by which a self-oscillating system regulates the flow of energy from a source. In Fig. 2.5.3 shows a diagram of the interaction of various elements of a self-oscillating system.

An example of a mechanical self-oscillating system is a clock mechanism with anchor progress (Fig. 2.5.4). The running wheel with oblique teeth is rigidly attached to a toothed drum, through which a chain with a weight is thrown. At the upper end of the pendulum is fixed anchor(anchor) with two plates of solid material, bent in a circular arc with the center on the axis of the pendulum. In hand watches, the weight is replaced by a spring, and the pendulum is replaced by a balancer - a handwheel connected to a spiral spring. The balancer performs torsional vibrations around its axis. The oscillatory system in a clock is a pendulum or balancer.

The source of energy is a raised weight or a wound spring. The device used to provide feedback is an anchor, which allows the running wheel to turn one tooth in one half-cycle. Feedback is provided by the interaction of the anchor with the running wheel. With each oscillation of the pendulum, a tooth of the running wheel pushes the anchor fork in the direction of movement of the pendulum, transferring to it a certain portion of energy, which compensates for energy losses due to friction. Thus, the potential energy of the weight (or twisted spring) is gradually, in separate portions, transferred to the pendulum.

Mechanical self-oscillating systems are widespread in life around us and in technology. Self-oscillations occur in steam engines, internal combustion engines, electric bells, strings of bowed musical instruments, air columns in the pipes of wind instruments, vocal cords when talking or singing, etc.

Figure 2.5.4. Clock mechanism with a pendulum.

Educational goals: formation of an idea of the process of scientific knowledge, organization and systematization of knowledge on the topic; developing an idea of the dependence of the oscillation period on body weight and spring stiffness; development of experimental skills, research skills.

Equipment: tape recorder, computers, program or (section “Mechanical vibrations and waves”, “Body vibrations on a spring”), § 31 of the textbook.

During the classes

1. Start of class

Teacher (begins the lesson with a poem by B. Pasternak: “In everything I want to get to the very essence<...>//Make the discovery”). What does the words “I made a discovery” mean to you guys? ( Listens to the answers.) Did I understand you correctly: if a person, through his hard work and perseverance, achieves the truth in something, then this means that he has made a discovery? Today we will also make small, but independent discoveries. So, the topic of our lesson is “Body vibrations on a spring.”

2. Repetition and generalization

Teacher. First, let's admire together our deep knowledge on the topic of Mechanical Vibrations. Write down the missing left-hand sides of the formulas in cards ( one student performs a task at the board):

(The class checks its notes, everyone gives themselves points on the self-control sheet according to the number of formulas they wrote correctly and the number of formulas found with errors.)

Now let’s pull out something valuable from the caches of memory. Here is a table with physical quantities, their units, and numbers. I will ask a question, and you will cross out the box with the correct answer:

Time interval during which one complete oscillation occurs Maximum deviation of the oscillating quantity from the equilibrium position Number of oscillations per unit time Unit of oscillation period Unit of oscillation frequency Unit of oscillation amplitude During what time the pendulum completed n= 20 oscillations if the oscillation period is 0.5 s? What is the frequency of these oscillations? The body oscillates along an axis X. Its coordinate changes with time according to the law x= 0.2cos0.63 t(SI). What is the amplitude of the body's vibrations? What is the cyclic frequency of these oscillations? A very soft large spring contracts in 2 s from its maximum stretch to its original state. What is the period of oscillation of the spring? If the length of the spring changes by 0.5 m, what is the distance traveled by the loose end of the spring during the period of oscillation?

(Correct answers “draw” the number “5” on the card. The guys put a mark on the self-control sheet - 1 point for the correct answer.)

The basis of any branch of physics is observation or experiment. Today I invite you to conduct research on mechanical vibrations. Break into four groups as desired. Each group takes a card with a task and completes it, and then tells what they did and what they received.

Task No. 1. Make a seconds pendulum (oscillation period 1 s). Devices and materials: thread, weight, ruler, stopwatch.

Task No. 2. Determine the period of oscillation of a meter-long string pendulum. What will it be equal to if the length of the thread is reduced by four times? Devices and materials: meter pendulum, stopwatch.

Task No. 3. Determine the period, frequency and cyclic frequency of the pendulum's oscillations. Write down the equation of oscillation of this pendulum. Devices and materials: ball, ruler, stopwatch, thread.

Task No. 4. Determine in practice the acceleration of gravity for a given area using a string pendulum. Devices and materials: thread, ball, ruler, stopwatch.

(The teacher evaluates the work of the groups. The guys put points on a self-control sheet: 1 point for conducting an experiment, 1 point for defending.)

3. Learning new material

Teacher. Now let’s move on to the topic of our lesson, “Body oscillations on a spring.” Let's try to establish the dependence of the period of free oscillations on the mass of the load, the stiffness of the spring and the amplitude of oscillations. ( The guys are divided into pairs at will, receive cards, during a computer experiment they establish these dependencies and write down the results and conclusions on the cards. .)

Establish the dependence of the period of free oscillations on the mass and stiffness of the spring

Fill the table

Draw a conclusion: if you increase the spring stiffness, then the period: decreases.

A, cm	5	7	10
T, With	1,4	1,4	1,4

Draw a conclusion: if you increase the amplitude of oscillations, then the period: does not change.

Write down the formula for the period of free oscillations

Use § 38 of the textbook V.A. Kasyanova"Physics-10":

Draw a conclusion: the period of free oscillation of a spring pendulum does not depend on amplitude of oscillations, and is completely determined by rigidity, mass (the own characteristics of the oscillatory system).

Check experimentally the dependence of the period of free oscillations on mass and stiffness.

I would like to guide you in your work with the words of A. Tolstoy: “Knowledge is only knowledge when it is acquired through the efforts of one’s thoughts, and not memory.” Good luck with your research!

(The guys establish dependencies, put 1 point for each formula on the self-control sheet.)

4. Consolidation, training, skill development

Teacher. Now let’s solve the problems on cards and check the answer using a computer experiment. The solution to the first problem is worth a maximum of 1 point, the second – 2 points.

Task 1. Determine the period of oscillation of a spring pendulum if the mass of the load is 0.5 kg and the spring stiffness is 10 N/m.

Task 2. Write the equation of motion of a spring pendulum x(t), If m= 1 kg, k= 10 N/m, A= 10 cm. Determine the coordinate at the moment of time t= 4 s.

Check the answer according to the graph, to do this, select the parameters, click Start and follow the readings t.

Creative task. Come up with, formulate and solve a problem, conduct a computer experiment and check your answer. Enter the teacher's assessment (up to 2 points) on the self-control sheet.

5. Reflection. Summarizing

Teacher. Let's summarize. What was the main thing? What was interesting? What new did you learn today? What have you learned? ( Listens to opinions. The guys count the points and give themselves marks: 24–25 points – “3”, 26–27 points – “4”, 28–29 points – “5”.)

DZ.§ 38, tasks 1, 2. Make up your own tasks for future students. Be sure to sign your works, the authorship will be preserved. And I want to end today’s lesson with the words of M. Faraday: “The art of the experimenter is to be able to ask nature questions and understand its answers.” And I think you succeeded today. The lesson is over. Thank you for the lesson. I wish you success. See you in the next lesson.

Literature

Physics in pictures 6.2. NC PHYSIKON, 1993. 1 electron. wholesale disk (DVD-ROM); [Electronic resource] URL: http://torrents.ru/forum/.
Open Physics 2.6: Part 1: LLC FISIKON, 1996–2005 [Electronic resource] URL: http://physics.ru
Kasyanov V.A. Physics: textbook. for general education institutions. 10 grades M.: Bustard, 2003. pp. 123–133.

Yana Vladimirovna Bocharnikova in 1990, she graduated from Far Eastern State University with a degree in physics, physics teacher, worked at the Khabarovsk Institute of Railway Transport Engineers, then taught computer science at a preschool educational institution for children 3–7 years old, taught physics at school and for 9 years now - at college. Winner of the city competition “Teacher of the Year-99” and the competition “Teacher of the Year-2005” in college, laureate of the regional competition “Teacher of the Year-2005”. In his work, he is guided by the words of S. Soloveichik: “To raise people with a deep sense of self-worth, full of self-respect and respect for others, people who are able to choose, to act independently - doesn’t this mean contributing to the strengthening and prosperity of the country?”

Student entries are highlighted here in gray font. – Ed.

Definition

Oscillation frequency($\nu$) is one of the parameters that characterize oscillations. This is the reciprocal of the oscillation period ($T$):

\[\nu =\frac(1)(T)\left(1\right).\]

Thus, the oscillation frequency is a physical quantity equal to the number of repetitions of oscillations per unit time.

\[\nu =\frac(N)(\Delta t)\left(2\right),\]

where $N$ is the number of complete oscillatory movements; $\Delta t$ is the time during which these oscillations 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 the hertz or reciprocal second:

\[\left[\nu \right]=с^(-1)=Hz.\]

Definition

Spring pendulum called a system that consists of an elastic spring to which a load is attached.

Let us assume that the mass of the load is $m$ and the elasticity coefficient 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 influence of elastic force if the system is taken out of equilibrium and left to its own devices. In this case, it is often believed that friction forces can be ignored.

A spring pendulum that oscillates freely is an example of a harmonic oscillator. Let him oscillate along the X axis. If the oscillations are small, Hooke’s law is satisfied, then we write the equation of motion 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 to equation (4) is a sine or cosine function of the form:

where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic frequency of oscillations of the spring pendulum, $A$ is the amplitude of oscillations; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ are the initial phases of oscillations.

From formula (3) and $(\omega )_0=\sqrt(\frac(k)(m))$, it follows that the oscillation frequency of the spring pendulum is equal to:

\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(6\right).\]

Formula (6) is valid if:

the spring in the pendulum is considered weightless;
the load attached to the spring is an absolutely rigid body;
there are no torsional vibrations.

Expression (6) shows that the oscillation frequency of the spring pendulum increases with decreasing mass of the load and increasing the elasticity coefficient 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 a dependence of the oscillation frequency on the amplitude appears.

Measuring the period of oscillation

Measuring time

reducing the amplitude by 2 times

Example 1

Exercise. The period of oscillation of a spring pendulum is $T=5\cdot (10)^(-3)s$. What is the oscillation frequency in this case? What is the cyclic frequency of vibration of this mass?

Solution. The oscillation frequency is the reciprocal of the oscillation period, therefore, to solve the problem it is enough to use the formula:

\[\nu =\frac(1)(T)\left(1.1\right).\]

Let's calculate the required frequency:

\[\nu =\frac(1)(5\cdot (10)^(-3))=200\ \left(Hz\right).\]

The cyclic frequency is related to the frequency $\nu $ as:

\[(\omega )_0=2\pi \nu \ \left(1.2\right).\]

Let's calculate the cyclic frequency:

\[(\omega )_0=2\pi \cdot 200\approx 1256\ \left(\frac(rad)(s)\right).\]

Answer.$1)\ \nu =200$ Hz. 2) $(\omega )_0=1256\ \frac(rad)(s)$

Example 2

Exercise. The mass of the load hanging on an 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:

\[(\nu )_2=\frac(1)(2\pi )\sqrt(\frac(k)(m+\Delta m))\ \left(2.2\right).\]

According to the conditions of the problem $(\nu )_2=\frac((\nu )_1)(n)$, we find the relation $\frac((\nu )_1)((\nu )_2):\frac((\nu )_1)((\nu )_2)=\sqrt(\frac(k)(m)\cdot \frac(m+\Delta m)(k))=\sqrt(1+\frac(\Delta m)( m))=n\ \left(2.3\right).$

Let us obtain from equation (2.3) the required mass of the load. To do this, let’s square both sides of expression (2.3) and express $m$:

Answer.$m=\frac(\Delta m)(n^2-1)$

Goal of the work. Familiarize yourself with the main characteristics of undamped and damped free mechanical vibrations.

Task. Determine the period of natural oscillations of a spring pendulum; check the linearity of the dependence of the square of the period on the mass; determine the spring stiffness; determine the period of damped oscillations and the logarithmic damping decrement of a spring pendulum.

Devices and accessories. A tripod with a scale, a spring, a set of weights of various weights, a vessel with water, a stopwatch.

1. Free oscillations of a spring pendulum. General information

Oscillations are processes in which one or more physical quantities that describe these processes periodically change. Oscillations can be described by various periodic functions of time. The simplest oscillations are harmonic oscillations - such oscillations in which the oscillating quantity (for example, the displacement of a load on a spring) changes over time according to the law of cosine or sine. Oscillations that occur after the action of an external short-term force on the system are called free.

If the load is removed from the equilibrium position by deflecting by an amount x, then the elastic force increases: F control = – kx 2= – k(x 1 + x). Having reached the equilibrium position, the load will have a speed different from zero and will pass the equilibrium position by inertia. As the movement continues, the deviation from the equilibrium position will increase, which will lead to an increase in the elastic force, and the process will repeat in the opposite direction. Thus, the oscillatory motion of the system is due to two reasons: 1) the desire of the body to return to the equilibrium position and 2) inertia, which does not allow the body to instantly stop in the equilibrium position. In the absence of friction forces, the oscillations would continue indefinitely. The presence of friction forces leads to the fact that part of the oscillation energy turns into internal energy and the oscillations gradually die out. Such oscillations are called damped.

Undamped free oscillations

First, let us consider the oscillations of a spring pendulum, which is not affected by friction forces - undamped free oscillations. According to Newton’s second law, taking into account the signs of projections onto the X axis

From the equilibrium condition, the displacement caused by gravity: . Substituting into equation (1), we obtain: Differential" href="/text/category/differentcial/" rel="bookmark">differential equation

https://pandia.ru/text/77/494/images/image008_28.gif" width="152" height="25 src=">. (3)

This equation is called harmonic equation. The greatest deviation of the load from the equilibrium position A 0 called the amplitude of oscillations. The quantity in the cosine argument is called oscillation phase. The constant φ0 represents the phase value at the initial time ( t= 0) and is called initial phase of oscillations. Magnitude

is it circular or cyclic? natural frequency related to period of oscillation T ratio https://pandia.ru/text/77/494/images/image012_17.gif" width="125" height="55">. (5)

Damped oscillations

Let us consider free oscillations of a spring pendulum in the presence of friction force (damped oscillations). In the simplest and at the same time most common case, the friction force is proportional to the speed υ movements:

Ftr = – rυ, (6)

Where r– a constant called the resistance coefficient. The minus sign shows that the friction force and speed are in opposite directions. Equation of Newton's second law in projection onto the X axis in the presence of elastic force and friction force

ma = – kx – rυ. (7)

This differential equation taking into account υ = dx/ dt can be written down

https://pandia.ru/text/77/494/images/image014_12.gif" width="59" height="48 src="> – attenuation coefficient; – cyclic frequency of free undamped oscillations of a given oscillatory system, i.e. in the absence of energy losses (β = 0). Equation (8) is called differential equation of damped oscillations.

To get the displacement dependence x from time t, it is necessary to solve the differential equation (8)..gif" width="172" height="27">, (9)

Where A 0 and φ0 – initial amplitude and initial phase of oscillations;
– cyclic frequency of damped oscillations at ω >> https://pandia.ru/text/77/494/images/image019_12.gif" width="96" height="27 src=">. (10)

On the graph of function (9), Fig. 2, the dotted lines show the change in amplitude (10) of damped oscillations.

Rice. 2. Displacement dependence X load from time to time t in the presence of friction force

To quantitatively characterize the degree of attenuation of oscillations, a value is introduced equal to the ratio of amplitudes that differ by a period, and is called damping decrement:

. (11)

The natural logarithm of this quantity is often used. This parameter is called logarithmic damping decrement:

The amplitude decreases in n times, then from equation (10) it follows that

From here we get the expression for the logarithmic decrement

If during the time t" amplitude decreases in e once ( e= 2.71 – the base of the natural logarithm), then the system will have time to complete the number of oscillations

Rice. 3. Installation diagram

The installation consists of a tripod 1 with measuring scale 2 . To a tripod with a spring 3 loads are suspended 4 of various masses. When studying damped oscillations in task 2, a ring is used to enhance the damping 5 , which is placed in a transparent container 6 with water.

In task 1 (performed without a vessel with water and a ring), to a first approximation, the damping of oscillations can be neglected and considered harmonic. As follows from formula (5) for harmonic oscillations, the dependence T 2 = f (m) – linear, from which the spring stiffness coefficient can be determined k according to the formula

where is the slope of the straight line T 2 from m.

Exercise 1. Determination of the dependence of the period of natural oscillations of a spring pendulum on the mass of the load.

1. Determine the period of oscillation of a spring pendulum at different values of the mass of the load m. To do this, use a stopwatch for each value m measure time three times t full n fluctuations ( n≥10) and according to the average time value https://pandia.ru/text/77/494/images/image030_6.gif" width="57 height=28" height="28">. Enter the results in Table 1.

2. Based on the measurement results, construct a graph of the square of the period T2 by weight m. From the slope of the graph, determine the spring stiffness k according to formula (16).

Table 1

Measurement results to determine the period of natural oscillations

3. Additional task. Estimate random, total and relative ε t time measurement errors for mass value m = 400 g.

Task 2. Determination of the logarithmic damping decrement of a spring pendulum.

1. Hang a mass on a spring m= 400 g with ring and place in a vessel with water so that the ring is completely submerged in water. Determine the period of damped oscillations for a given value m according to the method outlined in paragraph 1 of task 1. Repeat the measurements three times and enter the results on the left side of the table. 2.

2. Remove the pendulum from the equilibrium position and, noting its initial amplitude on a ruler, measure the time t" , during which the amplitude of oscillations decreases by 2 times. Take measurements three times. Enter the results on the right side of the table. 2.

table 2

Measurement results

to determine the logarithmic damping decrement

4. Test questions and assignments

1. What oscillations are called harmonic? Define their main characteristics.

2. What oscillations are called damped? Define their main characteristics.

3. Explain the physical meaning of the logarithmic damping decrement and damping coefficient.

4. Derive the time dependence of the speed and acceleration of a load on a spring performing harmonic oscillations. Provide graphs and analyze.

5. Derive the time dependence of kinetic, potential and total energy for a load oscillating on a spring. Provide graphs and analyze.

6. Obtain the differential equation of free vibrations and its solution.

7. Construct graphs of harmonic oscillations with initial phases π/2 and π/3.

8. Within what limits can the logarithmic damping decrement vary?

9. Give the differential equation of damped oscillations of a spring pendulum and its solution.

10. According to what law does the amplitude of damped oscillations change? Are damped oscillations periodic?

11. What motion is called aperiodic? Under what conditions is it observed?

12. What is the natural frequency of oscillations? How does it depend on the mass of the oscillating body for a spring pendulum?

13. Why is the frequency of damped oscillations less than the frequency of natural oscillations of the system?

14. A copper ball suspended from a spring performs vertical oscillations. How will the period of oscillation change if instead of a copper ball, an aluminum ball of the same radius is suspended from a spring?

15. At what value of the logarithmic damping decrement do the oscillations decay faster: at θ1 = 0.25 or θ2 = 0.5? Provide graphs of these damped oscillations.

Bibliography

1. Trofimova T. I. Physics course / . – 11th ed. – M.: Academy, 2006. – 560 p.

2. Savelyev I. V. General physics course: 3 volumes / . – St. Petersburg. : Lan, 2008. – T. 1. – 432 p.

3. Akhmatov A. S.. Laboratory workshop in physics / .
– M.: Higher. school, 1980. – 359 p.

Subject. Oscillations of a load on a spring. Mathematical
pendulum

Purpose of the lesson: to familiarize students with the laws of vibrations
spring and mathematical pendulums
Lesson type: learning new material
Lesson Plan
Knowledge check 5 min.1. What are harmonic vibrations?
2. Equation of harmonic vibrations.
3. What is the oscillation phase?
4. Graphs of harmonic vibrations
Demonstrations
5 min.1. Free oscillations of a spring pendulum.
Learning new things
material
25
min.
2. Dependence of the period of oscillation of the load on
spring from the elastic properties of the spring and mass
cargo
3. Free vibrations of the mathematical
pendulum.
4. Dependence of the oscillation period
mathematical pendulum from its length
1. The process of oscillation of a spring pendulum.
2. Period of oscillation of a spring pendulum.

4. Mathematical pendulum.
5. Period of mathematical oscillation
pendulum

Consolidation
studied
material
10
min.
1. We train to solve problems.
2. Test questions

LEARNING NEW MATERIAL
1. The process of oscillation of a spring pendulum
In order to describe vibrations (leaves and ears of air; air in
organ pipes and musical wind pipes
tools); for calculating vibration (vehicle bodies,
mounted on springs; foundations of buildings and machines),
Let's introduce a model of real oscillatory systems - spring
pendulum.

Consider the oscillations of a cart of mass m attached to
vertical wall with a spring of stiffness k.

We will assume that:
1) the friction force that acts on the cart is very small,
so you can ignore it. In this case, fluctuations
spring pendulum will be undamped;
2) deformation of the spring during body oscillations
are insignificant, therefore they can be considered elastic and
apply Hooke's law:

Let us consider the oscillations of a spring pendulum in more detail.
When the cart moves away from its equilibrium position by
distance A on the right, the spring is stretched and
the cart is subject to a maximum elastic force Fnp = kA.
Then the cart begins to move to the left with acceleration, which
changes: the elongation of the spring decreases and the elastic force
(and acceleration) also decrease. After a quarter period
the cart will return to its equilibrium position. At this moment the strength
elasticity and acceleration are zero, and the speed reaches
maximum value.
By inertia, the cart will continue to move, and a force will arise
elasticity increases. She will start to slow down
block and at a distance A from the equilibrium position the cart is on
the moment will stop. From the moment the vibrations began
half period.
For the next half of the period, the cart's movement will be exactly
like this, only in the opposite direction.
It is necessary to draw students' attention to the fact that, according to
Hooke's law, elastic force is directed against elongation
springs: the elastic force "pushed" the cart to position
balance.
Consequently, free oscillations of a spring pendulum
due to the following reasons:
1) the action of an elastic force on the body, always directed in
side of the equilibrium position;
2) the inertia of the oscillating body, due to which it does not
stops in the equilibrium position and continues
move in the same direction.
2. Period of oscillation of a spring pendulum
The first characteristic sign of oscillations of a spring pendulum
can be installed by gradually increasing the mass of suspended
to the weight springs. Hanging different weights from the spring
mass, we notice that with increasing mass there is a difficult period
load vibrations increase. For example, due to
heavy weight increase 4 times oscillation period
doubles:

The second characteristic sign can be established by changing
springs. After carrying out a series of measurements, it is easy to discover that
the load oscillates faster on a stiff spring and slower -
on soft, that is:
The third feature of a spring pendulum is that
that the period of its oscillations does not depend on the acceleration of free
falls. This is easy to verify using the method
“increasing gravity” due to a strong magnet,
which is placed under a load that oscillates.
Thus,
the period of oscillation of a spring pendulum does not depend on

Knowing the oscillation period, it is easy to calculate the frequency and
cyclic oscillation frequency:
3. Equation of harmonic vibrations
Let's consider the vibrations of the cart from the point of view of dynamics. On
three forces act on the stroller during movement: the reaction force
supports
, gravity m and elasticity force etc. Let us write
equation of Newton's second law in vector form:
Let's project this equation onto the horizontal and
vertical axis:
According to Hooke's law:

Thus we have:
This equation is called the equation of free vibrations
spring pendulum.
Let us denote: ω2 = k/m. Then the equation of motion of the load will be
have the form: ax = -ω2x. Equations of this type are called
differential equations.
The solution to this
equation is the function x = Acosωt.
4. Mathematical pendulum
To calculate the period of oscillation of a weight hanging on a thread,
it is necessary to “idealize” the problem a little. Firstly,
we will assume that the dimensions of the load are much smaller than the length of the thread,
and the thread is inextensible and weightless. Secondly, we will consider
The angle of deflection of the pendulum is quite small (no more than 10-15°).

dot.
Let's consider the oscillations of a mathematical pendulum. For this
take a small, but quite heavy, ball and
Let's hang it on a long, non-stretchable thread.
Considering the oscillations of a mathematical pendulum, we
we come to the conclusion that the reasons that determine
free vibrations, the same as in the case of a spring
pendulum (see Fig. a-e):

1) the action of forces on the ball, the resultant of which is always
directed towards the equilibrium position;
2) the inertia of the oscillating ball, due to which it
does not stop in the equilibrium position.
5. Period of oscillation of a mathematical pendulum
Let's prove
harmonic vibrations.
Let's write the equation of Newton's second law in projection onto the axis
OX (see figure):

What does a mathematical pendulum do?

Tx + mgx = max.
Since Tx = 0, then mgx = -mgsin and we get the equation:
-mgsin = max, or -gsin = ax.
The value of sin can be calculated from the triangle OAS - it
equal to the ratio of leg OA to hypotenuse OS. If the angles
small, OS ≈ l, where l is the length of the thread, and OA ≈ x, where x is the deviation
ball from its equilibrium position. Therefore sin = x/l.
Finally we get:

Denoting ω2 = g/l, we have equations for free oscillations
mathematical pendulum:
Cyclic frequency of oscillation of a mathematical pendulum:
Using the relation T = 2 /ω, we find the formula
for the period of oscillation of a mathematical pendulum:

pendulum.
It is known that in different parts of the globe the acceleration
free fall miscellaneous. It depends not only on the form
Earth, but also from the presence in its depths of heavy (metals) or
light (gas, oil) substances. And therefore the period
The pendulum will oscillate differently at different points. This
the property is used, in particular, during the search for deposits
mineral.

Question for students while presenting new material
1. How will the period of oscillation of a spring pendulum change?
due to changes in cargo mass? spring stiffness?
2. How will the period of oscillation of a spring pendulum change if
place a magnet underneath it?

increase the amplitude of oscillations.
4. Under what conditions does a mathematical pendulum oscillate?
can be considered harmonic?

5. Why does the ball oscillate on a long string?
stops at the moment of passing the position
balance?
6. How will the period of oscillation of a mathematical pendulum change,
what if the mass of the load is increased? decrease?

CONSTRUCTION OF LEARNED MATERIAL
1). We train to solve problems
1. A load suspended on a spring, being in equilibrium,
stretches the spring by 10 cm. Is this data sufficient?
to calculate the period of oscillation of a load on a spring?
2. When a load was suspended from the spring, it stretched by 20 cm.
The weight was pulled down and released. What is the period T of oscillations?
what arose?
3. A steel ball suspended from a spring makes
vertical vibrations. How will the period of oscillation change?
What if you hang a copper ball of the same radius from a spring?
4. Calculate the stiffness of the spring if suspended from it
a mass of 700 g undergoes 18 oscillations in 21 s.
5. What is the ratio of the lengths of two mathematical pendulums,
if one of them carries out 31 oscillations, and the second for exactly
such a period of time - 20 oscillations?
2). Control questions
1. Name the reasons for the oscillations of a spring pendulum.
2. You can use a spring pendulum to calculate
acceleration of free fall?
3. How will the period of oscillation of a spring pendulum change if
increase the mass of the load by 4 times and at the same time increase by 4
times the spring stiffness?
4. Name the main properties of a mathematical pendulum. Where
are they used?
5. What do spring and mathematical pendulums have in common?

What did we learn in class?
A spring pendulum is an oscillatory system
which is a body attached to a spring.
The period of oscillation of a spring pendulum does not depend on
acceleration of free fall and the less, the less
load mass and stiffer spring:
Frequency and cyclic frequency of oscillations of the spring
pendulum:
Equation of free oscillations of a spring pendulum:
A mathematical pendulum is an idealized
frictionless oscillatory system consisting of weightless and
inextensible thread on which the material is suspended
dot.
The period of free oscillations of a mathematical pendulum is not
depends on its mass, and is determined only by the length of the thread and
acceleration of gravity in the place where it is located
pendulum:
Equation of free oscillations of a mathematical pendulum:

Homework