How to find the oscillation period of a wave. Basic formulas in physics - vibrations and waves

As you study this section, please keep in mind that fluctuations of different physical nature are described from common mathematical positions. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.

It must be borne in mind that in any real oscillatory system there is resistance of the medium, i.e. the oscillations will be damped. To characterize the damping of oscillations, a damping coefficient and a logarithmic damping decrement are introduced.

If oscillations occur under the influence of an external, periodically changing force, then such oscillations are called forced. They will be undamped. The amplitude of forced oscillations depends on the frequency of the driving force. As the frequency of forced oscillations approaches the frequency of natural oscillations, the amplitude of forced oscillations increases sharply. This phenomenon is called resonance.

When moving on to the study of electromagnetic waves, you need to clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system emitting electromagnetic waves is an electric dipole. If a dipole undergoes harmonic oscillations, then it emits a monochromatic wave.

Formula table: oscillations and waves

Oscillation characteristics

Phase determines the state of the system, namely coordinate, speed, acceleration, energy, etc.

Cyclic frequency characterizes the rate of change in the phase of oscillations.

The initial state of the oscillatory system is characterized by initial phase

Oscillation amplitude A- this is the largest displacement from the equilibrium position

Period T- this is the period of time during which the point performs one complete oscillation.

Oscillation frequency is the number of complete oscillations per unit time t.

Frequency, cyclic frequency and period of oscillation are related as

Types of vibrations

Oscillations that occur in closed systems are called free or own fluctuations. Oscillations that occur under the influence of external forces are called forced. There are also self-oscillations(forced automatically).

If we consider oscillations according to changing characteristics (amplitude, frequency, period, etc.), then they can be divided into harmonic, fading, growing(as well as sawtooth, rectangular, complex).

During free oscillations in real systems, energy losses always occur. Mechanical energy is spent, for example, on performing work to overcome air resistance forces. Under the influence of friction, the amplitude of oscillations decreases, and after some time the oscillations stop. Obviously, the greater the force of resistance to movement, the faster the oscillations stop.

Forced vibrations. Resonance

Forced oscillations are undamped. Therefore, it is necessary to replenish energy losses for each oscillation period. To do this, it is necessary to influence the oscillating body with a periodically changing force. Forced vibrations occur with a frequency equal to the frequency of changes in the external force.

Forced vibrations

The amplitude of forced mechanical vibrations reaches its greatest value if the frequency of the driving force coincides with the frequency of the oscillatory system. This phenomenon is called resonance.

For example, if we periodically pull the cord in time with its own vibrations, we will notice an increase in the amplitude of its vibrations.

If you move a wet finger along the edge of a glass, the glass will make ringing sounds. Although it is not noticeable, the finger moves intermittently and transfers energy to the glass in short bursts, causing the glass to vibrate

The walls of the glass also begin to vibrate if a sound wave with a frequency equal to its own is directed at it. If the amplitude becomes very large, the glass may even break. Due to resonance, when F.I. Chaliapin sang, the crystal pendants of the chandeliers trembled (resonated). The occurrence of resonance can also be observed in the bathroom. If you softly sing sounds of different frequencies, a resonance will arise at one of the frequencies.

In musical instruments, the role of resonators is performed by parts of their bodies. A person also has his own resonator - this is the oral cavity, which amplifies the sounds produced.

The phenomenon of resonance must be taken into account in practice. In some cases it can be useful, in others it can be harmful. Resonance phenomena can cause irreversible damage in various mechanical systems, such as poorly designed bridges. Thus, in 1905, the Egyptian Bridge in St. Petersburg collapsed while a horse squadron was passing across it, and in 1940, the Tacoma Bridge in the USA collapsed.

The phenomenon of resonance is used when, with the help of a small force, it is necessary to obtain a large increase in the amplitude of vibrations. For example, the heavy tongue of a large bell can be swung by applying a relatively small force with a frequency equal to the natural frequency of the bell.

VIBRATION FREQUENCY, number of oscillations in 1 s. Denoted by . If T is the period of oscillation, then= 1/T; measured in hertz (Hz). Angular frequency  = 2 = 2/T rad/s.

PERIOD OF oscillation, the shortest period of time after which the oscillating system returns to the same state in which it was at the initial moment, chosen arbitrarily. Period is the reciprocal of the oscillation frequency. The concept of “period” is applicable, for example, in the case of harmonic oscillations, but is often used for weakly damped oscillations.

Circular or cyclic frequencyω

When the argument of the cosine or sine changes by 2π, these functions return to their previous value. Let us find the time period T during which the phase of the harmonic function changes by 2π.

ω(t + T) + α = ωt + α + 2π, or ωT = 2π.

The time T for one complete oscillation is called the oscillation period. Frequency ν is the reciprocal of the period

The unit of frequency is hertz (Hz), 1 Hz = 1 s -1.

The circular or cyclic frequency ω is 2π times greater than the oscillation frequency ν. Circular frequency is the rate of change of phase over time. Really:

.

AMPLITUDE (from the Latin amplitudo - value), the greatest deviation from the equilibrium value of a quantity that fluctuates according to a certain, including harmonic, law; see alsoHarmonic oscillations.

PHASE OF OSCILLATIONS argument of the function cos (ωt + φ), describing the harmonic oscillatory process (ω - circular frequency, t - time, φ - initial phase of oscillations, i.e. phase of oscillations at the initial moment of time t = 0)

Displacement, speed, acceleration of an oscillating system of particles.

Energy of harmonic vibrations.

Harmonic vibrations

An important special case of periodic oscillations are harmonic oscillations, i.e. such changes in a physical quantity that follow the law

Where . From a mathematics course we know that a function of type (1) varies from A to -A, and that it has the smallest positive period. Therefore, a harmonic oscillation of type (1) occurs with amplitude A and period.

Do not confuse cyclic frequency with oscillation frequency. There is a simple connection between them. Since, ah, then.

The quantity is called the phase of oscillation. At t=0 the phase is equal, therefore it is called the initial phase.

Note that for the same t:

where is the initial phase. It can be seen that the initial phase for the same oscillation is a value determined with an accuracy of up to. Therefore, from the set of possible values ​​of the initial phase, the initial phase value with the smallest absolute value or the smallest positive value is usually selected. But you don't have to do this. For example, given an oscillation , then it is convenient to write it in the form and work in the future with the last type of recording of this vibration.

It can be shown that vibrations of the form:

where and can be of any sign, with the help of simple trigonometric transformations it is always reduced to the form (1), and, and is not equal, generally speaking. Thus, oscillations of type (2) are harmonic with amplitude and cyclic frequency. Without giving a general proof, we will illustrate this with a specific example.

Let it be required to show that the oscillation

will be harmonic and find the amplitude, cyclic frequency, period and initial phase. Really,

-

We see that the fluctuation of the value of S was written down in the form (1). Wherein ,.

Try to see for yourself that

.

Naturally, the recording of harmonic oscillations in form (2) is no worse than the recording in form (1), and in a specific task there is usually no need to switch from recording in this form to recording in another form. You just need to be able to immediately find the amplitude, cyclic frequency and period, having in front of you any form of recording of a harmonic vibration.

Sometimes it is useful to know the nature of the change in the first and second time derivatives of the quantity S, which performs harmonic oscillations (oscillates according to the harmonic law). If , then differentiating S with respect to time t gives ,. It can be seen that S" and S"" also oscillate according to a harmonic law with the same cyclic frequency as the value of S and amplitudes, respectively. Let's give an example.

Let the x coordinate of a body performing harmonic oscillations along the x axis change according to the law, where x is in centimeters, time t is in seconds. It is required to write down the law of changes in the speed and acceleration of a body and find their maximum values. To answer the question posed, we note that the first time derivative of the quantity x is the projection of the body’s velocity onto the x-axis, and the second derivative of x is the projection of the acceleration onto the x-axis:,. Differentiating the expression for x with respect to time, we obtain ,. Maximum speed and acceleration values: .

Harmonic oscillations are oscillations performed according to the laws of sine and cosine. The following figure shows a graph of changes in the coordinates of a point over time according to the cosine law.

picture

Oscillation amplitude

The amplitude of a harmonic vibration is the greatest value of the displacement of a body from its equilibrium position. The amplitude can take on different values. It will depend on how much we displace the body at the initial moment of time from the equilibrium position.

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

x = Xm*cos(ω0*t).

Oscillation period

The period of oscillation is the time it takes to complete one complete oscillation. The period of oscillation is designated by the letter T. The units of measurement of the period correspond to the units of time. That is, in SI these are seconds.

Oscillation frequency is the number of oscillations performed per unit of time. The oscillation frequency is designated by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.

ν = 1/T.

Frequency units are in SI 1/sec. This unit of measurement is called Hertz. The number of oscillations in a time of 2*pi seconds will be equal to:

ω0 = 2*pi* ν = 2*pi/T.

Oscillation frequency

This quantity is called the cyclic frequency of oscillations. In some literature the name circular frequency appears. The natural frequency of an oscillatory system is the frequency of free oscillations.

The frequency of natural oscillations is calculated using the formula:

The frequency of natural vibrations depends on the properties of the material and the mass of the load. The greater the spring stiffness, the greater the frequency of its own vibrations. The greater the mass of the load, the lower the frequency of natural oscillations.

These two conclusions are obvious. The stiffer the spring, the greater the acceleration it will impart to the body when the system is thrown out of balance. The greater the mass of a body, the slower the speed of this body will change.

Free oscillation period:

T = 2*pi/ ω0 = 2*pi*√(m/k)

It is noteworthy that at small angles of deflection the period of oscillation of the body on the spring and the period of oscillation of the pendulum will not depend on the amplitude of the oscillations.

Let's write down the formulas for the period and frequency of free oscillations for a mathematical pendulum.

then the period will be equal

T = 2*pi*√(l/g).

This formula will be valid only for small deflection angles. From the formula we see that the period of oscillation increases with increasing length of the pendulum thread. The longer the length, the slower the body will vibrate.

The period of oscillation does not depend at all on the mass of the load. But it depends on the acceleration of free fall. As g decreases, the oscillation period will increase. This property is widely used in practice. For example, to measure the exact value of free acceleration.

1. Let us remember what is called the frequency and period of oscillations.

The time it takes a pendulum to complete one swing is called the period of oscillation.

The period is designated by the letter T and measured in seconds(With).

The number of complete oscillations in one second is called the oscillation frequency. Frequency is indicated by the letter n .

1 Hz = .

Unit of vibration frequency in Ш - hertz (1 Hz).

1 Hz - this is the frequency of such oscillations at which one complete oscillation occurs in 1 s.

The oscillation frequency and period are related by the relation:

n = .

2. The period of oscillation of the oscillatory systems we considered - mathematical and spring pendulums - depends on the characteristics of these systems.

Let's find out what the period of oscillation of a mathematical pendulum depends on. To do this, let's do an experiment. We will change the length of the thread of a mathematical pendulum and measure the time of several complete oscillations, for example 10. In each case, we will determine the period of oscillation of the pendulum by dividing the measured time by 10. Experience shows that the longer the length of the thread, the longer the period of oscillation.

Now let's place a magnet under the pendulum, thereby increasing the force of gravity acting on the pendulum, and measure the period of its oscillations. Note that the period of oscillation will decrease. Consequently, the period of oscillation of a mathematical pendulum depends on the acceleration of gravity: the greater it is, the shorter the period of oscillation.

The formula for the period of oscillation of a mathematical pendulum is:

Where l- length of the pendulum thread, g- acceleration of gravity.

3. Let us determine experimentally what determines the period of oscillation of a spring pendulum.

We will suspend weights of different masses from the same spring and measure the period of oscillation. Note that the greater the mass of the load, the longer the period of oscillation.

Then we will suspend the same load from springs of different stiffnesses. Experience shows that the greater the spring stiffness, the shorter the period of oscillation of the pendulum.

The formula for the period of oscillation of a spring pendulum is:

Where m- mass of cargo, k- spring stiffness.

4. The formulas for the period of oscillation of pendulums include quantities that characterize the pendulums themselves. These quantities are called parameters oscillatory systems.

If the parameters of the oscillatory system do not change during the oscillation process, then the period (frequency) of oscillation remains unchanged. However, in real oscillatory systems, friction forces act, so the period of real free oscillations decreases over time.

If we assume that there is no friction and the system performs free oscillations, then the period of oscillations will not change.

The free vibrations that a system could perform in the absence of friction are called natural vibrations.

The frequency of such oscillations is called natural frequency. It depends on the parameters of the oscillatory system.

Self-test questions

1. What is the period of oscillation of a pendulum called?

2. What is the frequency of oscillation of a pendulum? What is the unit of vibration frequency?

3. On what quantities and how does the period of oscillation of a mathematical pendulum depend?

4. On what quantities and how does the period of oscillation of a spring pendulum depend?

5. What vibrations are called natural vibrations?

Task 23

1. What is the period of oscillation of a pendulum if it completes 20 complete oscillations in 15 s?

2. What is the oscillation frequency if the oscillation period is 0.25 s?

3. What must be the length of the pendulum in a pendulum clock for its period of oscillation to be equal to 1 s? Count g= 10 m/s 2 ; p2 = 10.

4. What is the period of oscillation of a pendulum whose thread is 28 cm long on the Moon? The acceleration of gravity on the Moon is 1.75 m/s 2 .

5. Determine the period and frequency of oscillation of a spring pendulum if its spring stiffness is 100 N/m and the mass of the load is 1 kg.

6. How many times will the vibration frequency of a car on springs change if a load is placed in it, the mass of which is equal to the mass of the unloaded car?

Laboratory work No. 2

Study of vibrations
mathematical and spring pendulums

Goal of the work:

investigate on what quantities the period of oscillation of a mathematical and spring pendulum depends and on which does not depend.

Devices and materials:

tripod, 3 weights of different weights (ball, weight weighing 100 g, weight), thread 60 cm long, 2 springs of different stiffness, ruler, stopwatch, strip magnet.

Work order

1. Make a mathematical pendulum. Watch his hesitation.

2. Investigate the dependence of the period of oscillation of a mathematical pendulum on the length of the thread. To do this, determine the time of 20 complete oscillations of pendulums of length 25 and 49 cm. Calculate the period of oscillation in each case. Enter the results of measurements and calculations, taking into account the measurement error, into table 10. Draw a conclusion.

Table 10

3. Investigate the dependence of the period of oscillation of a pendulum on the acceleration of gravity. To do this, place a strip magnet under a 25 cm long pendulum. Determine the period of oscillation, compare it with the period of oscillation of a pendulum in the absence of a magnet. Draw a conclusion.

4. Show that the period of oscillation of a mathematical pendulum does not depend on the mass of the load. To do this, hang weights of different weights from a thread of constant length. For each case, determine the period of oscillation, keeping the amplitude the same. Draw a conclusion.

5. Show that the period of oscillation of a mathematical pendulum does not depend on the amplitude of the oscillations. To do this, deflect the pendulum first by 3 cm and then by 4 cm from the equilibrium position and determine the period of oscillation in each case. Enter the results of measurements and calculations in table 11. Draw a conclusion.

Table 11

6. Show that the period of oscillation of a spring pendulum depends on the mass of the load. By attaching weights of different masses to the spring, determine the period of oscillation of the pendulum in each case by measuring the time of 10 oscillations. Draw a conclusion.

7. Show that the period of oscillation of a spring pendulum depends on the spring stiffness. Draw a conclusion.

8. Show that the period of oscillation of a spring pendulum does not depend on the amplitude. Enter the results of measurements and calculations in Table 12. Draw a conclusion.

Table 12

Task 24

1 e.Explore the range of applicability of the mathematical pendulum model. To do this, change the length of the pendulum thread and the dimensions of the body. Check whether the period of oscillation depends on the length of the pendulum if the body is large and the length of the thread is small.

2. Calculate the lengths of second pendulums mounted on a pole ( g= 9.832 m/s 2), at the equator ( g= 9.78 m/s 2), in Moscow ( g= 9.816 m/s 2), in St. Petersburg ( g= 9.819 m/s 2).

3 * . How do temperature changes affect the movement of a pendulum clock?

4. How does the frequency of a pendulum clock change when going uphill?

5 * . A girl swings on a swing. Will the period of oscillation of the swing change if two girls sit on it? What if the girl swings not sitting, but standing?

Laboratory work No. 3*

Measuring gravity acceleration
using a mathematical pendulum

Goal of the work:

learn to measure the acceleration of gravity using the formula for the period of oscillation of a mathematical pendulum.

Devices and materials:

a tripod, a ball with a thread attached to it, a measuring tape, a stopwatch (or a watch with a second hand).

Work order

1. Hang the ball from a tripod on a 30 cm long thread.

2. Measure the time of 10 complete oscillations of the pendulum and calculate its period of oscillation. Enter the results of measurements and calculations in table 13.

3. Using the formula for the period of oscillation of a mathematical pendulum T= 2p, calculate the acceleration of gravity using the formula: g = .

4. Repeat the measurements, changing the length of the pendulum thread.

5. Calculate the relative and absolute error in changing the acceleration of free fall for each case using the formulas:

d g==+ ; D g = g d g.

Consider that the error in measuring length is equal to half the division value of a measuring tape, and the error in measuring time is equal to half the division value of a stopwatch.

6. Write down the value of the acceleration due to gravity in Table 13, taking into account the measurement error.

Table 13

Task 25

1. Will the error in measuring the period of oscillation of a pendulum change, and if so, how, if the number of oscillations is increased from 20 to 30?

2. How does increasing the length of the pendulum affect the accuracy of measuring the acceleration of gravity? Why?