Any periodically repeating movement is called oscillatory. Therefore, the dependences of the coordinates and speed of a body on time during oscillations are described by periodic functions of time. IN school course physicists consider such vibrations in which the dependencies and velocities of the body are trigonometric functions , or a combination thereof, where is a certain number. Such oscillations are called harmonic (functions And often called harmonic functions). To solve vibration problems included in the program of a single state exam in physics, you need to know the definitions of the main characteristics oscillatory motion: amplitude, period, frequency, circular (or cyclic) frequency and phase of oscillations. Let us give these definitions and connect the listed quantities with the parameters of the dependence of the body coordinates on time, which in the case of harmonic oscillations can always be represented in the form

where , and are some numbers.

The amplitude of oscillations is the maximum deviation of an oscillating body from its equilibrium position. Since the maximum and minimum values ​​of the cosine in (11.1) are equal to ±1, the amplitude of oscillations of the body oscillating (11.1) is equal to . The period of oscillation is the minimum time after which the movement of a body is repeated. For dependence (11.1), the period can be set from the following considerations. Cosine - periodic function with period . Therefore, the movement is completely repeated through such a value that . From here we get

The circular (or cyclic) frequency of oscillations is the number of oscillations performed per unit of time. From formula (11.3) we conclude that the circular frequency is the quantity from formula (11.1).

The oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinate on time. From formula (11.1) we see that the phase of oscillations of the body, the movement of which is described by dependence (11.1), is equal to . The value of the oscillation phase at time = 0 is called the initial phase. For dependence (11.1), the initial phase of oscillations is equal to . Obviously, the initial phase of oscillations depends on the choice of the time reference point (moment = 0), which is always conditional. By changing the origin of time, the initial phase of oscillations can always be “made” equal to zero, and the sine in formula (11.1) can be “turned” into a cosine or vice versa.

The program of the unified state exam also includes knowledge of formulas for the frequency of oscillations of spring and mathematical pendulums. A spring pendulum is usually called a body that can oscillate on a smooth horizontal surface under the action of a spring, the second end of which is fixed (left figure). A mathematical pendulum is a massive body, the dimensions of which can be neglected, oscillating on a long, weightless and inextensible thread (right figure). The name of this system, “mathematical pendulum,” is due to the fact that it represents an abstract mathematical model of real ( physical) pendulum. It is necessary to remember the formulas for the period (or frequency) of oscillations of spring and mathematical pendulums. For a spring pendulum

where is the length of the thread, is the acceleration free fall. Let's consider the application of these definitions and laws using the example of problem solving.

To find the cyclic frequency of oscillations of the load in task 11.1.1 Let's first find the period of oscillation, and then use formula (11.2). Since 10 m 28 s is 628 s, and during this time the load oscillates 100 times, the period of oscillation of the load is 6.28 s. Therefore, the cyclic frequency of oscillations is 1 s -1 (answer 2 ). IN problem 11.1.2 the load made 60 oscillations in 600 s, so the oscillation frequency is 0.1 s -1 (answer 1 ).

To understand the distance the load will travel in 2.5 periods ( problem 11.1.3), let's follow his movement. After a period, the load will return back to the point of maximum deflection, completing a complete oscillation. Therefore, during this time, the load will travel a distance equal to four amplitudes: to the equilibrium position - one amplitude, from the equilibrium position to the point of maximum deviation in the other direction - the second, back to the equilibrium position - the third, from the equilibrium position to the starting point - the fourth. During the second period, the load will again go through four amplitudes, and during the remaining half of the period - two amplitudes. Therefore, the distance traveled is equal to ten amplitudes (answer 4 ).

The amount of movement of the body is the distance from starting point to the final one. Over 2.5 periods in task 11.1.4 the body will have time to complete two full and half a full oscillation, i.e. will be at the maximum deviation, but on the other side of the equilibrium position. Therefore, the magnitude of the displacement is equal to two amplitudes (answer 3 ).

By definition, the oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinates of an oscillating body on time. Therefore the correct answer is problem 11.1.5 - 3 .

A period is the time of complete oscillation. This means that the return of a body back to the same point from which the body began to move does not mean that a period has passed: the body must return to the same point with the same speed. For example, a body, having started oscillations from an equilibrium position, will have time to deviate by a maximum amount in one direction, return back, deviate by a maximum in the other direction, and return back again. Therefore, during the period the body will have time to deviate by the maximum amount from the equilibrium position twice and return back. Consequently, the passage from the equilibrium position to the point of maximum deviation ( problem 11.1.6) the body spends a quarter of the period (answer 3 ).

Harmonic oscillations are those in which the dependence of the coordinates of the oscillating body on time is described by a trigonometric (sine or cosine) function of time. IN task 11.1.7 these are the functions and , despite the fact that the parameters included in them are designated as 2 and 2 . The function is a trigonometric function of the square of time. Therefore, vibrations of only quantities and are harmonic (answer 4 ).

During harmonic vibrations, the speed of the body changes according to the law , where is the amplitude of the speed oscillations (the time reference point is chosen so that the initial phase of the oscillations is equal to zero). From here we find the dependence kinetic energy bodies from time to time
(problem 11.1.8). Using the further known trigonometric formula, we get

From this formula it follows that the kinetic energy of a body changes during harmonic oscillations also according to the harmonic law, but with double the frequency (answer 2 ).

Behind the relationship between the kinetic energy of the load and the potential energy of the spring ( problem 11.1.9) is easy to follow from the following considerations. When the body is deflected by the maximum amount from the equilibrium position, the speed of the body is zero, and, therefore, the potential energy of the spring is greater than the kinetic energy of the load. On the contrary, when the body passes through the equilibrium position, the potential energy of the spring is zero, and therefore the kinetic energy is greater than the potential energy. Therefore, between the passage of the equilibrium position and the maximum deflection, the kinetic and potential energy are compared once. And since during a period the body passes four times from the equilibrium position to the maximum deflection or back, then during the period the kinetic energy of the load and the potential energy of the spring are compared with each other four times (answer 2 ).

Amplitude of speed fluctuations ( task 11.1.10) is easiest to find using the law of conservation of energy. At the point of maximum deflection, the energy of the oscillatory system is equal to the potential energy of the spring , where is the spring stiffness coefficient, is the vibration amplitude. When passing through the equilibrium position, the energy of the body is equal to the kinetic energy , where is the mass of the body, is the speed of the body when passing through the equilibrium position, which is the maximum speed of the body during the oscillation process and, therefore, represents the amplitude of the speed oscillations. Equating these energies, we find

(answer 4 ).

From formula (11.5) we conclude ( problem 11.2.2), that its period does not depend on the mass of a mathematical pendulum, and with an increase in length by 4 times, the period of oscillations increases by 2 times (answer 1 ).

A clock is an oscillatory process that is used to measure intervals of time ( problem 11.2.3). The words “clock is rushing” mean that the period of this process less than that what it should be. Therefore, to clarify the progress of these clocks, it is necessary to increase the period of the process. According to formula (11.5), to increase the period of oscillation of a mathematical pendulum, it is necessary to increase its length (answer 3 ).

To find the amplitude of oscillations in problem 11.2.4, it is necessary to represent the dependence of the body coordinates on time in the form of a single trigonometric function. For the function given in the condition, this can be done by introducing an additional angle. Multiplying and dividing this function by and using the addition formula trigonometric functions, we get

where is the angle such that . From this formula it follows that the amplitude of body oscillations is (answer 4 ).

The most important parameter characterizing mechanical, sound, electrical, electromagnetic and all other types of vibrations is period- the time during which one complete oscillation occurs. If, for example, the pendulum of a clock makes two complete oscillations in 1 s, the period of each oscillation is 0.5 s. The period of oscillation of a large swing is about 2 s, and the period of oscillation of a string can range from tenths to ten-thousandths of a second.

Figure 2.4 - Oscillation

Where: φ – oscillation phase, I– current strength, Ia– amplitude value of current strength (amplitude)

T– period of current fluctuation (period)

Another parameter characterizing fluctuations is frequency(from the word “often”) - a number showing how many complete oscillations per second are made by a clock pendulum, a sounding body, a current in a conductor, etc. The frequency of oscillations is estimated by a unit called the Hertz (abbreviated as Hz): 1 Hz is one oscillation per second. If, for example, a sounding string makes 440 complete vibrations in 1 s (at the same time it creates the tone “A” of the third octave), its vibration frequency is said to be 440 Hz. The alternating current frequency of the electric lighting network is 50 Hz. At this current, electrons in the wires of the network flow alternately 50 times in one direction and the same number of times in the opposite direction within a second, i.e. perform 50 complete oscillations in 1 s.

Larger units of frequency are kilohertz (written kHz), equal to 1000 Hz, and megahertz (written MHz), equal to 1000 kHz or 1,000,000 Hz.

Amplitude- the maximum value of displacement or change in a variable during oscillatory or wave motion. A non-negative scalar quantity, measured in units depending on the type of wave or vibration.

Figure 2.5 - Sinusoidal oscillation.

Where, y- wave amplitude, λ - wavelength.

For example:

the amplitude for mechanical vibration of a body (vibration), for waves on a string or spring, is the distance and is written in units of length;

The amplitude of sound waves and audio signals usually refers to the amplitude of the air pressure in the wave, but is sometimes described as the amplitude of the displacement relative to an equilibrium (the air or the speaker's diaphragm). Its logarithm is usually measured in decibels (dB);

For electromagnetic radiation the amplitude corresponds to the magnitude of the electric and magnetic fields.

The form of amplitude change is called envelope wave.

Sound vibrations

How do sound waves appear in air? Air consists of particles invisible to the eyes. When the wind blows, they can be transported over long distances. But they can also hesitate. For example, if we make a sharp movement with a stick in the air, we will feel a slight gust of wind and at the same time hear a faint sound. Sound this is the result of vibrations of air particles excited by the vibrations of the stick.

Let's do this experiment. Let's pull the string, for example, of a guitar, and then let it go. The string will begin to tremble - oscillate around its original resting position. Quite strong vibrations of the string are noticeable to the eye. Weak vibrations of the string can only be felt as a slight tickling if you touch it with your finger. While the string vibrates, we hear sound. As soon as the string calms down, the sound will fade away. The birth of sound here is the result of condensation and rarefaction of air particles. Oscillating from side to side, the string presses, as if pressing, air particles in front of it, forming areas of high pressure in a certain volume of it, and behind it, on the contrary, areas of low pressure. This is it sound waves. Spreading through the air at a speed of about 340 m/s, they carry a certain amount of energy. At the moment when the area of ​​​​increased pressure of the sound wave reaches the ear, it presses on the eardrum, bending it slightly inward. When the rarefied region of the sound wave reaches the ear, the eardrum bends slightly outward. The eardrum constantly vibrates in time with alternating areas of high and low air pressure. These vibrations are transmitted along the auditory nerve to the brain, and we perceive them as sound. The greater the amplitude of sound waves, the more energy they carry, the louder the sound we perceive.

Sound waves, like water or electrical vibrations, are represented by a wavy line - a sine wave. Its humps correspond to areas of high pressure, and its depressions correspond to areas of low air pressure. An area of ​​high pressure and a subsequent area of ​​low pressure form a sound wave.

By the frequency of vibration of a sounding body one can judge the tone or pitch of a sound. The higher the frequency, the higher the tone of the sound, and vice versa, the lower the frequency, the lower the tone of the sound. Our ear is capable of responding to a relatively small frequency band (section) sound vibrations - approximately 20 Hz to 20 kHz. Nevertheless, this frequency band accommodates the entire wide range of sounds created by the human voice and a symphony orchestra: from very low tones, similar to the sound of a beetle buzzing, to the barely perceptible high-pitched squeak of a mosquito. Oscillation frequency up to 20 Hz, called infrasonic, And above 20 kHz, called ultrasonic, we don't hear. And if the eardrum of our ear turned out to be capable of responding to ultrasonic vibrations, we could then hear the squeak of bats, the voice of a dolphin. Dolphins emit and hear ultrasonic vibrations with frequencies up to 180 kHz.

But one should not confuse the height, i.e. the tone of the sound with its strength. The pitch of a sound does not depend on the amplitude, but on the frequency of vibrations. A thick and long string of a musical instrument, for example, creates a low tone of sound, i.e. vibrates more slowly than a thin and short string, creating a high-pitched sound (Fig. 1).

Figure 2.6 - Sound waves

The higher the frequency of vibration of the string, the shorter the sound waves and the higher the pitch of the sound.

In electrical and radio engineering, alternating currents with frequencies ranging from several hertz to thousands of gigahertz are used. Broadcast radio antennas, for example, are fed by currents with frequencies ranging from approximately 150 kHz to 100 MHz.

These rapidly changing vibrations, called radio frequency vibrations, are the means by which sounds are transmitted wirelessly over long distances.

The entire huge range of alternating currents is usually divided into several sections - subranges.

Currents with a frequency from 20 Hz to 20 kHz, corresponding to vibrations that we perceive as sounds of different tones, are called currents(or fluctuations) audio frequency, and currents with a frequency above 20 kHz - ultrasonic frequency currents.

Currents with a frequency from 100 kHz to 30 MHz are called high frequency currents,

Currents with frequencies above 30 MHz - ultra-high and ultra-high frequency currents.

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

The same applies to anharmonic strictly periodic oscillations (and approximately - with varying degrees of success - to non-periodic oscillations, at least those close to periodicity).

In case we're talking about about oscillations of a harmonic oscillator with damping, the period is understood as the period of its oscillating component (ignoring damping), which coincides with twice the time interval between the nearest passages of the oscillating quantity through zero. In principle, this definition can be, with greater or less accuracy and usefulness, extended in some generalization to damped oscillations with other properties.

Designations: The usual standard notation for the period of oscillation is: T (\displaystyle T)(although others may apply, the most common is τ (\displaystyle \tau), Sometimes Θ (\displaystyle \Theta) etc.).

For wave processes, the period is also obviously related to the wavelength λ (\displaystyle \lambda)

Where v (\displaystyle v)- the speed of wave propagation (more precisely, the phase speed).

IN quantum physics the period of oscillation is directly related to energy (since in quantum physics the energy of an object - for example, a particle - is the frequency of oscillation of its wave function).

Theoretical finding Determining the period of oscillation of a particular physical system comes down, as a rule, to finding a solution to the dynamic equations (equations) that describe this system. For category linear systems(and approximately - for linearizable systems in the linear approximation, which is often very good) there are standard relatively simple mathematical methods that allow this to be done (if the physical equations themselves that describe the system are known).

For experimental determination period, clocks, stopwatches, frequency meters, stroboscopes, strobotachometers, and oscilloscopes are used. Also used are beats, heterodyning method in different types, the principle of resonance is used. For waves, you can measure the period indirectly - through the wavelength, for which interferometers, diffraction gratings, etc. are used. Sometimes sophisticated methods are required, specially developed for a particular difficult case(difficulties can arise from both the measurement of time itself, especially if we are talking about extremely small or, conversely, very large times, and the difficulties of observing a fluctuating value).

Encyclopedic YouTube

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  An idea of ​​the periods of oscillations of various physical processes gives the article Frequency intervals (considering that a period in seconds is the reciprocal of the frequency in hertz).

  Some idea of ​​the magnitude of the periods of various physical processes can also be given by the frequency scale of electromagnetic oscillations (see Electromagnetic spectrum).

  The periods of oscillation of sound audible by humans are in the range

  From 5·10 −5 to 0.2

  (its clear boundaries are somewhat arbitrary).

  Periods electromagnetic vibrations, corresponding to different colors of visible light - in the range

  From 1.1·10−15 to 2.3·10−15.

  Since, with extremely large and extremely small periods of oscillation, measurement methods tend to become more and more indirect (even to the point of smoothly flowing into theoretical extrapolations), it is difficult to name a clear upper and lower limit. lower limit for the period of oscillation measured directly. Some estimate for the upper bound can be given by the lifetime modern science(hundreds of years), and for the lower one - the period of oscillation of the wave function of the heaviest currently known particle ().

  Anyway border below can serve as the Planck time, which is so small that according to modern ideas not only can it hardly be physically measured at all, but it is also unlikely that in the more or less foreseeable future it will be possible to get closer to measuring quantities even of much greater orders of magnitude, and border on top- the existence of the Universe is more than ten billion years.

  Periods of oscillations of the simplest physical systems

  Spring pendulum

  Math pendulum

  T = 2 π l g (\displaystyle T=2\pi (\sqrt (\frac (l)(g))))

  Where l (\displaystyle l)- length of suspension (for example, thread), g (\displaystyle g)- acceleration of free fall.

  The period of small oscillations (on Earth) of a mathematical pendulum 1 meter long is equal to 2 seconds with good accuracy.

  Physical pendulum

  T = 2 π J m g l (\displaystyle T=2\pi (\sqrt (\frac (J)(mgl))))

  Where J (\displaystyle J)- moment of inertia of the pendulum relative to rotation axis, m (\displaystyle m) -

  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 have 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:

  T = 2p,

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

  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:

  T = 2p,

  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. Why equal to the period oscillations of a pendulum, the length of which is 28 cm, 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 oscillation 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

  Purpose of the work:

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

  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

  l, m	n	t d D t, s	Td D T, With
  0,25	20
  0,49	20

  3. Investigate the dependence of the period of oscillation of a pendulum on the acceleration of free fall. 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 the 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

  A, cm	n	t+D t, With	T+D T, With

  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

  A, cm	n	t+D t, With	T+D T, With

  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

  Purpose 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

  Experience no.	l d D l, m	n	t d D t, With	T d D T, With	g, m/s2	D g, m/s2	g d D g, m/s2

  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?