So it is with anharmonic strictly periodic oscillations (and approximately - with one success or another - and non-periodic oscillations, at least close to periodicity).
When it comes to 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 more or less accurately and usefully 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.).
T = 1 ν , ν = 1 T . (\displaystyle T=(\frac (1)(\nu )),\ \ \ \nu =(\frac (1)(T)).)For wave processes, the period is also obviously related to the wavelength λ (\displaystyle \lambda )
v = λ ν , T = λ v , (\displaystyle v=\lambda \nu ,\ \ \ T=(\frac (\lambda )(v)),)where v (\displaystyle v)- wave propagation velocity (more precisely, phase velocity).
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 the oscillation period of a particular physical system is reduced, as a rule, to finding a solution of dynamic equations (equation) that describes this system. For the category of linear systems (and approximately for linearizable systems in a 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, strobe tachometers, oscilloscopes are used. Beats are also used, the method of heterodyning in different forms, 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 also required, specially developed for a specific difficult case (difficulty can be both the measurement of time itself, especially when it comes to extremely short or vice versa very long times, and the difficulty of observing a fluctuating value).
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An idea about the periods of oscillations of various physical processes is given in the article Frequency Intervals (given that the period in seconds is the reciprocal of the frequency in hertz).
Some idea of the magnitudes 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 a sound audible to a person are in the range
From 5 10 −5 to 0.2
(its clear boundaries are somewhat arbitrary).
Periods of electromagnetic oscillations corresponding to different colors of visible light - in the range
From 1.1 10 −15 to 2.3 10 −15 .
Since, for extremely large and extremely small oscillation periods, measurement methods tend to become more and more indirect (up to a smooth flow into theoretical extrapolations), it is difficult to name a clear upper and lower bounds for the oscillation period measured directly. Some estimate for the upper limit can be given by the time of existence of modern science (hundreds of years), and for the lower one - by the oscillation period of the wave function of the heaviest particle known now ().
Anyway bottom border can serve as the Planck time, which is so small that, according to modern concepts, it is not only unlikely that it can be physically measured in any way at all, but it is unlikely that in the more or less foreseeable future it will be possible to approach the measurement of even much larger orders of magnitude, and top border- the time of existence of the Universe - more than ten billion years.
Periods of oscillations of the simplest physical systems
Spring pendulum
Mathematical pendulum
T = 2 π l g (\displaystyle T=2\pi (\sqrt (\frac (l)(g))))
where l (\displaystyle l)- the length of the suspension (for example, threads), g (\displaystyle g)- acceleration of gravity .
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)- the moment of inertia of the pendulum about the axis of rotation, m (\displaystyle m) -
The variety of oscillatory processes that surround us is so significant that you simply wonder - is there anything that does not oscillate? It is unlikely, because even a completely motionless object, say a stone that has been motionless for thousands of years, still performs oscillatory processes - it periodically heats up during the day, increasing, and cools down at night and decreases in size. And the closest example - trees and branches - tirelessly sway throughout their lives. But that is a stone, a tree. And if a 100-story building fluctuates in the same way from the pressure of the wind? It is known, for example, that the top deviates back and forth by 5-12 meters, why not a pendulum 500 m high. And how much does such a structure increase in size from temperature changes? Vibrations of machine bodies and mechanisms can also be included here. Just think, the plane you're flying in is constantly oscillating. Thinking about flying? It’s not worth it, because fluctuations are the essence of the world around us, you can’t get rid of them - they can only be taken into account and applied “for the sake of it”.
As usual, the study of the most complex areas of knowledge (and they are not simple) begins with an acquaintance with the simplest models. And there is no simpler and more understandable model of the oscillatory process than a pendulum. It is here, in the physics classroom, that we first hear such a mysterious phrase - “the period of oscillation of a mathematical pendulum”. The pendulum is a thread and a weight. And what is this special pendulum - mathematical? And everything is very simple, for this pendulum it is assumed that its thread has no weight, is inextensible, but oscillates under the influence of etc. all participants in the experiment. At the same time, the influence of some of them on the process is negligibly small. For example, it is a priori clear that the weight and elasticity of the pendulum thread under certain conditions do not have a noticeable effect on the oscillation period of a mathematical pendulum, as they are negligible, so their influence is excluded from consideration.
The definition of a pendulum, perhaps the simplest known, is as follows: the period is the time during which one complete oscillation takes place. Let's make a mark at one of the extreme points of the movement of the load. Now, every time the point closes, we count the number of complete oscillations and time, say, 100 oscillations. Determining the duration of one period is not difficult at all. Let us perform this experiment for a pendulum oscillating in one plane in the following cases:
Different initial amplitude;
different weight of cargo.
We will get a result that is stunning at first glance: in all cases, the period of oscillation of the mathematical pendulum remains unchanged. In other words, the initial amplitude and mass of a material point do not affect the duration of the period. For further presentation, there is only one inconvenience - because. the height of the load changes during movement, then the restoring force along the trajectory is variable, which is inconvenient for calculations. Let's cheat a little - swing the pendulum also in the transverse direction - it will begin to describe a cone-shaped surface, the period T of its rotation will remain the same, the speed V is a constant along which the load moves S = 2πr, and the restoring force is directed along the radius.
Then we calculate the period of oscillation of the mathematical pendulum:
T \u003d S / V \u003d 2πr / v
If the length of the thread l is much larger than the dimensions of the load (at least 15-20 times), and the angle of inclination of the thread is small (small amplitudes), then we can assume that the restoring force P is equal to the centripetal force F:
P \u003d F \u003d m * V * V / rOn the other hand, the moment of the restoring force and the load are equal, and then
P * l = r *(m*g), whence we obtain, given that P = F, the following equality: r * m * g/l = m*v*v/r
It is not difficult to find the speed of the pendulum: v = r*√g/l.
And now we recall the very first expression for the period and substitute the value of the speed:
Т=2πr/ r*√g/l
After trivial transformations, the formula for the oscillation period of a mathematical pendulum in its final form looks like this:
T \u003d 2 π √ l / g
Now, the previously experimentally obtained results of the independence of the period of oscillations from the mass of the load and the amplitude have been confirmed in an analytical form and do not seem so “amazing” at all, as they say, which was required to be proved.
Among other things, considering the last expression for the period of oscillation of a mathematical pendulum, one can see an excellent opportunity for measuring the acceleration of gravity. To do this, it is enough to assemble a certain reference pendulum at any point on the Earth and measure the period of its oscillations. So, quite unexpectedly, a simple and uncomplicated pendulum gave us a great opportunity to study the distribution of the density of the earth's crust, up to the search for deposits of earth's minerals. But that's a completely different story.
(lat. amplitude- magnitude) - this is the largest deviation of the oscillating body from the equilibrium position.
For a pendulum, this is the maximum distance that the ball moves from its equilibrium position (figure below). For oscillations with small amplitudes, this distance can be taken as the length of the arc 01 or 02, as well as the lengths of these segments.
The oscillation amplitude is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (in absolute value) ordinate of the sinusoidal curve, (see figure below).
Oscillation period.
Oscillation period- this is the smallest period of time after which the system, making oscillations, again returns to the same state in which it was at the initial moment of time, chosen arbitrarily.
In other words, the oscillation period ( T) is the time for which one complete oscillation takes place. For example, in the figure below, this is the time it takes for the weight of the pendulum to move from the rightmost point through the equilibrium point O to the leftmost point and back through the point O again to the far right.
For a full period of oscillation, therefore, the body travels a path equal to four amplitudes. The oscillation period is measured in units of time - seconds, minutes, etc. The oscillation period can be determined from the well-known oscillation graph (see figure below).
The concept of “oscillation period”, strictly speaking, is valid only when the values of the oscillating quantity are exactly repeated after a certain period of time, that is, for harmonic oscillations. However, this concept is also applied to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency is the number of oscillations per unit of time, for example, in 1 s.
The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, then this means that one oscillation is made for every second. The frequency and period of oscillations are related by the relations:
In the theory of oscillations, the concept is also used cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:
.Cyclic frequency is the number of oscillations per 2π seconds.
Harmonic oscillations - oscillations performed according to the laws of sine and cosine. The following figure shows a graph of the change in the coordinate of a point over time according to the law of cosine.
picture
Oscillation amplitude
The amplitude of a harmonic oscillation is the largest value of the displacement of the body from the 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 the sine and cosine can take values in the range from -1 to 1, the equation must contain the factor Xm, which expresses 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 for one complete oscillation. The period of oscillation is denoted by the letter T. The units of the period correspond to the units of time. That is, in SI it is seconds.
Oscillation frequency - the number of oscillations per unit time. The oscillation frequency is denoted by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.
v = 1/T.
Frequency units 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 value is called the cyclic oscillation frequency. In some literature, the name circular frequency is found. The natural frequency of an oscillatory system is the frequency of free oscillations.
The frequency of natural oscillations is calculated by the formula:
The frequency of natural oscillations depends on the properties of the material and the mass of the load. The greater the stiffness of the spring, the greater the frequency of natural oscillations. 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 unbalanced. The greater the mass of the body, the slower this speed of this body will change.
Period of free oscillations:
T = 2*pi/ ω0 = 2*pi*√(m/k)
It is noteworthy that at small deflection angles, 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
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 the length of the pendulum thread. The longer the length, the slower the body will oscillate.
The period of oscillation does not depend on the mass of the load. But it depends on the free fall acceleration. 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.
What is the period of oscillation? What is this quantity, what physical meaning does it have and how to calculate it? In this article, we will deal with these issues, consider various formulas by which the period of oscillations can be calculated, and also find out what relationship exists between such physical quantities as the period and frequency of oscillations of a body / system.
Definition and physical meaning
The period of oscillation is such a period of time in which the body or system makes one oscillation (necessarily complete). In parallel, we can note the parameter at which the oscillation can be considered complete. The role of such a condition is the return of the body to its original state (to the original coordinate). The analogy with the period of a function is very well drawn. Incidentally, it is a mistake to think that it takes place exclusively in ordinary and higher mathematics. As you know, these two sciences are inextricably linked. And the period of functions can be encountered not only when solving trigonometric equations, but also in various branches of physics, namely, we are talking about mechanics, optics and others. When transferring the period of oscillations from mathematics to physics, it should be understood simply as a physical quantity (and not a function), which has a direct dependence on the passing time.
What are the fluctuations?
Oscillations are divided into harmonic and anharmonic, as well as periodic and non-periodic. It would be logical to assume that in the case of harmonic oscillations, they occur according to some harmonic function. It can be either sine or cosine. In this case, the coefficients of compression-stretching and increase-decrease may also turn out to be in the case. Also, vibrations are damped. That is, when a certain force acts on the system, which gradually “slows down” the oscillations themselves. In this case, the period becomes shorter, while the frequency of oscillations invariably increases. The simplest experiment using a pendulum demonstrates such a physical axiom very well. It can be spring type, as well as mathematical. It doesn't matter. By the way, the oscillation period in such systems will be determined by different formulas. But more on that later. Now let's give examples.
Experience with pendulums
You can take any pendulum first, there will be no difference. The laws of physics are the laws of physics, that they are respected in any case. But for some reason, the mathematical pendulum is more to my liking. If someone does not know what it is: it is a ball on an inextensible thread that is attached to a horizontal bar attached to the legs (or the elements that play their role - to keep the system in balance). The ball is best taken from metal, so that the experience is clearer.
So, if you take such a system out of balance, apply some force to the ball (in other words, push it), then the ball will begin to swing on the thread, following a certain trajectory. Over time, you can notice that the trajectory along which the ball passes is reduced. At the same time, the ball begins to scurry back and forth faster and faster. This indicates that the oscillation frequency is increasing. But the time it takes for the ball to return to its original position decreases. But the time of one complete oscillation, as we found out earlier, is called a period. If one value decreases and the other increases, then they speak of inverse proportionality. So we got to the first moment, on the basis of which formulas are built to determine the period of oscillations. If we take a spring pendulum for testing, then the law will be observed there in a slightly different form. In order for it to be most clearly represented, we set the system in motion in a vertical plane. To make it clearer, it was first worth saying what a spring pendulum is. From the name it is clear that a spring must be present in its design. And indeed it is. Again, we have a horizontal plane on supports, to which a spring of a certain length and stiffness is suspended. To it, in turn, a weight is suspended. It can be a cylinder, a cube or another figure. It may even be some third-party item. In any case, when the system is taken out of equilibrium, it will begin to perform damped oscillations. The increase in frequency is most clearly seen in the vertical plane, without any deviation. On this experience, you can finish.
So, in their course, we found out that the period and frequency of oscillations are two physical quantities that have an inverse relationship.
Designation of quantities and dimensions
Usually, the oscillation period is denoted by the Latin letter T. Much less often, it can be denoted differently. The frequency is denoted by the letter µ (“Mu”). As we said at the very beginning, a period is nothing more than the time during which a complete oscillation occurs in the system. Then the dimension of the period will be a second. And since the period and frequency are inversely proportional, the frequency dimension will be unit divided by a second. In the record of tasks, everything will look like this: T (s), µ (1/s).
Formula for a mathematical pendulum. Task #1
As in the case with the experiments, I decided first of all to deal with the mathematical pendulum. We will not go into the derivation of the formula in detail, since such a task was not originally set. Yes, and the conclusion itself is cumbersome. But let's get acquainted with the formulas themselves, find out what kind of quantities they include. So, the formula for the period of oscillation for a mathematical pendulum is as follows:
Where l is the length of the thread, n \u003d 3.14, and g is the acceleration of gravity (9.8 m / s ^ 2). The formula should not cause any difficulties. Therefore, without additional questions, we will immediately proceed to solving the problem of determining the period of oscillation of a mathematical pendulum. A metal ball weighing 10 grams is suspended from an inextensible thread 20 centimeters long. Calculate the period of oscillation of the system, taking it for a mathematical pendulum. The solution is very simple. As in all problems in physics, it is necessary to simplify it as much as possible by discarding unnecessary words. They are included in the context in order to confuse the decisive one, but in fact they have absolutely no weight. In most cases, of course. Here it is possible to exclude the moment with “inextensible thread”. This phrase should not lead to a stupor. And since we have a mathematical pendulum, we should not be interested in the mass of the load. That is, the words about 10 grams are also simply designed to confuse the student. But we know that there is no mass in the formula, so with a clear conscience we can proceed to the solution. So, we take the formula and simply substitute the values \u200b\u200binto it, since it is necessary to determine the period of the system. Since no additional conditions were specified, we will round the values to the 3rd decimal place, as is customary. Multiplying and dividing the values, we get that the period of oscillation is 0.886 seconds. Problem solved.
Formula for a spring pendulum. Task #2
Pendulum formulas have a common part, namely 2n. This value is present in two formulas at once, but they differ in the root expression. If in the problem concerning the period of a spring pendulum, the mass of the load is indicated, then it is impossible to avoid calculations with its use, as was the case with the mathematical pendulum. But you should not be afraid. This is how the period formula for a spring pendulum looks like:
In it, m is the mass of the load suspended from the spring, k is the coefficient of spring stiffness. In the problem, the value of the coefficient can be given. But if in the formula of a mathematical pendulum you don’t particularly clear up - after all, 2 out of 4 values are constants - then a 3rd parameter is added here, which can change. And at the output we have 3 variables: the period (frequency) of oscillations, the coefficient of stiffness of the spring, the mass of the suspended load. The task can be oriented towards finding any of these parameters. Searching for a period again would be too easy, so we'll change the condition a bit. Find the stiffness of the spring if the full swing time is 4 seconds and the weight of the spring pendulum is 200 grams.
To solve any physical problem, it would be good to first make a drawing and write formulas. They are half the battle here. Having written the formula, it is necessary to express the stiffness coefficient. It is under our root, so we square both sides of the equation. To get rid of the fraction, multiply the parts by k. Now let's leave only the coefficient on the left side of the equation, that is, we divide the parts by T^2. In principle, the problem could be a little more complicated by setting not a period in numbers, but a frequency. In any case, when calculating and rounding (we agreed to round to the 3rd decimal place), it turns out that k = 0.157 N/m.
The period of free oscillations. Free period formula
The formula for the period of free oscillations is understood as those formulas that we examined in the two previously given problems. They also make up an equation of free oscillations, but there we are talking about displacements and coordinates, and this question belongs to another article.
1) Before taking on a task, write down the formula that is associated with it.
2) The simplest tasks do not require drawings, but in exceptional cases they will need to be done.
3) Try to get rid of roots and denominators if possible. An equation written in a line that does not have a denominator is much more convenient and easier to solve.
