Total energy of a mathematical pendulum formula. Harmonic vibrations

Definition

Mathematical pendulum- this is a special case of a physical pendulum, the mass of which is at one point.

Usually, a small ball (material point) with a large mass suspended on a long inextensible thread (suspension) is considered to be a mathematical pendulum. This is an idealized system that oscillates under the influence of gravity. Only for angles of the order of 50-100 is the mathematical pendulum a harmonic oscillator, that is, it performs harmonic oscillations.

Studying the swing of a chandelier on a long chain, Galileo studied the properties of a mathematical pendulum. He realized that the oscillation period of a given system does not depend on the amplitude at small deflection angles.

The formula for the oscillation period of a mathematical pendulum

Let the suspension point of the pendulum be fixed. A load suspended from a pendulum thread moves along an arc of a circle (Fig.1(a)) with acceleration, and some restoring force ($\overline(F)$) acts on it. This force changes as the load moves. As a result, the calculation of motion becomes complex. Let's introduce some simplifications. Let the pendulum oscillate not in a plane, but describe a cone (Fig. 1 (b)). The load in this case moves in a circle. The period of oscillations of interest to us will coincide with the period of the conical movement of the load. The period of revolution of a conical pendulum around the circumference is equal to the time that the weight spends on one turn around the circumference:

where $L$ is the circumference; $v$ - the speed of the cargo movement. If the angles of deviation of the thread from the vertical are small (small oscillation amplitudes), then it is assumed that the restoring force ($F_1$) is directed along the radius of the circle that the load describes. Then this force is equal to the centripetal force:

Consider similar triangles: AOB and DBC (Fig. 1 (b)).

We equate the right parts of expressions (2) and (3), we express the speed of movement of the load:

\[\frac(mv^2)(R)=mg\frac(R)(l)\ \to v=R\sqrt(\frac(g)(l))\left(4\right).\]

We substitute the resulting speed into formula (1), we have:

\ \

From formula (5) we see that the period of a mathematical pendulum depends only on the length of its suspension (the distance from the suspension point to the center of gravity of the load) and the free fall acceleration. Formula (5) for the period of a mathematical pendulum is called the Huygens formula; it is fulfilled when the suspension point of the pendulum does not move.

Using the dependence of the oscillation period of a mathematical pendulum on the free fall acceleration, the value of this acceleration is determined. To do this, measure the length of the pendulum, considering a large number of oscillations, find the period $T$, then calculate the acceleration of free fall.

Examples of problems with a solution

Example 1

Exercise. As you know, the magnitude of the acceleration of free fall depends on latitude. What is the acceleration of free fall at the latitude of Moscow if the period of oscillation of a mathematical pendulum of length $l=2.485\cdot (10)^(-1)$m is T=1 c?\textit()

Decision. As a basis for solving the problem, we take the formula for the period of a mathematical pendulum:

Let us express from (1.1) the free fall acceleration:

Let's calculate the desired acceleration:

Answer.$g=9.81\frac(m)(s^2)$

Example 2

Exercise. What will be the period of oscillation of a mathematical pendulum if the point of its suspension moves vertically downward 1) at a constant speed? 2) with acceleration $a$? The length of the thread of this pendulum is $l.$

Decision. Let's make a drawing.

1) The period of a mathematical pendulum whose suspension point moves uniformly is equal to the period of a pendulum with a fixed suspension point:

2) The acceleration of the pendulum's suspension point can be considered as the appearance of an additional force equal to $F=ma$, which is directed against the acceleration. That is, if the acceleration is directed upwards, then the additional force is directed downwards, which means that it is added to the force of gravity ($mg$). If the suspension point moves with downward acceleration, then the additional force is subtracted from the force of gravity.

The period of a mathematical pendulum that oscillates and for which the suspension point moves with acceleration, we find as:

Answer. 1) $T_1=2\pi \sqrt(\frac(l)(g))$; 2) $T_1=2\pi \sqrt(\frac(l)(g-a))$

(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 (modulo) 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.

As a concrete example of a body rotating about an axis, consider the motion of pendulums.

A physical pendulum is a rigid body that has a horizontal axis of rotation, around which it oscillates under the action of its weight (Fig. 119).

The position of the pendulum is completely determined by the angle of its deviation from the equilibrium position, and therefore, to determine the law of motion of the pendulum, it is sufficient to find the dependence of this angle on time.

Type equation:

is called the equation (law) of motion of the pendulum. It depends on the initial conditions, i.e. on the angle and angular velocity Thus,

The limiting case of a physical Pendulum is a mathematical pendulum representing (as mentioned earlier - Chapter 2, § 3) a material point connected to the horizontal axis around which it rotates by a rigid weightless rod (Fig. 120). The distance of a material point from the axis of rotation is called the length of the mathematical pendulum.

Equations of motion of physical and mathematical pendulums

We choose a system of coordinate axes so that the xy plane passes through the center of gravity of the body C and coincides with the swing plane of the pendulum, as shown in the drawing (Fig. 119). We direct the axis perpendicular to the plane of the drawing on us. Then, based on the results of the previous section, we write the equation of motion of a physical pendulum in the form:

where denotes the moment of inertia of the pendulum about its axis of rotation and

Therefore, you can write:

The active force acting on the pendulum is its weight, the moment of which relative to the weight gain axis will be:

where is the distance from the axis of rotation of the pendulum to its center of mass C.

Therefore, we arrive at the following equation of motion of a physical pendulum:

Since the mathematical pendulum is a special case of the physical one, the differential equation written above is also valid for the mathematical pendulum. If the length of a mathematical pendulum is equal to and its weight, then its moment of inertia relative to the axis of rotation is equal to

Since the distance of the center of gravity of the mathematical pendulum from the axis is equal to the final differential equation of motion of the mathematical pendulum can be written as:

Reduced length of a physical pendulum

Comparing equations (16.8) and (16.9), we can conclude that if the parameters of the physical and mathematical pendulums are related by the relation

then the laws of motion of the physical and mathematical pendulums are the same (under the same initial conditions).

The last relation indicates the length that a mathematical pendulum must have in order to move in the same way as the corresponding physical pendulum. This length is called the reduced length of the physical pendulum. The meaning of this concept lies in the fact that the study of the movement of a physical pendulum can be replaced by the study of the movement of a mathematical pendulum, which is the simplest mechanical scheme.

The first integral of the equation of motion of the pendulum

The equations of motion of physical and mathematical pendulums have the same form, therefore, the equation of their motion will be

Since the only force that is taken into account in this equation will be the force of gravity belonging to the potential force field, then the law of conservation of mechanical energy takes place.

The latter can be obtained by a simple trick, just multiply equation (16.10) by then

Integrating this equation, we get

Determining the integration constant C from the initial conditions, we find

Solving the last equation for we get

This relation is the first integral of the differential equation (16.10).

Determination of the support reactions of physical and mathematical pendulums

The first integral of the equations of motion allows us to determine the support reactions of the pendulums. As indicated in the previous paragraph, the reactions of the supports are determined from equations (16.5). In the case of a physical pendulum, the components of the active force along the coordinate axes and its moments relative to the axes will be:

The coordinates of the center of mass are determined by the formulas:

Then the equations for determining the reactions of the supports take the form:

The centrifugal moments of inertia of the body and the distance between the supports must be known according to the conditions of the problem. Angular acceleration in and angular velocity w are determined from equations (16.9) and (16.4) in the form:

Thus, equations (16.12) completely determine the components of the support reactions of a physical pendulum.

Equations (16.12) are further simplified if we consider a mathematical pendulum. Indeed, since the material point of the mathematical pendulum is located in the plane, then In addition, since one point is fixed, then Therefore, equations (16.12) turn into equations of the form:

From equations (16.13) using equation (16.9) it follows that the reaction of the support is directed along the thread I (Fig. 120). The latter is the obvious result. Therefore, projecting the components of equalities (16.13) onto the direction of the thread, we will find an equation for determining the reaction of the support of the form (Fig. 120):

Substituting the value here and taking into account that we write:

The last relation determines the dynamic response of the mathematical pendulum. Note that its static reaction will be

Qualitative study of the nature of the movement of the pendulum

The first integral of the equation of motion of the pendulum allows us to conduct a qualitative study of the nature of its motion. Namely, we write this integral (16.11) in the form:

During the movement, the radical expression must either be positive or vanish at some points. Let us assume that the initial conditions are such that

In this case, the radical expression does not vanish anywhere. Consequently, when moving, the pendulum will run through all the values ​​of the angle and the angular velocity of the pendulum has the same sign, which is determined by the direction of the initial angular velocity, or the angle will either increase all the time or decrease all the time, i.e. the pendulum will rotate in one side.

The directions of movement will correspond to one or another sign in the expression (16.11). A necessary condition for the implementation of such a movement is the presence of an initial angular velocity, since it is clear from inequality (16.14) that if then for any initial angle of deviation it is impossible to obtain such a movement of the pendulum.

Now let the initial conditions be such that

In this case, there are two such values ​​of the angle at which the radical expression vanishes. Let them correspond to the angles defined by the equality

And it will be somewhere in the range of change from 0 to . Further, it is obvious that when

the radical expression (16.11) will be positive, and if it is arbitrarily small, it will be negative.

Therefore, when the pendulum moves, its angle changes in the range:

At , the pendulum's angular velocity vanishes and the angle begins to decrease to . In this case, the sign of the angular velocity or the sign in front of the radical in expression (16.11) will change. When it reaches the value, the angular velocity of the pendulum again vanishes and the angle again begins to increase to the value

Thus, the pendulum will oscillate

Pendulum oscillation amplitude

When the pendulum oscillates, the maximum value of its deviation from the vertical is called the oscillation amplitude. It is equal to which is determined from the equality

As follows from the last formula, the oscillation amplitude depends on the initial data of the main characteristics of the pendulum or its reduced length.

In a particular case, when the pendulum is deviated from the equilibrium position and released without initial velocity, then it will be equal to , therefore, the amplitude does not depend on the reduced length.

The equation of motion of the pendulum in finite form

Let the initial speed of the pendulum be equal to zero, then the first integral of its equation of motion will be:

Integrating this equation, we find

We will count time from the position of the pendulum, corresponding then

We transform the integrand using the formula:

Then we get:

The resulting integral is called the elliptic integral of the first kind. It cannot be expressed in terms of a finite number of elementary functions.

The inversion of the elliptic integral (16.15) with respect to its upper limit represents the equation of motion of the pendulum:

This will be the well-studied Jacobi elliptic function.

Pendulum period

The time of one complete oscillation of the pendulum is called its period of oscillation. Let's denote it as T. Since the time of the pendulum's movement from position to position is the same as the time of movement from then T is determined by the formula:

We make a change of variables by setting

When changing within the range from 0 to , it will change from 0 to . Further,

and hence

The last integral is called the complete elliptic integral of the first kind (its values ​​are given in special tables).

At , the integrand tends to unity and .

Approximate formulas for small oscillations of a pendulum

In the case when the pendulum oscillations have a small amplitude (practically it should not exceed 20°), we can put

Then the differential equation of motion of the pendulum takes the form:

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 does not 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 spring stiffness, 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 up 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 to mean those formulas that we examined in the two previously given problems. They also make up an equation of free oscillations, but there we are already 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.

The period of oscillation of a physical pendulum depends on many circumstances: on the size and shape of the body, on the distance between the center of gravity and the point of suspension, and on the distribution of body mass relative to this point; therefore, calculating the period of a suspended body is a rather difficult task. The situation is simpler for the mathematical pendulum. From observations of such pendulums, the following simple laws can be established.

1. If, while maintaining the same length of the pendulum (the distance from the point of suspension to the center of gravity of the load), different loads are suspended, then the oscillation period will be the same, although the masses of the loads differ greatly. The period of a mathematical pendulum does not depend on the mass of the load.

2. If, when starting the pendulum, it is deflected through different (but not too large) angles, then it will oscillate with the same period, although with different amplitudes. As long as the amplitudes are not too large, the oscillations are close enough in their form to harmonic (§ 5) and the period of the mathematical pendulum does not depend on the amplitude of the oscillations. This property is called isochronism (from the Greek words "isos" - equal, "chronos" - time).

This fact was first established in 1655 by Galileo allegedly under the following circumstances. Galileo observed in the Pisa Cathedral the swinging of a chandelier on a long chain, which was pushed when ignited. During the course of the service, the amplitude of the swings gradually faded (§ 11), i.e., the amplitude of the oscillations decreased, but the period remained the same. Galileo used his own pulse as an indication of time.

We now derive a formula for the period of oscillation of a mathematical pendulum.

Rice. 16. Oscillations of a pendulum in a plane (a) and movement along a cone (b)

When the pendulum swings, the load moves accelerated along an arc (Fig. 16, a) under the action of a restoring force, which changes during movement. The calculation of the motion of a body under the action of a non-constant force is rather complicated. Therefore, for simplicity, we will proceed as follows.

Let us make the pendulum not oscillate in one plane, but describe the cone so that the load moves in a circle (Fig. 16, b). This movement can be obtained by adding two independent oscillations: one still in the plane of the drawing and the other in the perpendicular plane. Obviously, the periods of both of these plane oscillations are the same, since any oscillation plane is no different from any other. Consequently, the period of the complex movement - the rotation of the pendulum along the cone - will be the same as the period of the swing of the water plane. This conclusion can be easily illustrated by direct experience, taking two identical pendulums and telling one of them to swing in a plane, and the other to rotate along a cone.

But the period of revolution of the "conical" pendulum is equal to the length of the circle described by the load, divided by the speed:

If the angle of deviation from the vertical is small (small amplitudes), then we can assume that the restoring force is directed along the radius of the circle, i.e., equal to the centripetal force:

On the other hand, it follows from the similarity of triangles that . Since , then from here

Equating both expressions to each other, we get for the velocity of circulation

Finally, substituting this into the period expression, we find

So, the period of a mathematical pendulum depends only on the acceleration of free fall and on the length of the pendulum, i.e., the distance from the point of suspension to the center of gravity of the load. From the obtained formula it follows that the period of the pendulum does not depend on its mass and on the amplitude (provided that it is sufficiently small). In other words, we obtained by calculation those basic laws that were established earlier from observations.

But our theoretical derivation gives us more: it allows us to establish a quantitative relationship between the period of the pendulum, its length and the acceleration of free fall. The period of a mathematical pendulum is proportional to the square root of the ratio of the length of the pendulum to the acceleration due to gravity. The coefficient of proportionality is .

A very accurate way of determining this acceleration is based on the dependence of the period of the pendulum on the acceleration of free fall. By measuring the length of the pendulum and determining the period from a large number of oscillations, we can calculate using the formula obtained. This method is widely used in practice.

It is known (see Volume I, §53) that the acceleration of free fall depends on the geographical latitude of the place (at the pole, and at the equator). Observations on the swing period of a certain reference pendulum make it possible to study the distribution of free fall acceleration over latitude. This method is so accurate that even more subtle differences in meaning on the earth's surface can be detected with its help. It turns out that even on the same parallel, the values ​​\u200b\u200bare different at different points on the earth's surface. These anomalies in the distribution of gravitational acceleration are associated with the uneven density of the earth's crust. They are used to study the distribution of density, in particular, to detect the occurrence of any minerals in the thickness of the earth's crust. Extensive gravimetric changes, which made it possible to judge the occurrence of dense masses, were carried out in the USSR in the region of the so-called Kursk magnetic anomaly (see Volume II, § 130) under the guidance of the Soviet physicist Pyotr Petrovich Lazarev. In combination with data on the anomaly of the earth's magnetic field, these gravimetric data made it possible to establish the distribution of the occurrence of iron masses, which determine the Kursk magnetic and gravitational anomalies.