10.4. Energy conservation law for harmonic vibrations

10.4.1. Conservation of energy at mechanical harmonic vibrations

Conservation of energy during oscillations of a mathematical pendulum

With harmonic vibrations, the total mechanical energy of the system is conserved (remains constant).

Total mechanical energy of a mathematical pendulum

E = W k + W p,

where W k - kinetic energy, W k = = mv 2/2; W p - potential energy, W p = mgh; m is the mass of the cargo; g - free fall acceleration module; v - modulus of cargo speed; h - the height of the lifting of the load above the equilibrium position (Fig. 10.15).

With harmonic oscillations, the mathematical pendulum passes through a number of successive states, therefore, it is advisable to consider the energy of the mathematical pendulum in three positions (see Fig.10.15):

Rice. 10.15

1) in equilibrium position

potential energy is zero; the total energy coincides with the maximum kinetic energy:

E = W k max;

2) in extreme position(2) the body is raised above the initial level to the maximum height h max, so the potential energy is also maximum:

W p max = m g h max;

kinetic energy is zero; the total energy coincides with the maximum potential energy:

E = W p max;

3) in intermediate position(3) the body has an instantaneous velocity v and is raised above the initial level to a certain height h, therefore the total energy is the sum

E = m v 2 2 + m g h,

where mv 2/2 - kinetic energy; mgh - potential energy; m is the mass of the cargo; g - free fall acceleration module; v - modulus of cargo speed; h is the height of the load lifting above the equilibrium position.

With harmonic oscillations of a mathematical pendulum, the total mechanical energy is conserved:

E = const.

The values ​​of the total energy of the mathematical pendulum in its three positions are reflected in table. 10.1.

№	Position	W p	W k	E = W p + W k
1	Equilibrium	0	m v max 2/2	m v max 2/2
2	Extreme	mgh max	0	mgh max
3	Intermediate (instant)	mgh	mv 2/2	mv 2/2 + mgh

The values ​​of the total mechanical energy presented in the last column of the table. 10.1, have equal values ​​for any position of the pendulum, which is a mathematical expression:

m v max 2 2 = m g h max;

m v max 2 2 = m v 2 2 + m g h;

m g h max = m v 2 2 + m g h,

where m is the mass of the cargo; g - free fall acceleration module; v is the modulus of the instantaneous speed of the load in position 3; h - the height of the lifting of the load above the equilibrium position in position 3; v max - modulus of maximum cargo speed in position 1; h max is the maximum lifting height of the load above the equilibrium position in position 2.

Thread deflection angle mathematical pendulum from the vertical (Fig.10.15) is determined by the expression

cos α = l - h l = 1 - h l,

where l is the length of the thread; h is the height of the load lifting above the equilibrium position.

Maximum angle deviation α max is determined by the maximum lifting height of the load above the equilibrium position h max:

cos α max = 1 - h max l.

Example 11. The period of small oscillations of a mathematical pendulum is 0.9 s. At what maximum angle from the vertical will the thread deviate if, passing through the equilibrium position, the ball moves at a speed of 1.5 m / s? There is no friction in the system.

Solution . The figure shows two positions of the mathematical pendulum:

• equilibrium position 1 (characterized by the maximum speed of the ball v max);
• extreme position 2 (characterized by the maximum height of the ball rise h max above the equilibrium position).

The desired angle is determined by the equality

cos α max = l - h max l = 1 - h max l,

where l is the length of the pendulum thread.

We find the maximum height of the pendulum ball rise above the equilibrium position from the law of conservation of total mechanical energy.

The total energy of the pendulum in the equilibrium position and in the extreme position is determined by the following formulas:

• in equilibrium position -

E 1 = m v max 2 2,

where m is the mass of the ball of the pendulum; v max is the modulus of the ball's velocity in the equilibrium position (maximum speed), v max = 1.5 m / s;

• in extreme position -

E 2 = mgh max,

where g is the modulus of gravitational acceleration; h max is the maximum height of the ball rise above the equilibrium position.

The law of conservation of total mechanical energy:

m v max 2 2 = m g h max.

Let us express from this the maximum height of the ball rise above the equilibrium position:

h max = v max 2 2 g.

The length of the thread is determined from the formula for the oscillation period of a mathematical pendulum

T = 2 π l g,

those. thread length

l = T 2 g 4 π 2.

Substitute h max and l in the expression for the cosine of the desired angle:

cos α max = 1 - 2 π 2 v max 2 g 2 T 2

and we will make the calculation taking into account the approximate equality π 2 = 10:

cos α max = 1 - 2 ⋅ 10 ⋅ (1.5) 2 10 2 ⋅ (0.9) 2 = 0.5.

It follows that the maximum deflection angle is 60 °.

Strictly speaking, at an angle of 60 °, the oscillations of the ball are not small and it is inappropriate to use the standard formula for the oscillation period of a mathematical pendulum.

Conservation of energy during oscillations of a spring pendulum

Total mechanical energy of a spring pendulum consists of kinetic energy and potential energy:

E = W k + W p,

where W k - kinetic energy, W k = mv 2/2; W p - potential energy, W p = k (Δx) 2/2; m is the mass of the cargo; v - modulus of cargo speed; k - coefficient of rigidity (elasticity) of the spring; Δx - deformation (tension or compression) of the spring (Fig. 10.16).

In the International System of Units, the energy of a mechanical oscillatory system is measured in joules (1 J).

With harmonic vibrations, the spring pendulum passes through a number of successive states, therefore it is advisable to consider the energy of the spring pendulum in three positions (see Fig.10.16):

1) in equilibrium position(1) the speed of the body has a maximum value v max, so the kinetic energy is also maximum:

W k max = m v max 2 2;

the potential energy of the spring is zero, since the spring is not deformed; the total energy coincides with the maximum kinetic energy:

E = W k max;

2) in extreme position(2) the spring has a maximum deformation (Δx max), so the potential energy also has a maximum value:

W p max = k (Δ x max) 2 2;

the kinetic energy of the body is zero; the total energy coincides with the maximum potential energy:

E = W p max;

3) in intermediate position(3) the body has an instantaneous velocity v, the spring has at this moment some deformation (Δx), therefore the total energy is the sum

E = m v 2 2 + k (Δ x) 2 2,

where mv 2/2 - kinetic energy; k (Δx) 2/2 - potential energy; m is the mass of the cargo; v - modulus of cargo speed; k - coefficient of rigidity (elasticity) of the spring; Δx - deformation (tension or compression) of the spring.

When the load of the spring pendulum is displaced from the equilibrium position, it is acted upon by restoring force, the projection of which on the direction of movement of the pendulum is determined by the formula

F x = −kx,

where x is the displacement of the weight of the spring pendulum from the equilibrium position, x = ∆x, ∆x is the deformation of the spring; k - coefficient of rigidity (elasticity) of the pendulum spring.

With harmonic oscillations of a spring pendulum, the total mechanical energy is conserved:

E = const.

The values ​​of the total energy of the spring pendulum in its three positions are shown in Table. 10.2.

№	Position	W p	W k	E = W p + W k
1	Equilibrium	0	m v max 2/2	m v max 2/2
2	Extreme	k (Δx max) 2/2	0	k (Δx max) 2/2
3	Intermediate (instant)	k (Δx) 2/2	mv 2/2	mv 2/2 + k (Δx) 2/2

The values ​​of the total mechanical energy presented in the last column of the table have equal values ​​for any position of the pendulum, which is a mathematical expression total mechanical energy conservation law:

m v max 2 2 = k (Δ x max) 2 2;

m v max 2 2 = m v 2 2 + k (Δ x) 2 2;

k (Δ x max) 2 2 = m v 2 2 + k (Δ x) 2 2,

where m is the mass of the cargo; v is the modulus of the instantaneous speed of the load in position 3; Δx - deformation (tension or compression) of the spring in position 3; v max - modulus of maximum cargo speed in position 1; Δx max - maximum deformation (tension or compression) of the spring in position 2.

Example 12. A spring pendulum performs harmonic oscillations. How many times is its kinetic energy greater than the potential at the moment when the displacement of the body from the equilibrium position is a quarter of the amplitude?

Solution . Let's compare the two positions of the spring pendulum:

• extreme position 1 (characterized by the maximum displacement of the pendulum load from the equilibrium position x max);
• intermediate position 2 (characterized by intermediate values ​​of displacement from the equilibrium position x and velocity v →).

The total energy of the pendulum in the extreme and intermediate positions is determined by the following formulas:

• in extreme position -

E 1 = k (Δ x max) 2 2,

where k is the coefficient of rigidity (elasticity) of the spring; ∆x max - vibration amplitude (maximum displacement from the equilibrium position), ∆x max = A;

• in an intermediate position -

E 2 = k (Δ x) 2 2 + m v 2 2,

where m is the mass of the load of the pendulum; ∆x - displacement of the load from the equilibrium position, ∆x = A / 4.

The total mechanical energy conservation law for a spring pendulum is as follows:

k (Δ x max) 2 2 = k (Δ x) 2 2 + m v 2 2.

We divide both sides of the written equality by k (∆x) 2/2:

(Δ x max Δ x) 2 = 1 + m v 2 2 ⋅ 2 k Δ x 2 = 1 + W k W p,

where W k is the kinetic energy of the pendulum in an intermediate position, W k = mv 2/2; W p is the potential energy of the pendulum in an intermediate position, W p = k (∆x) 2/2.

Let us express the required energy ratio from the equation:

W k W p = (Δ x max Δ x) 2 - 1

and calculate its value:

W k W p = (A A / 4) 2 - 1 = 16 - 1 = 15.

At the specified time, the ratio of the kinetic and potential energies of the pendulum is 15.

A mechanical system that consists of a material point (body) hanging on an inextensible weightless thread (its mass is negligible compared to the weight of a body) in a uniform gravity field is called a mathematical pendulum (another name is an oscillator). There are other types of this device. A weightless rod can be used instead of a thread. A mathematical pendulum can clearly reveal the essence of many interesting phenomena. With a small amplitude of oscillation, its movement is called harmonic.

General information about the mechanical system

The formula for the oscillation period of this pendulum was derived by the Dutch scientist Huygens (1629-1695). This contemporary of I. Newton was very fond of this mechanical system. In 1656 he created the first pendulum clock. They measured time with exceptional precision for those times. This invention became the most important stage in the development of physical experiments and practical activities.

If the pendulum is in a balanced position (hanging vertically), it will be balanced by the thread tension force. A plane pendulum on an inextensible thread is a system with two degrees of freedom with a constraint. When you change just one component, the characteristics of all its parts change. So, if the thread is replaced with a rod, then this mechanical system will have only 1 degree of freedom. What properties does a mathematical pendulum have? In this simplest system, chaos arises under the influence of periodic disturbances. In the case when the suspension point does not move, but oscillates, a new equilibrium position appears at the pendulum. With rapid up and down vibrations, this mechanical system takes on a stable upside-down position. It also has its own name. It is called the Kapitsa pendulum.

Pendulum properties

The mathematical pendulum has very interesting properties. All of them are confirmed by well-known physical laws. The period of oscillation of any other pendulum depends on various circumstances, such as the size and shape of the body, the distance between the suspension point and the center of gravity, and the distribution of mass relative to a given point. That is why determining the period of a hanging body is a rather difficult task. It is much easier to calculate the period of a mathematical pendulum, the formula of which will be given below. As a result of observations of such mechanical systems, it is possible to establish the following patterns:

If, keeping the same length of the pendulum, we suspend different weights, then the period of their oscillations will be the same, although their masses will be very different. Consequently, the period of such a pendulum does not depend on the mass of the load.

If, when starting the system, the pendulum is deflected by not too large, but different angles, then it will begin to oscillate with the same period, but at different amplitudes. As long as the deviations from the center of equilibrium are not too great, the oscillations in their form will be close enough to harmonic ones. The period of such a pendulum does not depend in any way on the oscillatory amplitude. This property of this mechanical system is called isochronism (translated from the Greek "chronos" - time, "isos" - equal).

Period of the mathematical pendulum

This indicator represents a period Despite the complex wording, the process itself is very simple. If the length of the thread of the mathematical pendulum is L, and the acceleration of gravity is g, then this value is equal to:

The period of small natural oscillations does not in any way depend on the mass of the pendulum and the amplitude of oscillations. In this case, the pendulum moves like a mathematical one with a reduced length.

Oscillations of a mathematical pendulum

A mathematical pendulum oscillates, which can be described by a simple differential equation:

x + ω2 sin x = 0,

where x (t) is an unknown function (this is the angle of deviation from the lower equilibrium position at time t, expressed in radians); ω is a positive constant, which is determined from the parameters of the pendulum (ω = √g / L, where g is the acceleration of gravity and L is the length of the mathematical pendulum (suspension).

The equation of small vibrations near the equilibrium position (harmonic equation) looks like this:

x + ω2 sin x = 0

Oscillatory movements of the pendulum

A mathematical pendulum that makes small oscillations moves along a sinusoid. The second-order differential equation meets all the requirements and parameters of such a motion. To determine the trajectory, it is necessary to set the speed and coordinate, from which independent constants are then determined:

x = A sin (θ 0 + ωt),

where θ 0 is the initial phase, A is the vibration amplitude, ω is the cyclic frequency determined from the equation of motion.

Mathematical pendulum (formulas for large amplitudes)

This mechanical system, which oscillates with a significant amplitude, obeys more complex laws of motion. For such a pendulum, they are calculated by the formula:

sin x / 2 = u * sn (ωt / u),

where sn is the Jacobi sine, which for u< 1 является периодической функцией, а при малых u он совпадает с простым тригонометрическим синусом. Значение u определяют следующим выражением:

u = (ε + ω2) / 2ω2,

where ε = E / mL2 (mL2 is the energy of the pendulum).

Determination of the oscillation period of a nonlinear pendulum is carried out according to the formula:

where Ω = π / 2 * ω / 2K (u), K is an elliptic integral, π - 3,14.

Pendulum movement along the separatrix

A separatrix is ​​the trajectory of a dynamical system with a two-dimensional phase space. The mathematical pendulum moves along it non-periodically. At an infinitely distant moment of time, it falls from the extreme upper position to the side with zero speed, then gradually picks it up. Ultimately, it stops, returning to its original position.

If the amplitude of oscillations of the pendulum approaches the number π , this indicates that the motion on the phase plane approaches the separatrix. In this case, under the influence of a small forcing periodic force, the mechanical system exhibits chaotic behavior.

When the mathematical pendulum deviates from the equilibrium position with a certain angle φ, the tangential force of gravity Fτ = -mg sin φ arises. The minus sign means that this tangent component is directed in the direction opposite to the deviation of the pendulum. When x denotes the displacement of a pendulum along an arc of a circle with a radius L, its angular displacement is φ = x / L. The second law for projections and forces will give the desired value:

mg τ = Fτ = -mg sin x / L

Based on this ratio, it can be seen that this pendulum is a nonlinear system, since the force that tends to return it to the equilibrium position is always proportional not to the displacement x, but to sin x / L.

Only when the mathematical pendulum performs small oscillations is it a harmonic oscillator. In other words, it becomes a mechanical system capable of performing harmonic vibrations. This approximation is practically valid for angles of 15-20 °. Oscillations of a pendulum with large amplitudes are not harmonic.

Newton's Law for Small Oscillations of a Pendulum

If a given mechanical system performs small vibrations, Newton's 2nd law will look like this:

mg τ = Fτ = -m * g / L * x.

Based on this, we can conclude that the mathematical pendulum is proportional to its displacement with a minus sign. This is the condition due to which the system becomes a harmonic oscillator. The modulus of the aspect ratio between displacement and acceleration is equal to the square of the angular frequency:

ω02 = g / L; ω0 = √ g / L.

This formula reflects the natural frequency of small oscillations of this type of pendulum. Based on this,

T = 2π / ω0 = 2π√ g / L.

Calculations based on the law of conservation of energy

The properties of a pendulum can also be described using the law of conservation of energy. It should be borne in mind that the pendulum in the gravity field is equal to:

E = mg∆h = mgL (1 - cos α) = mgL2sin2 α / 2

Full is equal to kinetic or maximum potential: Epmax = Ekmsx = E

After the law of conservation of energy is written down, take the derivative of the right and left sides of the equation:

Since the derivative of constants is 0, then (Ep + Ek) "= 0. The derivative of the sum is equal to the sum of the derivatives:

Ep "= (mg / L * x2 / 2)" = mg / 2L * 2x * x "= mg / L * v + Ek" = (mv2 / 2) = m / 2 (v2) "= m / 2 * 2v * v "= mv * α,

hence:

Mg / L * xv + mva = v (mg / L * x + m α) = 0.

Based on the last formula, we find: α = - g / L * x.

Practical application of the mathematical pendulum

Acceleration varies with latitude because the density of the earth's crust is not the same across the planet. Where rocks with higher density occur, it will be slightly higher. The acceleration of a mathematical pendulum is often used for geological exploration. Various minerals are searched for in it. Simply by counting the number of oscillations of the pendulum, you can find coal or ore in the bowels of the Earth. This is due to the fact that such fossils have a density and mass greater than the loose rocks underlying them.

The mathematical pendulum was used by such outstanding scientists as Socrates, Aristotle, Plato, Plutarch, Archimedes. Many of them believed that this mechanical system could influence the fate and life of a person. Archimedes used a mathematical pendulum in his calculations. Nowadays, many occultists and psychics use this mechanical system to fulfill their prophecies or search for missing people.

The famous French astronomer and naturalist K. Flammarion also used a mathematical pendulum for his research. He claimed that with his help he was able to predict the discovery of a new planet, the appearance of the Tunguska meteorite and other important events. During World War II, a specialized Pendulum Institute worked in Germany (Berlin). Nowadays, the Munich Institute of Parapsychology is engaged in similar research. The employees of this institution call their work with the pendulum "radioesthesia".

Definition

Mathematical pendulum is an oscillatory system, which is a special case of a physical pendulum, the entire mass of which is concentrated at one point, the center of mass of the pendulum.

Usually a mathematical pendulum is represented as a ball suspended on a long, weightless and inextensible thread. It is an idealized system that vibrates harmonically under the influence of gravity. A good approximation of a mathematical pendulum is a massive small ball oscillating on a thin long string.

Galileo was the first to study the properties of a mathematical pendulum, considering the swing of a chandelier on a long chain. He found that the period of oscillation of a mathematical pendulum does not depend on the amplitude. If, when starting the mint, you deflect it at different small angles, then its oscillations will occur with the same period, but with different amplitudes. This property is called isochronism.

Equation of motion of a mathematical pendulum

The mathematical pendulum is a classic example of a harmonic oscillator. It performs harmonic oscillations, which are described by the differential equation:

\ [\ ddot (\ varphi) + (\ omega) ^ 2_0 \ varphi = 0 \ \ left (1 \ right), \]

where $ \ varphi $ is the angle of deflection of the thread (suspension) from the equilibrium position.

The solution to equation (1) is the function $ \ varphi (t): $

\ [\ varphi (t) = (\ varphi) _0 (\ cos \ left ((\ omega) _0t + \ alpha \ right) \ left (2 \ right), \) \]

where $ \ alpha $ is the initial phase of oscillations; $ (\ varphi) _0 $ - vibration amplitude; $ (\ omega) _0 $ - cyclic frequency.

The oscillation of a harmonic oscillator is an important example of periodic motion. The oscillator serves as a model in many problems of classical and quantum mechanics.

Cyclic frequency and period of oscillation of a mathematical pendulum

The cyclic frequency of a mathematical pendulum depends only on the length of its suspension:

\ [\ (\ omega) _0 = \ sqrt (\ frac (g) (l)) \ left (3 \ right). \]

The oscillation period of the mathematical pendulum ($ T $) in this case is:

Expression (4) shows 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 acceleration of gravity.

Energy equation for a mathematical pendulum

When considering the vibrations of mechanical systems with one degree of freedom, it is often not Newton's equation of motion that is taken as the initial one, but the energy equation. Since it is easier to compose, and it is an equation of the first order in time. Let's assume that there is no friction in the system. The law of conservation of energy for a mathematical pendulum performing free oscillations (small oscillations) can be written as:

where $ E_k $ is the kinetic energy of the pendulum; $ E_p $ - potential energy of the pendulum; $ v $ is the speed of the pendulum; $ x $ - linear displacement of the weight of the pendulum from the equilibrium position along an arc of a circle of radius $ l $, while the angle - displacement is related to $ x $ as:

\ [\ varphi = \ frac (x) (l) \ left (6 \ right). \]

The maximum value of the potential energy of a mathematical pendulum is:

Maximum kinetic energy:

where $ h_m $ is the maximum lifting height of the pendulum; $ x_m $ - maximum deviation of the pendulum from the equilibrium position; $ v_m = (\ omega) _0x_m $ - maximum speed.

Examples of tasks with a solution

Example 1

Exercise. What is the maximum lifting height of the ball of a mathematical pendulum if its speed of movement when passing the equilibrium position was $ v $?

Solution. Let's make a drawing.

Let the potential energy of the ball be zero in its equilibrium position (point 0). At this point, the ball's velocity is maximum and is equal to $ v $ by the condition of the problem. At the point of maximum ascent of the ball over the equilibrium position (point A), the ball's velocity is zero, the potential energy is maximum. Let us write the law of conservation of energy for the considered two positions of the ball:

\ [\ frac (mv ^ 2) (2) = mgh \ \ left (1.1 \ right). \]

From equation (1.1) we find the desired height:

Answer.$ h = \ frac (v ^ 2) (2g) $

Example 2

Exercise. What is the acceleration of gravity if a mathematical pendulum with length $ l = 1 \ m $ oscillates with a period equal to $ T = 2 \ s $? Consider the oscillations of a mathematical pendulum small. \ Textit ()

Solution. As a basis for solving the problem, we take the formula for calculating the period of small oscillations:

Let us express the acceleration from it:

Let's calculate the acceleration of gravity:

Answer.$ g = 9.87 \ \ frac (m) (s ^ 2) $

Mathematical pendulum is a material point suspended on a weightless and inextensible thread located in the gravity field of the Earth. A mathematical pendulum is an idealized model that correctly describes a real pendulum only under certain conditions. A real pendulum can be considered mathematical if the length of the thread is much greater than the dimensions of the body suspended from it, the weight of the thread is negligible compared to the mass of the body, and the deformations of the thread are so small that they can be completely neglected.

In this case, the oscillatory system is formed by a thread, a body attached to it and the Earth, without which this system could not serve as a pendulum.

where a NS – acceleration, g - acceleration of gravity, NS- offset, l Is the length of the pendulum thread.

This equation is called the equation of free oscillations of a mathematical pendulum. It correctly describes the considered fluctuations only when the following assumptions are fulfilled:

2) only small oscillations of a pendulum with a small swing angle are considered.

Free vibrations of any systems in all cases are described by similar equations.

The reasons for free oscillations of a mathematical pendulum are:

1. The action on the pendulum of the force of tension and the force of gravity, which prevents its displacement from the equilibrium position and forces it to descend again.

2. The inertia of the pendulum, due to which it, while maintaining its speed, does not stop in the equilibrium position, but passes through it further.

The period of free oscillations of a mathematical pendulum

The period of free oscillations of a mathematical pendulum does not depend on its mass, but is determined only by the length of the thread and the acceleration of gravity in the place where the pendulum is located.

Conversion of energy with harmonic vibrations

During harmonic oscillations of the spring pendulum, the potential energy of the elastically deformed body transforms into its kinetic energy, where k – coefficient of elasticity, NS - the modulus of displacement of the pendulum from the equilibrium position, m is the mass of the pendulum, v is its speed. According to the harmonic vibration equation:

, .

Total energy of the spring pendulum:

.

Total energy for a mathematical pendulum:

In the case of a mathematical pendulum

Energy transformations during oscillations of a spring pendulum occur in accordance with the law of conservation of mechanical energy ( ). When the pendulum moves down or up from the equilibrium position, its potential energy increases, while its kinetic energy decreases. When the pendulum passes the equilibrium position ( NS= 0), its potential energy is zero and the kinetic energy of the pendulum has the greatest value, equal to its total energy.

Thus, in the process of free oscillations of the pendulum, its potential energy is converted into kinetic, kinetic into potential, potential then back into kinetic, etc. But the total mechanical energy remains unchanged.

Forced vibrations. Resonance.

Oscillations occurring under the action of an external periodic force are called forced hesitation... An external periodic force, called forcing, imparts additional energy to the oscillatory system, which is used to replenish energy losses due to friction. If the driving force changes in time according to the sine or cosine law, then the forced oscillations will be harmonic and undamped.

Unlike free oscillations, when the system receives energy only once (when the system is removed from the equilibrium state), in the case of forced oscillations, the system continuously absorbs this energy from a source of external periodic force. This energy makes up for the losses spent on overcoming friction, and therefore the total energy of the oscillatory system no remains unchanged.

The frequency of the forced vibrations is equal to the frequency of the driving force... In the case when the frequency of the driving force υ coincides with the natural frequency of the oscillating system υ 0 , there is a sharp increase in the amplitude of forced oscillations - resonance. Resonance arises due to the fact that when υ = υ 0 an external force, acting in time with free oscillations, is always co-directed with the speed of the oscillating body and performs positive work: the energy of the oscillating body increases, and the amplitude of its oscillations becomes large. The graph of the dependence of the amplitude of forced vibrations A T on the frequency of the driving force υ shown in the figure, this graph is called the resonance curve:

The phenomenon of resonance plays an important role in a number of natural, scientific and industrial processes. For example, it is necessary to take into account the phenomenon of resonance when designing bridges, buildings and other structures that experience vibration under load, otherwise, under certain conditions, these structures may be destroyed.

If the body, attached to the spring (Figure 4), is deflected from the equilibrium position by a distance A, for example, to the left, then it, having passed through the equilibrium position, will deflect to the right. This follows from the law of conservation of energy.

The potential energy of a compressed or stretched spring is

where k is the stiffness of the spring and x is its elongation. In the extreme left position, the elongation of the spring x = - A, therefore, the potential energy is

The kinetic energy at this moment is equal to zero, because the velocity is equal to zero. This means that potential energy is the total mechanical energy of the system at this moment. If we agree that the friction force is zero, and the other forces are balanced, then our system can be considered closed and its total energy cannot change during motion. When the body in its motion is in the extreme right position (x = A), its kinetic energy will again be equal to zero and the total energy is again equal to the potential one. And the total energy cannot change. Hence, it is again equal to

This means that the body will deviate to the right by a distance equal to A.

In the equilibrium position, on the contrary, the potential energy is zero, because the spring is not deformed, x = 0. In this position, the total energy of the body is equal to its kinetic energy

where m is the mass of the body and is its speed (it is maximum at this moment). But this kinetic energy must also have an equal value. Consequently, during oscillatory motion, the transformation of kinetic energy into potential energy and vice versa occurs. At any point between the positions of equilibrium and maximum deflection, the body has both kinetic energy and potential, but their sum, i.e. the total energy in any position of the body is equal to. The total mechanical energy W of an oscillating body is proportional to the square of the amplitude and its oscillations

Pendulums. Mathematical pendulum

A pendulum is any body suspended so that its center of gravity is below the suspension point. This means that the load suspended on a rope is an oscillatory system similar to the pendulum of a wall clock. Any system capable of free vibrations has a stable equilibrium position. For a pendulum, this is the position in which the center of gravity is on the vertical below the suspension point. If we take the pendulum out of this position or push it, then it will begin to oscillate, deviating in one direction or the other from the equilibrium position. We know that the greatest deviation from the equilibrium position, to which the pendulum reaches, is called the amplitude of oscillations. The amplitude is determined by the initial deflection or push with which the pendulum was set in motion. This property - the dependence of the amplitude on the conditions at the beginning of the movement - is characteristic not only of free oscillations of a pendulum, but in general for free oscillations of very many oscillatory systems.

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 weight relative to this point; therefore, calculating the period of a suspended body is a rather difficult task. The situation is simpler for a mathematical pendulum. A mathematical pendulum is a weight suspended from a thin thread, the dimensions of which are much less than the length of the thread, and its mass of manna is greater than the mass of the thread. This means that the body (load) and the thread must be such that the load can be considered a material point, and the thread is weightless. From observations of such pendulums, the following simple laws can be established.

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

2. Sida, acting on the body at any point of the trajectory, is directed towards the equilibrium position, and at the equilibrium point itself is equal to zero.

3. The force is proportional to the deviation of the body from the equilibrium position.

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4. If, when starting the pendulum, we deflect it at different (but not too large) angles, then it will oscillate with the same period, albeit with different amplitudes. As long as the amplitudes are not too large, the oscillations are close enough in their form to harmonic, 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 (in an Orthodox church, a central chandelier, a lamp with many candles or icon lamps) on a long chain, which was pushed when ignited. During the divine service, the swing swing gradually faded (Chapter 8), that is, the swing amplitude decreased, but the period remained the same. Galileo used his own pulse as an indicator of time.

This property of the pendulum turned out to be not only surprising, but also useful. Galileo proposed using a pendulum as a regulator in a clock. In Galileo's time, clocks were powered by a weight, and a crude device such as windmill blades was used to adjust the stroke, which used air resistance. A pendulum could be used for counting equal time intervals, because small oscillations occur in the same time as large ones caused by random gusts of wind. A century after Galileo, pendulum clocks came into use, but mariners still needed accurate clocks to measure longitude at sea. A prize was announced for the creation of such a marine clock that would allow time to be measured with sufficient accuracy. The award went to Garisson for the chronometer, which used a flywheel (balance) and a special spring to regulate the stroke.

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

When the pendulum swings, the load moves accelerated along the arc VA (Fig. 5, a) under the action of the returning force P 1, which changes during movement.

The calculation of body movement under the influence of a non-constant force is rather complicated. Therefore, for simplicity, we will proceed as follows.

Let us force the pendulum to perform not oscillation in one plane, but to describe the cone so that the load moves in a circle (Fig. 5, b). This movement can be obtained as a result of the addition of two independent vibrations: one - still in the plane of the drawing and the other - in the perpendicular plane. Obviously, the periods of both these plane oscillations are the same, since any oscillation plane is no different from any other. Consequently, the period of complex motion - the revolution of the pendulum along a cone - will be the same as the period of swing in one plane. This conclusion can be easily illustrated by direct experiment, 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 returning force P 1 is directed along the radius of the BC circle, that is, it is equal to the centripetal force:

On the other hand, from the similarity of the triangles OBC and DBE, it follows that BE: BD = CB: OB. Since OB = l, CB = r, BE = P 1, hence

Equating both expressions Р 1 to each other, we get for the speed of circulation

Finally, substituting this into the expression for the period T, we find

So, the period of a mathematical pendulum depends only on the acceleration of gravity g and on the length of the pendulum l, that is, the distance from the suspension point to the center of gravity of the load. It follows from the obtained formula that the period of the pendulum does not depend on its mass and amplitude (provided that it is small enough). In other words, those basic laws that were established earlier from observations were obtained by calculating.

But this theoretical conclusion gives us more: it allows us to establish a quantitative relationship between the period of the pendulum, its length and the acceleration of gravity. The period of a mathematical pendulum is proportional to the square root of the ratio of the length of the pendulum to the acceleration of gravity. The aspect ratio is 2?

The dependence of the period of the pendulum on the acceleration of gravity is a very accurate way of determining this acceleration. Having measured the length of the pendulum l and having determined the period T from a large number of oscillations, we can calculate using the resulting formula g. This method is widely used in practice.

pendulum oscillation resonance coordinate