When studying this topic, they solve problems in kinematics and dynamics of elastic vibrations. It is useful in this case to compare the elastic oscillations with the oscillations of the pendulum already considered in order to reveal both their general and specific features.

Solving problems requires the application of Newton's second law, Hooke's law and formulas for the kinematics of harmonic oscillatory motion.

The period of elastic harmonic oscillations of a body with a mass is determined by the formula (No. 758). This formula allows you to determine the period of various harmonic oscillations, if the value is known. For elastic oscillations, this is the stiffness coefficient, and for oscillations of a mathematical pendulum (No. 748).

In problems of energy transformations in oscillatory motion, one mainly considers the transformation of kinetic energy into potential energy. But for the case of damped oscillations, the transformation of mechanical energy into internal energy is also taken into account. Kinetic energy of elastic vibrations

Potential energy

Will the oscillations of bodies of different masses on the same spring also differ? Check your answer with experience.

Answer. A body of greater mass will have a longer period of oscillation. From the formula it follows that with the same elastic force, a body of greater mass will have less acceleration and, therefore, will move more slowly. This can be checked by oscillating weights of different masses suspended on a dynamometer.

757(e). A weight was hung on the spring and then supported so that the spring would not stretch. Describe how the load will move if the support supporting it is removed. Check your answer with experience.

Solution, Let's release the load to fall freely down. Then he will stretch the spring by an amount that can be determined from the relation

According to the law of conservation of energy, during the reverse upward movement, the load rises to a height will oscillate with an amplitude h. If the load is suspended on a spring, it will stretch it by an amount

Therefore, the position in which the load hangs at rest is the center around which oscillations occur. This conclusion is easy to check on a "soft" long spring, for example, from the "Archimedes' bucket" device.

758. A body with a mass under the action of a spring having rigidity oscillates without friction in a horizontal plane along the rod a (Fig. 238). Determine the period of oscillation of the body using the law of conservation of energy.

Solution. In the extreme position, all the energy of the body is potential, and on the average - kinetic. According to the law of conservation of energy

For the equilibrium position Therefore,

759(e). Determine the stiffness coefficient of the rubber thread and calculate the period of oscillation of the mass suspended on it. Check your answer with experience.

Solution. To answer the vorros problem, students must have a rubber thread, a weight of 100 V, a ruler and a stopwatch.

Having suspended the load on the thread, first calculate the value numerically equal to the force that stretches the thread per unit length. In one of the experiments, the following data were obtained. The initial length of the thread cm, the final Where cm

By measuring the time of 10-20 full oscillations of the load with a stopwatch, they make sure that the period found by the calculations coincides with that obtained from experience.

760. Using the solution of problems 757 and 758, determine the oscillation period of the car on the springs, if its static draft is equal to

Solution.

Consequently,

We have obtained an interesting formula by which it is easy to determine the period of elastic oscillations of the body, knowing only the value

761(e). Using the formula, calculate and then test by experience the oscillation period on the spring from the “Archimedes bucket” of loads weighing 100, 300, 400 g.

762. Using the formula, get the formula for the period of oscillation of a mathematical pendulum.

Solution. For a mathematical pendulum, therefore

763. Using the condition and solution of problem 758, find the law according to which the elastic force of the spring changes, and write down the equations of this harmonic oscillatory motion, if in the extreme position the body had energy

Solution.

Let us assume that the Oscillation amplitude A is determined from the formula

Similarly, substituting the value of mass, amplitude and period into the general formulas for displacement, velocity and acceleration, we obtain:

The acceleration formula could also be obtained using the force formula

764. A mathematical pendulum having mass and length was deflected by 5 cm. What is the speed of acceleration a and potential energy it will have a distance of cm from the equilibrium position?

>> Energy conversion during harmonic vibrations

§24 CONVERSION OF ENERGY AT HARMONIC OSCILLATIONS

Let us consider the transformation of energy during harmonic oscillations in two cases: there is no friction in the system; there is friction in the system.

Energy transformations in systems without friction. By shifting the ball attached to the spring (see Fig. 3.3), to the right by a distance x m, we inform the oscillatory system of potential energy:

When the ball moves to the left, the deformation of the spring becomes smaller, and the potential energy of the system decreases. But at the same time, the speed increases and, consequently, the kinetic energy increases. At the moment the ball passes the equilibrium position, the potential energy of the oscillatory system becomes equal to zero (W n = 0 at x = 0). The kinetic energy reaches its maximum.

After passing the equilibrium position, the speed of the ball begins to decrease. Consequently, the kinetic energy also decreases. The potential energy of the system increases again. At the extreme left point, it reaches a maximum, and the kinetic energy becomes equal to zero. Thus, during oscillations, there is a periodic transition of potential energy into kinetic energy and vice versa. It is easy to see that the same transformations of mechanical energy from one of its forms to another occur in the case of a mathematical pendulum.

The total mechanical energy during vibrations of a body attached to a spring is equal to the sum of the kinetic and potential energies of the oscillatory system:

Kinetic and potential energies change periodically. But the total mechanical energy of an isolated system, in which there are no resistance forces, remains (according to the law of conservation of mechanical energy) unchanged. It is equal to either the potential energy at the moment of maximum deviation from the equilibrium position, or the kinetic energy at the moment when the body passes the equilibrium position:

The energy of an oscillating body is directly proportional to the square of the amplitude of the coordinate oscillations or the square of the amplitude of the velocity oscillations (see formula (3.26)).

damped vibrations. Free vibrations of a weight attached to a spring, or a pendulum, are harmonic only when there is no friction. But the forces of friction, or, more precisely, the forces of resistance environment, although perhaps small, always act on an oscillating body.

The resistance forces do negative work and thereby reduce the mechanical energy of the system. Therefore, over time, the maximum deviations of the body from the equilibrium position become smaller and smaller. In the end, after the supply of mechanical energy is exhausted, the oscillations will stop altogether. Oscillations in the presence of resistance forces are damped.

The plot of body coordinates versus time for damped oscillations is shown in Figure 3.10. A similar graph can be drawn by the oscillating body itself, such as a pendulum.

Figure 3.11 shows a pendulum with a sandbox. A pendulum draws a graph of the dependence of its coordinate on time on a sheet of cardboard moving uniformly under it with a stream of sand. This is a simple method of time sweep of oscillations, which gives a fairly complete picture of the process of oscillatory motion. With a small resistance, the attenuation of oscillations over several periods is small. If, on the other hand, a sheet of thick paper is attached to the suspension threads to increase the resistance force, the attenuation will become significant.

In cars, special ones are used to dampen body vibrations when driving on rough roads. When the body vibrates, the piston associated with it moves in a cylinder filled with liquid. The liquid flows through the holes in the piston, which leads to the appearance of large resistance forces and the rapid damping of oscillations.

The energy of an oscillating body in the absence of friction forces remains unchanged.

If resistance forces act on the bodies of the system, then the oscillations are damped.

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Mechanical oscillations are body movements that repeat exactly or approximately at regular intervals. The main characteristics of mechanical vibrations are: displacement, amplitude, frequency, period. Displacement is the deviation of a body from its equilibrium position. Amplitude - the module of maximum deviation from the equilibrium position. Frequency - the number of complete oscillations per unit time. Period - the time of one complete oscillation, i.e. the minimum period of time after which the process is repeated. The period and frequency are related by the relation: v = 1/T. The simplest type of oscillatory motion is harmonic oscillations, in which the oscillating value changes with time according to the law of sine or cosine (Fig. 9). Free vibrations are those that occur due to the initially imparted energy with the subsequent absence of external influences on the system that oscillates. For example, fluctuations of the load on the thread (Fig. 10). Let us consider the process of energy conversion using the example of load oscillations on a thread (see Fig. 10). When the pendulum deviates from the equilibrium position, it rises to a height h relative to the zero level, therefore, at point A, the pendulum
has a potential energy mgh. When moving to the equilibrium position, to point O, the height decreases to zero, and the speed of the load increases, and at point O all the potential energy mgh will turn into kinetic energy mv ^ 2/2. In the equilibrium position, the kinetic energy is at its maximum and the potential energy is at its minimum. After passing the equilibrium position, the kinetic energy is converted into potential energy, the speed of the pendulum decreases and, at the maximum deviation from the equilibrium position, becomes equal to zero. During oscillatory motion, periodic transformations of its kinetic and potential energy always occur.
With free mechanical vibrations, energy is inevitably lost to overcome the resistance forces. If oscillations occur under the action of a periodic external force, then such oscillations are called forced. For example, parents swing a child on a swing, a piston moves in a car engine cylinder, an electric razor knife and a needle oscillate. sewing machine. The nature of forced oscillations depends on the nature of the action of the external force, on its magnitude, direction, frequency of action and does not depend on the size and properties of the oscillating body. For example, the foundation of the motor, on which it is fixed, performs forced oscillations with a frequency determined only by the number of revolutions of the motor, and does not depend on the dimensions of the foundation.

When the frequency of the external force coincides with the frequency of the natural oscillations of the body, the amplitude of the forced oscillations increases sharply. This phenomenon is called mechanical resonance. Graphically, the dependence of the amplitude of forced oscillations on the frequency of the external force is shown in Figure 11.
The phenomenon of resonance can cause the destruction of machines, buildings, bridges, if their natural frequencies coincide with the frequency of a periodically acting force. Therefore, for example, engines in automobiles are mounted on special shock absorbers, and military units are prohibited from moving along the bridge to keep pace.
In the absence of friction, the amplitude of forced oscillations at resonance should increase indefinitely with time. In real systems, the amplitude in the steady state resonance is determined by the condition of energy losses during the period and the work of the external force for the same time. The less friction, the greater the amplitude at resonance.

When a mathematical pendulum oscillates, the total energy of the system is the sum of the kinetic energy of a material point (ball) and the potential energy of a material point in the field of gravitational forces. When a spring pendulum oscillates, the total energy is the sum of the kinetic energy of the ball and the potential energy of the elastic deformation of the spring:

When passing the equilibrium position both in the first and in the second pendulum, the kinetic energy of the ball reaches its maximum value, the potential energy of the system is equal to zero. During oscillations, the kinetic energy is periodically converted into the potential energy of the system, while the total energy of the system remains unchanged if there are no resistance forces (the law of conservation of mechanical energy). For example, for a spring pendulum one can write:

In an oscillatory circuit (Fig. 14.1.c), the total energy of the system is the sum of the energy of a charged capacitor ( electric field energy) and the energy of a coil with current ( magnetic field energy. When the capacitor charge is maximum, the current in the coil is zero (see formulas 14.11 and 14.12 ), the energy of the electric field of the capacitor is maximum, the energy of the magnetic field of the coil is zero. At the time when the charge of the capacitor is zero, the current in the coil is maximum, the energy of the electric field of the capacitor is zero, the energy of the magnetic field of the coil is maximum. As well as in mechanical oscillators, in the oscillatory circuit, the energy of the electric field is periodically converted into the energy of the magnetic field, while the total energy of the system remains unchanged if there is no active resistance R. You can write:

. (14.15)

If in the process of oscillations external resistance forces act on a mathematical or spring pendulum, and there is active resistance in the oscillating circuit circuit R, the energy of oscillations, and hence the amplitude of oscillations will decrease. Such fluctuations are called damped oscillations , figure 14.2 shows a graph of the dependence of the fluctuating value of X on time.

Rice. 14.3

§ 16. Alternating electric current.

We are already familiar with direct current sources, we know what they are for, we know the laws of direct current. But of much greater practical importance in our lives is alternating electric current, which is used in everyday life, in production and other areas of human activity. The strength of the current and the voltage of the alternating current (for example, in the lighting network of our apartment) change over time according to the harmonic law. The frequency of industrial alternating current is 50 Hz. AC sources are diverse in their design and characteristics. A wire frame rotating in a constant uniform magnetic field can be considered as the simplest model of an alternating current generator. In Fig. 14.3, the frame rotates around the vertical axis OO, perpendicular to the magnetic field lines, with a constant angular velocity . Corner α between the vector and the normal varies according to the law , the magnetic flux through the surface S, limited by the frame, changes with time, an induction emf appears in the frame.

Energy transformations during harmonic vibrations.

When a mathematical pendulum oscillates, the total energy of the system is the sum of the kinetic energy of a material point (ball) and the potential energy of a material point in the field of gravitational forces. When a spring pendulum oscillates, the total energy is the sum of the kinetic energy of the ball and the potential energy of the elastic deformation of the spring:

When passing through the equilibrium position both in the first and in the second pendulum, the kinetic energy of the ball reaches its maximum value, the potential energy of the system is zero. During oscillations, a periodic transformation of kinetic energy into the potential energy of the system occurs, while the total energy of the system remains unchanged if there are no resistance forces (the law of conservation of mechanical energy). For example, for a spring pendulum we can write:

In an oscillatory circuit (Fig. 14.1.c), the total energy of the system is the sum of the energy of a charged capacitor ( electric field energy) and the energy of a coil with current ( magnetic field energy. When the capacitor charge is maximum, the current in the coil is zero (see formulas 14.11 and 14.12 ), the energy of the electric field of the capacitor is maximum, the energy of the magnetic field of the coil is zero. At the time when the charge of the capacitor is zero, the current in the coil is maximum, the energy of the electric field of the capacitor is zero, the energy of the magnetic field of the coil is maximum. As well as in mechanical oscillators, in the oscillatory circuit, the energy of the electric field is periodically converted into the energy of the magnetic field, while the total energy of the system remains unchanged if there is no active resistance R. You can write:

. (14.15)

If during the oscillation process external resistance forces act on a mathematical or spring pendulum, and there is active resistance in the oscillating circuit circuit R, the energy of oscillations, and hence the amplitude of oscillations will decrease. Such fluctuations are called damped oscillations , figure 14.2 shows a graph of the dependence of the fluctuating value of X on time.

Rice. 14.3

§ 16. Alternating electric current.

We are already familiar with direct current sources, we know what they are for, we know the laws of direct current. But of much greater practical importance in our lives is alternating electric current, which is used in everyday life, in production and other areas of human activity. The strength of the current and the voltage of the alternating current (for example, in the lighting network of our apartment) change over time according to the harmonic law. The frequency of industrial alternating current is 50 Hz. AC sources are diverse in their design and characteristics. A wire frame rotating in a constant uniform magnetic field can be considered as the simplest model of an alternating current generator. In Fig. 14.3, the frame rotates around the vertical axis OO, perpendicular to the magnetic field lines, with a constant angular velocity . Corner α between the vector and the normal varies according to the law , the magnetic flux through the surface S, limited by the frame, changes with time, an induction emf appears in the frame.