Change and transformation of energy during harmonic vibrations. Harmonic vibrations. Conversion of energy during harmonic vibrations. Potential energy formula
Changes in time according to a sinusoidal law:
Where X- the value of the fluctuating quantity at the moment of time t, A- amplitude , ω - circular frequency, φ is the initial phase of oscillations, ( φt + φ ) is the total phase of oscillations . At the same time, the values A, ω And φ - permanent.
For mechanical vibrations with an oscillating value X are, in particular, displacement and speed, for electrical oscillations - voltage and current strength.
Harmonic oscillations occupy a special place among all types of oscillations, since this is the only type of oscillation whose shape is not distorted when passing through any homogeneous medium, i.e., waves propagating from a source of harmonic oscillations will also be harmonic. Any non-harmonic vibration can be represented as a sum (integral) of various harmonic vibrations (in the form of a spectrum of harmonic vibrations).
Energy transformations during harmonic vibrations.
In the process of oscillations, there is a transition of potential energy Wp into kinetic W k and vice versa. In the position of maximum deviation from the equilibrium position, the potential energy is maximum, the kinetic energy is zero. As we return to the equilibrium position, the speed of the oscillating body increases, and with it the kinetic energy also increases, reaching a maximum in the equilibrium position. The potential energy then drops to zero. Further-neck movement occurs with a decrease in speed, which drops to zero when the deflection reaches its second maximum. Potential energy here increases to its initial (maximum) value (in the absence of friction). Thus, the oscillations of the kinetic and potential energies occur with a double (compared to the oscillations of the pendulum itself) frequency and are in antiphase (i.e., there is a phase shift between them equal to π ). Total vibration energy W remains unchanged. For a body oscillating under the action of an elastic force, it is equal to:
Where v m- the maximum speed of the body (in the equilibrium position), x m = A- amplitude.
Due to the presence of friction and resistance of the medium, free oscillations damp out: their energy and amplitude decrease with time. Therefore, in practice, not free, but forced oscillations are used more often.
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.
Hence,
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 a mass and a length is deflected by 5 cm. What is its acceleration rate a and what potential energy will it have at a distance cm from the equilibrium position?
53. Conversion of energy during harmonic vibrations. Forced vibrations. Resonance.
When the mathematical pendulum deviates from the equilibrium position, its potential energy increases, because the distance to the earth increases. When moving to the equilibrium position, the speed of the pendulum increases, and the kinetic energy increases, due to a decrease in the potential reserve. In the equilibrium position, kinetic energy is maximum, potential energy is minimum. In the position of maximum deviation - vice versa. With spring - the same, but not the potential energy in the Earth's gravitational field, but the potential energy of the spring is taken. Free vibrations always turn out to be damped, i.e. with decreasing amplitude, because energy is spent on interaction with surrounding bodies. The energy loss in this case is equal to the work of external forces during the same time. The amplitude depends on the frequency of the force change. It reaches its maximum amplitude at the frequency of oscillations of the external force, which coincides with the natural frequency of oscillations of the system. The phenomenon of an increase in the amplitude of forced oscillations under the described conditions is called resonance. Since at resonance, the external force performs the maximum positive work for the period, the resonance condition can be defined as the condition for maximum energy transfer to the system.
54. Propagation of oscillations in elastic media. Transverse and longitudinal waves. Wavelength. Relation of the wavelength to the speed of its propagation. Sound waves. Sound speed. Ultrasound
Excitation of oscillations in one place of the medium causes forced oscillations of neighboring particles. The process of propagation of vibrations in space is called a wave. Waves in which vibrations occur perpendicular to the direction of propagation are called transverse waves. Waves in which vibrations occur along the direction of wave propagation are called longitudinal waves. Longitudinal waves can arise in all media, transverse waves - in solids under the action of elastic forces during deformation or surface tension forces and gravity forces. The speed of propagation of oscillations v in space is called the speed of the wave. The distance l between points closest to each other, oscillating in the same phases, is called the wavelength. The dependence of the wavelength on the speed and period is expressed as , or . When waves occur, their frequency is determined by the source oscillation frequency, and the speed is determined by the medium where they propagate, therefore waves of the same frequency can have different lengths in different media. The processes of compression and rarefaction in the air propagate in all directions and are called sound waves. Sound waves are longitudinal. The speed of sound, like the speed of any wave, depends on the medium. In air, the speed of sound is 331 m/s, in water - 1500 m/s, in steel - 6000 m/s. Sound pressure is additional pressure in a gas or liquid caused by a sound wave. The intensity of sound is measured by the energy carried by sound waves per unit of time through a unit area of a section perpendicular to the direction of wave propagation, and is measured in watts per square meter. The intensity of a sound determines its loudness. The pitch of the sound is determined by the frequency of vibrations. Ultrasound and infrasound are called sound vibrations that lie beyond the limits of hearing with frequencies of 20 kilohertz and 20 hertz, respectively.
55. Free electromagnetic oscillations in the circuit. Energy conversion in an oscillatory circuit. Natural frequency of oscillations in the circuit.
An electrical oscillatory circuit is a system consisting of a capacitor and a coil connected in a closed circuit. When a coil is connected to a capacitor, a current is generated in the coil and the energy of the electric field is converted into the energy of a magnetic field. The capacitor does not discharge instantly, because. this is prevented by the EMF of self-induction in the coil. When the capacitor is completely discharged, the self-induction EMF will prevent the current from decreasing, and the energy of the magnetic field will turn into electric energy. The current arising in this case will charge the capacitor, and the sign of the charge on the plates will be opposite to the original. After that, the process is repeated until all the energy is spent on heating the circuit elements. Thus, the energy of the magnetic field in the oscillatory circuit is converted into electric energy and vice versa. For the total energy of the system, it is possible to write the relations: 
, whence for an arbitrary moment of time 
. As is known, for a complete chain . Assuming that in the ideal case R»0, we finally get , or . The solution to this differential equation is the function 
, Where . The value of w is called its own circular (cyclic) frequency of oscillations in the circuit.
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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 p = 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)).
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 of the 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 fading.
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, then the attenuation will become significant.
In cars, special shock absorbers 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.
