Belarus transformation of energy at harmonic oscillations. Forced vibrations
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.
Problem solving requires the application of Newton's second law, Hooke's law and kinematic formulas 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 by which the force of elasticity 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?
fluctuations are any processes or movements that repeat at regular intervals.
Free vibrations arise in the system under the action of its internal forces after being removed from the equilibrium position.
Conditions for the occurrence of free oscillations:
1 . After removing the system from the equilibrium position, a force must arise that seeks to return it to the equilibrium position;
2 . Friction and resistance in the system should be sufficiently small.
Harmonic vibrations- these are periodic changes in a physical quantity depending on time, occurring according to the law of sine or cosine.
damped vibrations are oscillations that occur when friction and resistance forces in the system are taken into account.
Oscillation amplitude (A) is the modulus of the greatest displacement of the body from the equilibrium position.
Oscillation period (T) is the time for one complete oscillation. The unit of measure is [c].
T = t /N , where t is the time, N is the number of oscillations.
Oscillation frequency (ν) is the number of oscillations per unit time.
The unit of measure is [Hz].
Cyclic (circular) frequency (ω 0) is the number of oscillations in 2π seconds. Units of measurement - [rad / s]. ω 0 = 2π ν = 2π/Т.
Harmonic vibration equation x \u003d A sin (ω 0 t + φ 0), x \u003d A cos (ω 0 t + φ 0),
φ - initial phase (units - [glad]).
Examples of harmonic oscillations are the oscillations of mathematical and spring pendulums.
Mathematical pendulum is a material point suspended on a long weightless inextensible thread. The scheme of forces acting on a mathematical pendulum is shown in the figure.
F \u003d F t + F control
For a mathematical pendulum, the cyclic frequency
oscillations ω 0 = √g/l
oscillation period Т = 2π√l/g,
where l is the length of the thread,
g is the free fall acceleration.
Spring pendulum is a body of mass m, oscillating on a spring with a stiffness coefficient k. For spring pendulum
cyclic oscillation frequency ω 0 = √k / m,
oscillation period Т = 2π√m / k.
When the springs are connected in series, the overall stiffness coefficient
to total = (k 1 ∙ k 2) / (k 1 + k 2).
With a parallel connection of the springs, the total stiffness coefficient k total \u003d k 1 + k 2.
The law of conservation of energy during harmonic oscillations:
E max sweat = E sweat + E kin = E max kin;
where E max sweat is the maximum potential energy,
E sweat - potential energy,
E kin - kinetic energy,
E max kin - maximum kinetic energy.
Forced vibrations are vibrations that occur under the action of an external, periodically acting force. Forced oscillations are characterized by the phenomenon of resonance.
Resonance is a sharp increase in amplitude
forced oscillations at coincidence
the frequency of the action of an external force with a frequency
natural oscillations of the system.
An increase in the amplitude of forced
vibrations at resonance is expressed by
more distinct, the less friction in the system.
Curve 2 in the figure corresponds to
more friction in the system,
curve 1 - less friction. Rice. 14.12
Self-oscillations are called oscillations that are undamped due to the presence of an energy source inside the system. Systems in which self-oscillations exist are called self-oscillating systems. In this case, the energy supply to the oscillatory system is regulated by the system itself with the help of a regulator via the feedback channel.
Mechanical vibrations propagate in elastic media. If any particle of the medium begins to oscillate, then due to the interaction between the particles of the medium, the oscillations begin to propagate in all directions, hence a wave arises.
Wave are vibrations that propagate in space over time.
The wave is called longitudinal, if particle oscillations occur along the direction of wave propagation. Longitudinal waves can propagate in solid, liquid and gaseous media.
The wave is called transverse, if particle oscillations occur perpendicular to the direction of wave propagation. Transverse waves can only propagate in a solid medium.
Wavelength (λ)- this is the distance between two points closest to each other, oscillating in the same phases. In one period, a wave propagates in space over a distance equal to the wavelength.
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.
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§ 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 . Injection 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.
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 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.
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§ 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 . Injection 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.
