Earth, stand and suspended body (see Fig. 3) form a oscillatory system called a physical pendulum. Racks, two springs and the body M (see Fig. 4) form a oscillatory system, which is usually called a horizontal spring pendulum. All oscillatory systems are inherent in a number of common properties. Consider the main ones.
1 Each oscillatory system has a state of stable equilibrium. The physical pendulum is a position in which the center of the suspension body is located on one vertical with the point of the suspension. The vertical spring pendulum is a position in which the force of gravity is equalized by the force of elasticity of the spring. The horizontal spring pendulum is a position in which both springs are deformed the same.
2 After the oscillatory system is derived from the position of a stable equilibrium, a force returning the system to a steady position. The origin of this force may be different. Thus, the physical pendulum is the resultant F of gravity G and the strength of the elasticity T (Fig. 5), and spring pendulum is the power of the elasticity of the springs (Fig. 6).
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3 Returning to a steady state, the oscillating system cannot immediately stop. In the mechanical oscillatory systems, the inertness of the oscillating body interferes. The listed properties lead to the fact that if the oscillatory system in one way or another to derive from the state of stable equilibrium, then there will be fluctuations in it in the absence of external forces. The fluctuations arising could continue indefinitely for a long time if there were no friction in the oscillatory system (resistance). It is these, ideal oscillatory systems we will consider in many cases. The ideal oscillatory system has two decisive signs:
a) there is no friction (resistance) and, therefore, no irreversible energy transformations;
b) the parameters of such an oscillatory system (the length of the thread, the mass of the oscillating body, the rigidity of the spring) is constant.
An example of an ideal vibrational system can serve as a so-called mathematical pendulum, which is a load of small sizes, suspended on a flexible weightless and unpretentious spring. The length of the thread and the mass of the cargo in the process of oscillation of the pendulum remain unchanged. If you consider the thread infinitely thin and perfectly flexible, and the size of the cargo is infinitely small, point, then the friction will not be friction in the oscillations of the mathematical pendulum.
In real oscillatory systems there are friction, and the system parameters in the process of oscillatory movement change slightly. So, the pendulum, which is a cargo of final sizes, suspended on a silk thread, cannot be considered in the full sense of the ideal vibrational system, since in the process of its oscillatory movement there is a resistance of the air and friction at the point of the suspension, and the length of the thread changes (albeit very insignificant) . But with small oscillations of such a pendulum, the resistance of the air is small, and the length of the thread changes so insignificant that this pendulum can be considered an almost ideal oscillatory system. This also applies to the spring pendulum. It can be considered an ideal oscillatory system if the mass of the oscillating body and the rigidity of the spring is constant, and the friction so little that can not be considered.
1 Free oscillations.Oscillations occurring in an oscillatory system that is not subject to periodic external forces is called free oscillations. For the occurrence of free oscillations to the oscillatory system, it should be carried out from outside the short-term impact that displays the system from the equilibrium state (deviation from the average position of the pendulum, clamped in the vice of the steel line, strings, etc.).
2 Oscillogram of oscillations. If the pendulum cargo will serve as a vascular with ink, in which there is a narrow hole, then with pendulum oscillations.
Mechanical oscillations – these are movements that are definitely or approximately repeated at certain time intervals. (eg, vibration of branches on a tree, a pendulum of hours, a car on springs and so on)
Oscillations are there free and forced.
Oscillations arising in the system under the action of internal forces are calledfree . All free oscillations faded. (eg: string oscillation, after hitting)
Fluctuations made by bodies under the action of external periodically changing forces are calledforced (eg: oscillation of a metal blank when working with a blacksmith hammer).
The conditions for the emergence of free oscillations :
- When the body is removed from the equilibrium position in the system, force should occur, seeking to return it to the equilibrium position;
- Friction forces in the system should be very small (i.e., strive for zero).
… E. kin → E. R → E. kin →…
On the example of fluctuations of the body on the thread see turning energy . In 1 position, we observe the equilibrium of the oscillatory system. Speed \u200b\u200band, therefore, the kinetic energy of the body is maximal. With the deviation of the pendulum from the equilibrium position, it rises to height h. relative to the zero level, therefore, at the point and the pendulum has potential energy E R. . When moving to the position of equilibrium, to the point O, the height is reduced to zero, and the speed of cargo increases, and at the point of all potential energy E R. will turn into kinetic energy E Kin . In the equilibrium position, the kinetic energy has the maximum value, and the potential energy is minimal. After passing the position of the equilibrium on inertia, the conversion of kinetic energy into the potential, the speed of the pendulum decreases and at the maximum
Definition
Oscillatory move - This is a movement, accurately or approximately repeated after the same intervals, in which the body is repeatedly and in different directions the position passes.
The oscillatory movement along with the progressive and rotational is one of the species.
The physical system (or body), in which vibrations occur during the deviation from the equilibrium position, is called a oscillating system. Figure 1 shows examples of oscillatory systems: a) thread + ball + land; b) load + spring; c) stretched string.
Fig.1. Examples of oscillatory systems: a) thread + ball + earth; b) load + spring; c) stretched string
If there are no losses associated with action in the oscillatory system, the oscillations will continue indefinitely. Such oscillatory systems are called ideal. In real oscillatory systems, there are always loss of energy caused by the forces of resistance, as a result of which the oscillations cannot continue indefinitely, i.e. are fading.
Free oscillations are fluctuations arising in the system under the influence of domestic forces. - oscillations arising in the system under the action of external periodic.
The conditions for the emergence of free oscillations in the system
- the system should be in the stable position: with the deviation of the system from the equilibrium position, the force should occur, seeking to return the system to the equilibrium position - returning;
- the presence of an excess mechanical energy compared with its energy in the equilibrium position;
- the excessive, obtained by the system by shifting it from the equilibrium position, should not be fully spent on overcoming the friction forces when returning to the equilibrium position, i.e. The system must be small enough.
Examples of solving problems
Example 1.
| The task | Which of the above movements are an example of mechanical oscillations: a) the movement of dragonfly wings; b) the movement of a parachute, descending to the ground; c) the movement of the earth around the Sun; d) the movement of grass in the wind; e) the movement of the ball at the bottom of the spherical bowl; g) Move the swing? In which cases of oscillations are forced and why? |
| Answer | An example is the following cases: a) the movement of dragonfly wings; d) the movement of grass in the wind; e) the movement of the ball at the bottom of the spherical bowl; g) the movement of the swing. In all these cases, the body makes movement repeating over time, passing the same provisions in direct and in reverse order. Earth, turning around the sun, makes a repeating movement, but it does not change the direction of its movement, so the case c) the movement of the earth around the Sun; Not an example of mechanical oscillations. Forced oscillations are cases a) movement of dragonfly wings; and d) the movement of grass in the wind. In both cases, the oscillations are performed under the action of external force (in the first case, the forces of the muscles of the dragonfly, in the second case - the strength of the wind). In the case of g) the movement of the swing swings will be forced if there is a swing from time to time. If you bring the swing from the position of the equilibrium and release, the oscillations will be free. |
Example 2.
| The task | The oscillations of which of the bodies below will be free: a) the piston in the engine cylinder; b) a sewing machine needle; c) the branch of the tree after the bird flew from it; d) string of musical instrument; e) the end of the compass arrow; e) phone membrane during conversation; g) bowls of lever scales? |
| Answer | The oscillations will be free in cases: c) the tree branch after the bird flew from it; d) string of musical instrument; e) the end of the arrow of the compass and g) bowls of lever scales. In all these cases, the external effort only displays the system from the equilibrium position, the fluctuations in the system are performed under the influence of domestic forces. In cases of c), and d), this is the strength of elasticity, in the case of e) - the power of the magnetic field of the Earth in the case of g) is |
(or own oscillations) - these are oscillations of the oscillatory system, performed only due to the initially reported energy ( potential or kineti-Chesky) In the absence of external influences.
Potential or kinetic energy can be reported, for example, in mechanical systems through the initial displacement or initial speed.
Free fluid fluctuating bodies always interact with other bodies and together with them type the system of bodies, which is called oscillatory system.
For example, a spring, a ball and a vertical stand, to which the upper end of the spring is attached (see fig. Below) are included in the oscillatory system. Here the ball freely slides in the string (friction force is negligible). If you take the ball to the right and give it yourself, it will make free oscillations near the position of the equilibrium (points ABOUT) Due to action forces of elasticity Springs aimed at the position of equilibrium.
Another classic example of a mechanical oscillatory system is a mathematical pendulum (see Fig. Below). In this case, the ball performs free fluctuations under the action of two forces: forces of gravity And the strengths of the thread (the oscillatory system also includes the Earth). Their referring is aimed at the position of equilibrium.
Forces acting between the bodies of the oscillatory system are called internal forces. External forces Called the forces acting on the system from the side of the bodies that are not included in it. From this point of view, free oscillations can be defined as oscillations in the system under the action of the internal forces after the system is derived from the equilibrium position.
The conditions for the emergence of free oscillations are:
1) the emergence of force in them returning the system to the position of a stable equilibrium, after it was derived from this state;
2) no friction in the system.
Dynamics of free oscillations.
Body fluctuations under the action of elasticity. The equation of the oscillatory movement of the body under the action of the force of elasticity F. () can be obtained taking into account second law Newton (F \u003d MA) I. law Guka. (F UPR \u003d -KX), where m. - the mass of the ball, and - the acceleration acquired by the ball under the action of the force of elasticity, k. - Spring rigidity coefficient, h. - Body displacement from position equilibrium (both equations are recorded in the projection on the horizontal axis Oh). Equating the right parts of these equations and considering the acceleration but - This is the second derivative of the coordinate h. (displacements), we get:
.
Similar to acceleration expression but We get differentiating (v \u003d -V M SIN ω 0 T \u003d -V M x M cos (ω 0 t + π / 2)):
a \u003d -a m cos ω 0 t,
where a m \u003d ω 2 0 x m - Acceleration amplitude. Thus, the amplitude of the speed of harmonic cola is proportional to the frequency, and the amplitude of the acceleration is the square of the oscillation frequency.
