Properties of a spring pendulum
Definition 1
An ideal spring pendulum is a spring, the mass of which can be neglected, with a body with a point mass attached to it. In this case, one or both ends of the spring are fixed, and the friction force can be neglected.
Such a construction can be considered only as a mathematical model. Examples of real spring pendulums (cylindrical spirals wound from elastic wire) are all kinds of devices that dampen vibrations: shock absorbers, suspensions, springs, etc. Spring pendulums, although of a slightly different design (in the form of flat spirals) are used in mechanical watches.
The properties of springs depend on the substance from which they are made (as a rule, it is a special spring steel), the diameter of the wire, the shape of its section, the diameter of the spring cylinder, and its length. Together, these indicators determine the key characteristic of the spring - its stiffness.
The spring stores energy during longitudinal tension or compression due to elastic deformations in the crystal lattice of its substance.
Remark 1
When stretched or compressed too much, the spring material loses its elastic properties. Such deformation is called plastic or residual.
Formula for calculating the oscillation frequency
If a spring with a load attached to it is subjected to longitudinal elastic deformation and then released, it will begin to perform reciprocating harmonic oscillations, during which the movement of the load attached to it is described by the formula:
$x = A \cdot \cos(\omega_0 \cdot t + \phi)$
Here $A$ is the oscillation amplitude, $\phi$ is the initial phase, $\omega_0$ is the natural cyclic oscillation frequency of the spring pendulum, calculated as
$\omega_0 = \sqrt(\frac(k)(m))$ > $0$,
- $k$ - spring stiffness,
- $m$ is the mass of the body fixed on it.
The cyclic frequency differs in that it characterizes not the number of complete cycles per unit of time, but the number of radians "passed" by the point oscillating according to the harmonic law.
The oscillation period of a spring pendulum is calculated as
Free vibrations are made under the action of the internal forces of the system after the system has been brought out of equilibrium.
In order to free vibrations were made according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position should be proportional to the displacement of the body from the equilibrium position and directed in the direction opposite to the displacement (see § 2.1):
Forces of any other physical nature that satisfy this condition are called quasi-elastic .
Thus, a load of some mass m attached to the stiffening spring k, the second end of which is fixed motionless (Fig. 2.2.1), constitute a system capable of performing free harmonic oscillations in the absence of friction. The mass on the spring is called linear harmonic oscillator.
The circular frequency ω 0 of free vibrations of a load on a spring is found from Newton's second law:
With a horizontal arrangement of the spring-load system, the force of gravity applied to the load is compensated by the reaction force of the support. If the load is suspended on a spring, then the force of gravity is directed along the line of movement of the load. In the equilibrium position, the spring is stretched by an amount x 0 equal to
Therefore, Newton's second law for a load on a spring can be written as
Equation (*) is called the equation of free vibrations . It should be noted that the physical properties of the oscillatory system determine only the natural frequency of oscillations ω 0 or the period T . Such parameters of the oscillation process as amplitude x m and the initial phase φ 0 are determined by the way in which the system was brought out of equilibrium at the initial moment of time.
If, for example, the load was displaced from the equilibrium position by a distance Δ l and then at time t= 0 released without initial speed, then x m = ∆ l, φ 0 = 0.
If, however, the initial speed ± υ 0 was imparted to the load, which was in the equilibrium position, with the help of a sharp push, then,
So the amplitude x m free oscillations and its initial phase φ 0 are determined initial conditions .
There are many varieties of mechanical oscillatory systems that use the forces of elastic deformations. On fig. 2.2.2 shows the angular analogue of a linear harmonic oscillator. A horizontally located disk hangs on an elastic thread fixed in its center of mass. When the disk rotates through an angle θ, a moment of forces arises M elastic torsion strain:
where I = I C - moment of inertia of the disk about the axis passing through the center of mass, ε - angular acceleration.
By analogy with the load on the spring, you can get:
Free vibrations. Mathematical pendulum
Mathematical pendulum called a body of small size, suspended on a thin inextensible thread, the mass of which is negligible compared to the mass of the body. In the equilibrium position, when the pendulum hangs on a plumb line, the force of gravity is balanced by the force of the thread tension. When the pendulum deviates from the equilibrium position by a certain angle φ, a tangential component of gravity appears F τ = - mg sin φ (Fig. 2.3.1). The minus sign in this formula means that the tangential component is directed in the direction opposite to the pendulum deflection.
If denoted by x linear displacement of the pendulum from the equilibrium position along the arc of a circle of radius l, then its angular displacement will be equal to φ = x / l. Newton's second law, written for the projections of the acceleration and force vectors on the direction of the tangent, gives:
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This relation shows that the mathematical pendulum is a complex non-linear system, since the force tending to return the pendulum to its equilibrium position is proportional to the non-displacement x, a
Only in case small fluctuations when close can be replaced by a mathematical pendulum is a harmonic oscillator, i.e., a system capable of performing harmonic oscillations. In practice, this approximation is valid for angles of the order of 15-20°; while the value differs from no more than 2%. Pendulum oscillations at large amplitudes are not harmonic.
For small oscillations of a mathematical pendulum, Newton's second law is written as
This formula expresses natural frequency of small oscillations of a mathematical pendulum .
Hence,
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Any body mounted on a horizontal axis of rotation is capable of performing free oscillations in the gravitational field and, therefore, is also a pendulum. Such a pendulum is called physical (Fig. 2.3.2). It differs from the mathematical one only in the distribution of masses. In a position of stable equilibrium, the center of mass C of the physical pendulum is below the axis of rotation O on the vertical passing through the axis. When the pendulum deviates by an angle φ, a moment of gravity arises, tending to return the pendulum to the equilibrium position:
and Newton's second law for a physical pendulum becomes (see §1.23)
Here ω 0 - natural frequency of small oscillations of a physical pendulum .
Hence,
Therefore, the equation expressing Newton's second law for a physical pendulum can be written as
Finally, for the circular frequency ω 0 of free oscillations of the physical pendulum, the following expression is obtained:
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Energy transformations during free mechanical vibrations
With free mechanical vibrations, the kinetic and potential energies change periodically. At the maximum deviation of the body from the equilibrium position, its velocity, and hence the kinetic energy, vanishes. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a spring, the potential energy is the energy of the elastic deformation of the spring. For a mathematical pendulum, this is the energy in the Earth's gravitational field.
When the body in its motion passes through the equilibrium position, its speed is maximum. The body skips the equilibrium position according to the law of inertia. At this moment, it has the maximum kinetic and minimum potential energy. An increase in kinetic energy occurs at the expense of a decrease in potential energy. With further movement, the potential energy begins to increase due to the decrease in kinetic energy, etc.
Thus, during harmonic oscillations, a periodic transformation of kinetic energy into potential energy and vice versa occurs.
If there is no friction in the oscillatory system, then the total mechanical energy during free vibrations remains unchanged.
For spring load(see §2.2):
In real conditions, any oscillatory system is under the influence of friction forces (resistance). In this case, part of the mechanical energy is converted into the internal energy of the thermal motion of atoms and molecules, and the vibrations become fading (Fig. 2.4.2).
The damping rate of the oscillations depends on the magnitude of the friction forces. The time interval τ during which the oscillation amplitude decreases in e≈ 2.7 times, called decay time .
The frequency of free oscillations depends on the damping rate of the oscillations. As the friction forces increase, the natural frequency decreases. However, the change in the natural frequency becomes noticeable only at sufficiently large friction forces, when the natural oscillations quickly decay.
An important characteristic of an oscillatory system that performs free damped oscillations is quality factor Q. This parameter is defined as a number N total oscillations made by the system during the damping time τ, multiplied by π:
Thus, the quality factor characterizes the relative loss of energy of the oscillatory system due to the presence of friction over a time interval equal to one oscillation period.
Forced vibrations. Resonance. Self-oscillations
Oscillations that occur under the influence of an external periodic force are called forced.
The external force performs positive work and provides an influx of energy to the oscillatory system. It does not allow oscillations to fade, despite the action of friction forces.
A periodic external force can vary in time according to various laws. Of particular interest is the case when an external force, changing according to a harmonic law with a frequency ω, acts on an oscillatory system capable of performing natural oscillations at a certain frequency ω 0 .
If free oscillations occur at a frequency ω 0, which is determined by the parameters of the system, then steady forced oscillations always occur at frequency ω of the external force.
After the beginning of the impact of external force on the oscillatory system, some time Δ t to establish forced oscillations. The settling time is equal in order of magnitude to the decay time τ of free oscillations in the oscillatory system.
At the initial moment, both processes are excited in the oscillatory system - forced oscillations at a frequency ω and free oscillations at a natural frequency ω 0 . But free vibrations are damped due to the inevitable presence of friction forces. Therefore, after some time, only stationary oscillations at the frequency ω of the external driving force remain in the oscillatory system.
Consider, as an example, forced vibrations of a body on a spring (Fig. 2.5.1). An external force is applied to the free end of the spring. It forces the free (left in Fig. 2.5.1) end of the spring to move according to the law
If the left end of the spring is displaced by a distance y, and the right one - at a distance x from their original position, when the spring was not deformed, then the elongation of the spring Δ l equals:
In this equation, the force acting on the body is represented as two terms. The first term on the right side is the elastic force tending to return the body to the equilibrium position ( x= 0). The second term is the external periodic impact on the body. This term is called compelling force.
The equation expressing Newton's second law for a body on a spring in the presence of an external periodic action can be given a rigorous mathematical form, if we take into account the relationship between the acceleration of the body and its coordinate: Then will be written in the form
Equation (**) does not take into account the action of friction forces. Unlike free oscillation equations(*) (see §2.2) forced vibration equation(**) contains two frequencies - the frequency ω 0 of free oscillations and the frequency ω of the driving force.
The steady forced vibrations of the load on the spring occur at the frequency of the external action according to the law
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Amplitude of forced vibrations x m and the initial phase θ depend on the ratio of the frequencies ω 0 and ω and on the amplitude y m external force.
At very low frequencies, when ω<< ω 0 , движение тела массой m, attached to the right end of the spring, repeats the movement of the left end of the spring. Wherein x(t) = y(t), and the spring remains practically undeformed. The external force applied to the left end of the spring does not do work, since the modulus of this force at ω<< ω 0 стремится к нулю.
If the frequency ω of the external force approaches the natural frequency ω 0 , there is a sharp increase in the amplitude of forced oscillations. This phenomenon is called resonance . Amplitude dependence x m forced oscillations from the frequency ω of the driving force is called resonant characteristic or resonance curve(Fig. 2.5.2).
At resonance, the amplitude x m load fluctuations can be many times greater than the amplitude y m oscillations of the free (left) end of the spring, caused by an external action. In the absence of friction, the amplitude of forced oscillations at resonance should increase indefinitely. In real conditions, the amplitude of steady-state forced oscillations is determined by the condition: the work of an external force during the period of oscillations must be equal to the loss of mechanical energy over the same time due to friction. The less friction (i.e., the higher the quality factor Q oscillatory system), the greater the amplitude of forced oscillations at resonance.
For oscillatory systems with a not very high quality factor (< 10) резонансная частота несколько смещается в сторону низких частот. Это хорошо заметно на рис. 2.5.2.
The phenomenon of resonance can cause the destruction of bridges, buildings and other structures, if the natural frequencies of their oscillations coincide with the frequency of a periodically acting force, which has arisen, for example, due to the rotation of an unbalanced motor.
Forced vibrations are undamped fluctuations. The inevitable losses of energy due to friction are compensated by the supply of energy from an external source of a periodically acting force. There are systems in which undamped oscillations arise not due to periodic external influences, but as a result of the ability of such systems to regulate the flow of energy from a constant source. Such systems are called self-oscillating, and the process of undamped oscillations in such systems - self-oscillations . In a self-oscillatory system, three characteristic elements can be distinguished - an oscillatory system, an energy source and a feedback device between the oscillatory system and the source. As an oscillatory system, any mechanical system capable of performing its own damped oscillations (for example, a pendulum of a wall clock) can be used.
The energy source can be the deformation energy of the spring or the potential energy of the load in the gravitational field. The feedback device is a mechanism by which the self-oscillating system regulates the flow of energy from the source. On fig. 2.5.3 shows a diagram of the interaction of various elements of a self-oscillating system.
An example of a mechanical self-oscillating system is a clockwork with anchor move (Fig. 2.5.4). A running wheel with oblique teeth is rigidly fastened to a toothed drum, through which a chain with a weight is thrown. Attached to the upper end of the pendulum anchor(anchor) with two plates of hard material, curved along an arc of a circle with a center on the axis of the pendulum. In a wristwatch, the weight is replaced by a spring, and the pendulum is replaced by a balancer - a handwheel fastened to a spiral spring. The balancer performs torsional vibrations around its axis. The oscillatory system in the clock is a pendulum or balancer.
The source of energy is a weight lifted up or a wound spring. The feedback device is an anchor that allows the running wheel to turn one tooth in one half cycle. Feedback is provided by the interaction of the anchor with the running wheel. With each oscillation of the pendulum, the travel wheel tooth pushes the anchor fork in the direction of the pendulum movement, transferring to it a certain portion of energy, which compensates for the energy losses due to friction. Thus, the potential energy of the weight (or twisted spring) is gradually, in separate portions, transferred to the pendulum.
Mechanical self-oscillatory systems are widespread in the life around us and in technology. Self-oscillations are made by steam engines, internal combustion engines, electric bells, strings of bowed musical instruments, air columns in the pipes of wind instruments, vocal cords when talking or singing, etc.
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| Figure 2.5.4. Clock mechanism with a pendulum. |
An oscillatory movement is any periodically repeating movement. Therefore, the dependences of the coordinate and velocity of the body on time during oscillations are described by periodic functions of time. In the school physics course, such oscillations are considered in which the dependences and velocities of the body are trigonometric functions 
, 
or a combination of them, where is some number. Such oscillations are called harmonic (functions 
and 
often called harmonic functions). To solve problems for oscillations included in the program of the unified state exam in physics, you need to know the definitions of the main characteristics of oscillatory motion: amplitude, period, frequency, circular (or cyclic) frequency and phase of oscillations. Let us give these definitions and connect the enumerated quantities with the parameters of the dependence of the body coordinate on time , which in the case of harmonic oscillations can always be represented as
where , and are some numbers.
The amplitude of oscillation is the maximum deviation of an oscillating body from the equilibrium position. Since the maximum and minimum value of the cosine in (11.1) is equal to ±1, then the amplitude of oscillations of the body that oscillates (11.1) is equal to . The period of oscillation is the minimum time after which the movement of the body repeats. For dependence (11.1), the period can be set from the following considerations. Cosine is a periodic function with period . Therefore, the movement is completely repeated through such a value that . From here we get
Circular (or cyclic) oscillation frequency is the number of oscillations per unit of time. From formula (11.3) we conclude that the circular frequency is the value from formula (11.1).
The oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinate on time. From formula (11.1) we see that the phase of oscillations of the body, the motion of which is described by dependence (11.1), is equal to 
. The value of the oscillation phase at time = 0 is called the initial phase. For dependence (11.1) the initial phase of oscillations is equal to the value . Obviously, the initial phase of the oscillations depends on the choice of the time reference point (moment = 0), which is always conditional. By changing the origin of the time reference, the initial phase of the oscillations can always be "made" equal to zero, and the sine in the formula (11.1) is "turned" into a cosine or vice versa.
The program of the unified state exam also includes knowledge of the formulas for the oscillation frequency of the spring and mathematical pendulums. It is customary to call a spring pendulum a body that can oscillate on a smooth horizontal surface under the action of a spring, the second end of which is fixed (left figure). A mathematical pendulum is a massive body, the dimensions of which can be neglected, oscillating on a long, weightless and inextensible thread (right figure). The name of this system - "mathematical pendulum" is due to the fact that it is an abstract mathematical real model ( physical) of the pendulum. It is necessary to remember the formulas for the period (or frequency) of oscillations of the spring and mathematical pendulums. For spring pendulum
where is the length of the thread, is the free fall acceleration. Consider the application of these definitions and laws on the example of problem solving.
To find the cyclic frequency of the load in task 11.1.1 let us first find the oscillation period, and then use the formula (11.2). Since 10 m 28 s is 628 s, and during this time the load makes 100 oscillations, the period of oscillation of the load is 6.28 s. Therefore, the cyclic oscillation frequency is 1 s -1 (the answer 2 ). AT task 11.1.2 the load made 60 oscillations in 600 s, so the oscillation frequency is 0.1 s -1 (the answer 1 ).
To understand which way the cargo will go in 2.5 periods ( task 11.1.3), follow its movement. After a period, the load will return back to the point of maximum deflection, making a complete oscillation. Therefore, during this time, the load will cover a distance equal to four amplitudes: to the equilibrium position - one amplitude, from the equilibrium position to the point of maximum deviation in the other direction - the second, back to the equilibrium position - the third, from the equilibrium position to the starting point - the fourth. During the second period, the load will again pass four amplitudes, and for the remaining half of the period - two amplitudes. Therefore, the distance traveled is equal to ten amplitudes (the answer 4 ).
The amount of movement of the body is the distance from the start point to the end point. For 2.5 periods in task 11.1.4 the body will have time to complete two full and half full oscillations, i.e. will be at the maximum deviation, but on the other side of the equilibrium position. Therefore, the amount of displacement is equal to two amplitudes (the answer 3 ).
By definition, the phase of oscillations is an argument of a trigonometric function, which describes the dependence of the coordinate of an oscillating body on time. Therefore the correct answer is task 11.1.5 - 3 .
The period is the time of complete oscillation. This means that the return of the body back to the same point from which the body began to move does not mean that the period has passed: the body must return to the same point with the same speed. For example, a body, having started oscillations from an equilibrium position, will have time to deviate by the maximum amount in one direction, go back, deviate to the maximum in the other direction and come back again. Therefore, during the period, the body will have time to deviate twice by the maximum value from the equilibrium position and return back. Therefore, the passage from the equilibrium position to the point of maximum deviation ( task 11.1.6) the body spends the fourth part of the period (the answer 3 ).
Such oscillations are called harmonic, in which the dependence of the coordinate of the oscillating body on time is described by a trigonometric (sine or cosine) function of time. AT task 11.1.7 these are the functions and , despite the fact that the parameters included in them are denoted as 2 and 2 . The function is the trigonometric function of the square of time. Therefore, fluctuations of only quantities and are harmonic (the answer 4 ).
With harmonic oscillations, the speed of the body changes according to the law 
, where is the amplitude of the speed oscillations (the time reference is chosen so that the initial phase of the oscillations would be equal to zero). From here we find the dependence of the kinetic energy of the body on time 
(task 11.1.8). Using the well-known trigonometric formula, we obtain
It follows from this formula that the kinetic energy of the body changes during harmonic oscillations also according to the harmonic law, but with a doubled frequency (the answer is 2 ).
Behind the ratio between the kinetic energy of the load and the potential energy of the spring ( task 11.1.9) can be easily traced from the following considerations. When the body is deviated by the maximum amount from the equilibrium position, the speed of the body is zero, and, therefore, the potential energy of the spring is greater than the kinetic energy of the load. In contrast, when the body passes the equilibrium position, the potential energy of the spring is zero, and therefore the kinetic energy is greater than the potential energy. Therefore, between the passage of the equilibrium position and the maximum deviation, the kinetic and potential energies are compared once. And since during the period the body passes four times from the equilibrium position to the maximum deviation or vice versa, then during the period the kinetic energy of the load and the potential energy of the spring are compared with each other four times (the answer is 2 ).
Amplitude of speed fluctuations ( task 11.1.10) is easiest to find by the law of conservation of energy. At the point of maximum deflection, the energy of the oscillatory system is equal to the potential energy of the spring 
, where is the spring stiffness coefficient, is the oscillation amplitude. When passing through the equilibrium position, the energy of the body is equal to the kinetic energy 
, where is the mass of the body, is the speed of the body when passing through the equilibrium position, which is the maximum speed of the body in the process of oscillation and, therefore, represents the amplitude of the speed oscillations. Equating these energies, we find
(answer 4 ).
From formula (11.5) we conclude ( task 11.2.2) that its period does not depend on the mass of the mathematical pendulum, and with an increase in length by 4 times, the oscillation period increases by 2 times (the answer is 1 ).
The clock is an oscillatory process that is used to measure time intervals ( task 11.2.3). The words clock "rush" mean that the period of this process is less than what it should be. Therefore, to clarify the course of these clocks, it is necessary to increase the period of the process. According to formula (11.5), in order to increase the period of oscillation of a mathematical pendulum, it is necessary to increase its length (the answer is 3 ).
To find the amplitude of oscillations in task 11.2.4, it is necessary to represent the dependence of the body coordinate on time in the form of a single trigonometric function. For the function given in the condition, this can be done by introducing an additional angle. Multiplying and dividing this function by 
and using the formula for adding trigonometric functions, we get
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where is an angle such that 
. From this formula it follows that the amplitude of body oscillations is 
(answer 4
).
where k is the coefficient of elasticity of the body, m- weight of cargo
Mathematical pendulum called a system consisting of a material point of mass m, suspended on a weightless inextensible thread that oscillates under the action of gravity (Fig. 5.13, b).
The period of oscillation of a mathematical pendulum
where l is the length of the mathematical pendulum, g is the free fall acceleration.
physical pendulum a rigid body is called, which oscillates under the action of gravity around the horizontal axis of the suspension, which does not pass through the center of mass of the body (Fig. 5.13, c).
,
where J is the moment of inertia of the oscillating body about the oscillation axis; d is the distance of the center of mass of the pendulum from the axis of oscillation; - reduced length of the physical pendulum.
When two identically directed harmonic oscillations of the same period are added, a harmonic oscillation of the same period is obtained with amplitude
Resulting initial phase, obtained by adding two vibrations, :
, (5.50)
where A 1 and A 2 are the amplitudes of the oscillation terms, φ 1 and φ 2 are their initial phases.
When adding two mutually perpendicular oscillations of the same period resulting motion trajectory equation looks like:
If, in addition to the elastic force, a friction force acts on a material point, then the oscillations will be damped, and the equation for such an oscillation will have the form
, (5.52)
where is called the damping factor ( r is the drag coefficient).
The ratio of two amplitudes spaced apart in time equal to the period is called
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Among various electrical phenomena, a special place is occupied by electromagnetic oscillations, in which electrical quantities periodically change and are accompanied by mutual transformations of electric and magnetic fields. It is used to excite and maintain electromagnetic oscillations. oscillatory circuit- a circuit consisting of an inductor L connected in series, a capacitor with a capacitance C and a resistor with a resistance R (Fig. 5.14).
Period T of electromagnetic oscillations in an oscillatory circuit
. (5.54)
If the resistance of the oscillatory circuit is small, i.e.<<1/LC, то период колебаний колебательного контура определяется Thomson's formula
If the circuit resistance R is not equal to zero, then the oscillations will be fading. Wherein potential difference across the capacitor plates changes over time according to the law
, (5.56)
where δ is the attenuation coefficient, U 0 is the amplitude value of the voltage.
Attenuation factor oscillations in the oscillatory circuit
where L is the loop inductance, R is the resistance.
Logarithmic damping decrement is the ratio of two amplitudes spaced apart in time, equal to the period
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Resonance called the phenomenon of a sharp increase in the amplitude of forced oscillations when the frequency of the driving force ω approaches a frequency equal to or close to the natural frequency ω 0 of the oscillatory system (Fig. 5.15.).
Resonance Condition:
. (5.59)
The time interval during which the amplitude of damped oscillations decreases in e times, is called relaxation time
To characterize the attenuation of oscillatory circuits, a quantity called the quality factor of the circuit is often used. Q circuit Q called the number of complete oscillations N, multiplied by the number π, after which the amplitude decreases in e once
. (5.61)
If the damping factor is zero, then the oscillations will be undamped, voltage will change according to the law
. (5.62)
In the case of direct current, the ratio of voltage to current is called the resistance of the conductor. Similarly, with alternating current, the ratio of the amplitude of the active component of the voltage U a to the current amplitude i 0 is called active resistance chain X
In the circuit under consideration, it is equal to the DC resistance. Active resistance always generates heat.
Attitude
. (5.64)
called circuit reactance.
The presence of reactance in the circuit is not accompanied by the release of heat.
Full resistance called the geometric sum of active and reactive resistance
, (5.65)
AC circuit capacitance X c is called the ratio
Inductive reactance
Ohm's law for alternating current is written in the form
where I eff and U ef - effective values of current and voltage associated with their amplitude values I 0 and U 0 by the relations
If the circuit contains active resistance R, capacitance C and inductance L connected in series, then phase shift between voltage and current is determined by the formula
. (5.70)
If the active resistance R and the inductance are connected in parallel in the AC circuit, then circuit impedance is determined by the formula
, (5.71)
and phase shift between voltage and current is determined by the following relation
, (5.72)
where υ is the oscillation frequency.
AC power is determined by the following relation
. (5.73)
Wavelength is related to the period by the following relation
where c=3·10 8 m/s is the speed of sound propagation.
Examples of problem solving
Problem 5.1. Along a piece of straight wire with a length l\u003d 80 cm current flows I \u003d 50 A. Determine the magnetic induction B of the field created by this current at point A, equidistant from the ends of the wire segment and located at a distance r 0 \u003d 30 cm from its middle.
where dB is the magnetic induction created by a wire element of length d l with current I at the point determined by the radius vector r; μ 0 is the magnetic constant, μ is the magnetic permeability of the medium in which the wire is located (in our case, since the medium is air, μ = 1).
Vectors from different current elements are co-directed (Fig.), so expression (1) can be rewritten in scalar form:
where α is the angle between the radius vector and current element dl.
Substituting expression (4) into (3), we obtain
Note that with a symmetrical location of point A relative to the wire segment cos α 2 = - cos α 1 .
With this in mind, formula (7) takes the form
Substituting formula (9) into (8), we obtain
Problem 5.2. Two parallel infinitely long wires D and C, through which currents flow in one direction, electric currents with a force of I \u003d 60 A, are located at a distance d \u003d 10 cm from each other. Determine the magnetic induction of the field created by conductors with current at point A (Fig.), A distance from the axis of one conductor at a distance of r 1 \u003d 5 cm, from the other - r 2 \u003d 12 cm.
We find the modulus of the magnetic induction vector by the cosine theorem:
where α is the angle between the vectors B 1 and B 2 .
Magnetic inductions B 1 and B 2 are expressed, respectively, in terms of current I and distances r 1 and r 2 from the wires to point A:
It can be seen from the figure that α = Ð DAC (as angles with respectively perpendicular sides).
From the triangle DAC, using the cosine theorem, we find cosα
Let's check whether the right side of the obtained equality gives a unit of magnetic field induction (T)
Calculations:
Answer: B = 3.08 10 -4 T.
Problem 5.3. A current I = 80 A flows through a thin conducting ring with a radius R = 10 cm. Find the magnetic induction at point A, equidistant from all points of the ring at a distance r = 20 cm.
determined by the radius vector .
where integration is over all elements d l rings.
Let us decompose the vector dB into two components dB ┴ , perpendicular to the plane of the ring, and dB|| , parallel to the plane of the ring, i.e.
where 
and (because d l is perpendicular to r and hence sinα = 1).
With this in mind, formula (3) takes the form
Let's check whether the right side of equality (5) gives a unit of magnetic induction
Calculations:
Tl.
Answer: B = 6.28 10 -5 T.
Problem 5.4. A long wire with current I = 50 A is bent at an angle α = 2π/3. Determine the magnetic induction at point A (Fig. to problem 5.4., a). Distance d = 5 cm.
The vector is co-directed with the vector and is determined by the right screw rule. In Figure 5.4., b, this direction is marked with a cross in a circle (that is, perpendicular to the drawing plane, from us).
Calculations:
Tl.
Answer: B = 3.46 10 -5 T.
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Task 5.5. Two infinitely long wires are crossed at a right angle (Fig. to problem 5.5., a). Currents I 1 \u003d 80 A and I 2 \u003d 60 A flow through the wires. The distance d between the wires is 10 cm. Determine the magnetic induction B at point A, equally distant from both wires.
| Given: I 1 \u003d 80 A I 2 \u003d 60 A d \u003d 10 cm \u003d 0.1 m | Solution: In accordance with the principle of superposition of magnetic fields, the magnetic induction at point A will be equal to the geometric sum of the magnetic inductions and created by the currents I 1 and I 2 . |
| Find: B - ? |
It follows from the figure that the vectors B 1 and B 2 are mutually perpendicular (their directions are found according to the gimlet rule and are shown in two projections in the figure for problem 5.5.,b).
The strength of the magnetic field, according to (5.8), created by an infinitely long straight conductor,
where μ is the relative magnetic permeability of the medium (in our case, μ = 1).
Substituting formula (2) into (3), we find the magnetic inductions B 1 and B 2 created by currents I 1 and I 2
Substituting formula (4) into (1), we obtain
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Let's check whether the right side of the obtained equality gives a unit of magnetic induction (T):
Calculations:
Tl.
Answer: B = 4 10 -6 T.
Problem 5.6. An infinitely long wire is bent as shown in the figure for problem 5.6, a. Radius R arc of a circle is 10 cm. Determine the magnetic induction of the field created at the point O current I = 80 A flowing through this wire.
In our case, the wire can be divided into three parts (Fig. to problem 5.6, b): two straight wires (1 and 3), with one end going to infinity, and an arc of a semicircle (2) of radius R.
Considering that the vectors are directed in accordance with the gimlet rule perpendicular to the plane of the drawing from us, then the geometric summation can be replaced by the algebraic one:
In our case, the magnetic field at point O is created by only half of this circular current, so
In our case, r 0 = R, α 1 = π/2 (cos α 1 = 0), α 2 → π (cos α 2 = -1).
 |
Let's check whether the right side of the obtained equality gives a unit of magnetic induction (T):
Calculations:
Tl.
Answer: B = 3.31 10 -4 T.
Problem 5.7. On two parallel straight wires of length l= 2.5cm each, distanced d= 20 cm apart, the same currents flow I = 1 kA. Calculate the strength of the interaction of currents.
Current I 1 creates a magnetic field at the location of the second wire (with current I 2). Let's draw a line of magnetic induction (dotted line in the figure) through the second wire and tangentially to it - the vector of magnetic induction B 1.
Figure for task 5.7
The magnetic induction module B 1 is determined by the relation
Since the vector d l is perpendicular to the vector B 1 , then sin(d l,B) = 1 and then
We find the force F of the interaction of wires with current by integrating:
Let's check whether the right side of the resulting equality gives a unit of force (N):
Calculation:
N.
Answer: F = 2.5 N.
Since the Lorentz force is perpendicular to the velocity vector, it will tell the particle (proton) normal acceleration a n.
According to Newton's second law,
| , | (1) |
where m is the proton mass.
In the figure, the trajectory of the proton is aligned with the plane of the drawing and the (arbitrary) direction of the vector is given. We direct the Lorentz force perpendicular to the vector to the center of the circle (vectors a n and F are co-directed). Using the left hand rule, we determine the direction of the magnetic field lines (the direction of the vector ).
A spring pendulum is a material point of mass , attached to an absolutely elastic weightless spring with stiffness 
. There are two simplest cases: horizontal (Fig. 15, a) and vertical (Fig. 15, b) pendulums.
a)
Horizontal pendulum(Fig. 15a). When shifting cargo 
out of equilibrium 
by the amount 
acts on it in a horizontal direction. restoring elastic force
(Hooke's law).
It is assumed that the horizontal support on which the load slides 
during its vibrations, it is absolutely smooth (no friction).
b) vertical pendulum(fig.15, b). The equilibrium position in this case is characterized by the condition:
where 
- the magnitude of the elastic force acting on the load 
when the spring is statically stretched 
under the influence of gravity 
.
|
a |
|
Fig.15. Spring pendulum: a- horizontal and b– vertical
If the spring is stretched and the load is released, it will begin to oscillate vertically. If the offset at some point in time is 
,
then the elastic force will now be written as 
.
In both cases considered, the spring pendulum performs harmonic oscillations with a period
(27)
and cyclic frequency
.
(28)
Using the example of considering a spring pendulum, we can conclude that harmonic oscillations are a movement caused by a force that increases in proportion to the displacement 
. Thus, if the restoring force looks like Hooke's law
(she got the namequasi-elastic force
), then the system must perform harmonic oscillations. At the moment of passing the equilibrium position, the restoring force does not act on the body, however, the body skips the equilibrium position by inertia and the restoring force changes direction to the opposite.
Mathematical pendulum
Fig.16. Mathematical pendulum
, which performs small oscillations under the action of gravity (Fig. 16).
Oscillations of such a pendulum at small deflection angles 
(not exceeding 5º) can be considered harmonic, and the cyclic frequency of the mathematical pendulum:
,
(29)
and the period:
.
(30)
2.3. Body energy during harmonic vibrations
The energy imparted to the oscillatory system during the initial push will be periodically transformed: the potential energy of the deformed spring will be converted into the kinetic energy of the moving load and vice versa.
Let the spring pendulum perform harmonic oscillations with the initial phase 
, i.e. 
(fig.17).
Fig.17. Law of conservation of mechanical energy
when the spring pendulum oscillates
At the maximum deviation of the load from the equilibrium position, the total mechanical energy of the pendulum (the energy of a deformed spring with stiffness 
) is equal to 
. When passing through the equilibrium position ( 
) the potential energy of the spring will become equal to zero, and the total mechanical energy of the oscillatory system will be determined as 
.
Figure 18 shows the dependences of the kinetic, potential and total energy in cases where harmonic oscillations are described by trigonometric functions of the sine (dashed line) or cosine (solid line).
Fig.18. Graphs of the time dependence of the kinetic
and potential energy for harmonic oscillations
From the graphs (Fig. 18) it follows that the frequency of change in kinetic and potential energy is twice as high as the natural frequency of harmonic oscillations.