Vibrations and waves
A. Amplitude
B. cyclic frequency
C. initial phase
Initial phase of harmonic oscillations material point defines
A. oscillation amplitude
B. deviation of the point from the equilibrium position at the initial moment of time
C. period and frequency of oscillation
D. maximum speed when the point passes through the equilibrium position
E. full stock mechanical energy points
3 For harmonic oscillation, shown in the figure, the oscillation frequency is ...
The body performs harmonic oscillations with a circular frequency of 10 s-1. If the body, when it passes the equilibrium position, has a speed of 0.2 m / s, then the amplitude of the body's oscillations is equal to
5. Which of the following statements is correct:
A. With harmonic vibrations, the restoring force
B. Directly proportional to displacement.
C. Inversely proportional to displacement.
D. Proportional to the square of the offset.
E. Does not depend on bias.
6. The equation of free harmonic undamped oscillations has the form:
7. The equation of forced oscillations has the form:
8. The equation of free damped oscillations has the form:
9. True (and) is (are) the following of the following expressions:
A. The damping coefficient of harmonic damped oscillations does not depend on the kinematic or dynamic viscosity of the medium in which such oscillations occur.
B. The natural frequency of oscillations is equal to the frequency of damped oscillations.
C. The damped oscillation amplitude is a function of time (A(t)).
D. Damping breaks the periodicity of oscillations, so damped oscillations are not periodic.
10. If the mass of a load of 2 kg, suspended on a spring and performing harmonic oscillations with a period T, is increased by 6 kg, then the oscillation period will become equal to ...
11. The speed of passage of the equilibrium position by a load of mass m, oscillating on a spring of stiffness k with oscillation amplitude A, is equal to ...
12. The mathematical pendulum made 100 oscillations in 314 C. The length of the pendulum is ...
13. The expression that determines the total energy E of the harmonic oscillation of a material point has the form ...
Which of the following quantities remain unchanged in the process of harmonic oscillations: 1) speed; 2) frequency; 3) phase; 4) period; 5) potential energy; 6) total energy.
D. all values change
Indicate all the correct statements. 1) Mechanical vibrations can be free and forced. 2) Free vibrations can occur only in an oscillatory system. 3) Free vibrations can occur not only in an oscillatory system. 4) Forced oscillations can occur only in an oscillatory system. 5) Forced oscillations can occur not only in an oscillatory system. 6) Forced oscillations can occur cannot occur in an oscillatory system.
A. All statements are true
B. 3, 6, 8 and 7
E. All statements are not true
What is called the amplitude of oscillations?
A. Offset.
B. Deflection of bodies A.
C. Movement of bodies A.
D. The greatest deviation of the body from the equilibrium position.
What is the letter for frequency?
What is the speed of the body when passing through the equilibrium position?
A. Equal to zero.
C. Minimal A.
D. Max A.
What are the properties of oscillatory motion?
A. Save.
B. Change.
C. Repeat.
D. Slow down.
E. Answers A - D are not correct
What is the period of oscillation?
A. Time of one complete oscillation.
B. Oscillation time until the body stops completely A.
C. The time taken to deflect the body from the equilibrium position.
D. Answers A - D are not correct.
What letter represents the period of oscillation?
What is the speed of the body when passing the point of maximum deflection?
A. Equal to zero.
B. The same for any position of the bodies A.
C. Minimal A.
D. Max A.
E. Answers A - E are not correct.
What is the value of the acceleration at the point of equilibrium?
A. Max.
B. Minimum.
C. The same for any position of the bodies A.
D. Equal to zero.
E. Answers A - E are not correct.
The oscillating system is
A. a physical system in which there are fluctuations when deviating from the equilibrium position
B. a physical system in which oscillations occur when deviated from the equilibrium position
C. a physical system in which, when deviated from the equilibrium position, oscillations arise and exist
D. a physical system in which oscillations do not arise and do not exist when deviated from the equilibrium position
The pendulum is
A. a body suspended by a thread or spring
AT. solid, which, under the action of applied forces, oscillates
C. None of the answers are correct.
D. a rigid body that, under the action of applied forces, oscillates around a fixed point or around an axis.
Choose the correct answer(s) to the following question: What determines the frequency of oscillation of a spring pendulum? 1) on its mass; 2) on the acceleration of free fall; 3) on the stiffness of the spring; 4) on the amplitude of oscillations?
Indicate which of the following waves are longitudinal: 1) sound waves in gases; 2) ultrasonic waves in liquids; 3) waves on the surface of water; 4) radio waves; 5) light waves in transparent crystals
Which of the following parameters determines the period of oscillations mathematical pendulum: 1) the mass of the pendulum; 2) thread length; 3) free fall acceleration at the location of the pendulum; 4) oscillation amplitudes?
The sound source is
A. any oscillating body
B. bodies oscillating with a frequency of more than 20,000 Hz
C. bodies oscillating with a frequency of 20 Hz to 20,000 Hz
D. bodies oscillating at a frequency below 20 Hz
49. The volume of sound is determined by ...
A. the amplitude of the vibrations of the sound source
B. the frequency of vibration of the sound source
C. the period of oscillation of the sound source
D. the speed of the sound source
What kind of wave is sound?
A. longitudinal
B. transverse
S. has a longitudinal-transverse character
53. To find the speed of sound, you need ...
A. wavelength divided by the frequency of the sound source
B. wavelength divided by the period of oscillation of the sound source
C. wavelength times the oscillation period of the sound source
D. Oscillation period divided by wavelength
What is hydromechanics?
A. the science of fluid motion;
V. the science of the equilibrium of liquids;
C. the science of the interaction of fluids;
D. the science of the balance and movement of fluids.
What is a liquid?
A. a physical substance capable of filling voids;
B. a physical substance that can change shape under the influence of force or maintain its volume;
C. a physical substance that can change its volume;
D. a physical substance capable of flowing.
The pressure is determined
A. the ratio of the force acting on the liquid to the area of impact;
B. the product of the force acting on the liquid and the area of impact;
C. the ratio of the area of influence to the value of the force acting on the liquid;
D. the ratio of the difference between the acting forces and the impact area.
Point out the correct statements
A. An increase in the flow velocity of a viscous fluid due to the inhomogeneity of pressure across the cross section of the pipe creates a swirl and the movement becomes turbulent.
B. In turbulent fluid flow, the Reynolds number is less than the critical one.
C. The nature of the flow of fluid through the pipe does not depend on the speed of its flow.
D. Blood is a Newtonian fluid.
Point out the correct statements
A. In a laminar fluid flow, the Reynolds number is less than the critical one.
B. The viscosity of Newtonian fluids does not depend on the velocity gradient.
C. The capillary method for determining viscosity is based on Stokes' law.
D. When the temperature of a liquid rises, its viscosity does not change.
Point out the correct statements
A. When determining the viscosity of a liquid by the Stokes method, the movement of a ball in a liquid must be uniformly accelerated.
B. The Reynolds number is a similarity criterion: when modeling the circulatory system: the correspondence between the model and nature is observed when the Reynolds number is the same for them.
C. Hydraulic resistance is the greater, the lower the viscosity of the liquid, the length of the pipe and the larger its cross-sectional area.
D. If the Reynolds number is less than the critical one, then the fluid motion is turbulent, if it is greater, then it is laminar.
Point out the correct statements
A. Stokes' law is derived under the assumption that the walls of the vessel do not affect the motion of the ball in the liquid.
B. When heated, the viscosity of the liquid decreases.
C. During the flow of a real fluid, its individual layers act on each other with forces perpendicular to the layers.
D. Under given external conditions, the more liquid flows through a horizontal pipe of constant cross section, the greater its viscosity.
02. Electrodynamics
1. Power lines electric field are called:
1. locus of points with the same intensity
2. lines, at each point of which the tangents coincide with the direction of the tension vector
3. lines connecting points with the same intensity
3. An electrostatic field is called:
1. electric field of stationary charges
2. a special kind of matter, through which all bodies that have mass interact
3. a special kind of matter through which all interact elementary particles
1. the energy characteristic of the field, the magnitude of the vector
2. the energy characteristic of the field, the value of the scalar
3. the force characteristic of the field, the value of the scalar
4. the force characteristic of the field, the magnitude of the vector
7. At each point of the electric field created by several sources, the intensity is:
1. algebraic difference of the field strengths of each of the sources
2. algebraic sum field strengths of each source
3. the geometric sum of the field strengths of each of the sources
4. the scalar sum of the field strengths of each of the sources
8. At each point of the electric field created by several sources, the potential of the electric field is:
1. algebraic potential difference of the fields of each of the sources
2. the geometric sum of the potentials of the fields of each of the sources
3. the algebraic sum of the potentials of the fields of each of the sources
10. The unit of measurement of the dipole moment of the current dipole in the SI system is:
13. The work of the electric field to move a charged body from point 1 to point 2 is:
1. product of mass and tension
2. the product of the charge and the potential difference at points 1 and 2
3. product of charge and intensity
4. product of mass and potential difference at points 1 and 2
15. A system of two point electrodes located in a weakly conductive medium with a constant potential difference between them is called:
1. electric dipole
2. current dipole
3. electrolytic bath
16. The source of the electrostatic field are (indicate incorrect):
1. single charges
2. charge systems
3. electric current
4. charged bodies
17. The magnetic field is called:
1. one of the components of electro magnetic field, through which the fixed electric charges
2. a special kind of matter, through which bodies that have mass interact
3. one of the components of the electromagnetic field, through which moving electric charges interact
18. electromagnetic field called:
1. a special kind of matter through which electric charges interact
2. space in which forces act
3. a special kind of matter, through which bodies with mass interact
19. Variable electric shock is called electric current.
1. changing only in magnitude
2. changing both in size and direction
3. the magnitude and direction of which do not change with time
20. The current strength in a sinusoidal alternating current circuit is in phase with the voltage if the circuit consists of:
1. made of ohmic resistance
2. made of capacitance
3. made of inductive reactance
24. The impedance of an alternating current circuit is called:
1. AC circuit impedance
2. reactive component of the AC circuit
3. ohmic component of the AC circuit
27. Current carriers in metals are:
1. electrons
4. electrons and holes
28. Current carriers in electrolytes are:
1. electrons
4. electrons and holes
29. Conductivity of biological tissues is:
1. e
2. perforated
3. ionic
4. electron-hole
31. An irritating effect on the human body has:
1. high frequency alternating current
2. constant current
3. low frequency current
4. all listed species currents
32. A sinusoidal electric current is an electric current in which, according to the harmonic law, it changes over time:
1. amplitude value of the current strength
2. instantaneous current value
3. effective current value
34. Electrophysiotherapy uses:
1. exclusively alternating currents of high frequency
2. exclusively direct currents
3. exclusively impulse currents
4. all listed types of currents
It's called impedance. . .
1. dependence of the circuit resistance on the frequency of the alternating current;
2. active resistance of the circuit;
3. circuit reactance;
4. circuit impedance.
A stream of protons flying in a straight line enters a uniform magnetic field, the induction of which is perpendicular to the direction of particle flight. Which of the trajectories will the flow move in a magnetic field?
1. Around the circumference
2. In a straight line
3. By parabola
4. Along the helix
5. By hyperbole
Using a coil connected to a galvanometer and a bar magnet, Faraday's experiments are modeled. How does the reading of the galvanometer change if the magnet is introduced into the coil first slowly and then much faster?
1. the readings of the galvanometer will increase
2. no change will happen
3. readings of the galvanometer will decrease
4. the galvanometer needle will deviate in the opposite direction
5. everything is determined by the magnetization of the magnet
A resistor, a capacitor, and a coil are connected in series in an alternating current circuit. The amplitude of voltage fluctuations on the resistor is 3 V, on the capacitor 5 V, on the coil 1 V. What is the amplitude of the voltage fluctuations on the three elements of the circuit.
174. An electromagnetic wave is emitted ... .
3. resting charge
4. electric shock
5. other reasons
What is the arm of a dipole?
1. distance between the poles of the dipole;
2. the distance between the poles, multiplied by the amount of charge;
3. The shortest distance from the axis of rotation to the line of action of the force;
4.distance from the axis of rotation to the line of action of the force.
Under the action of a uniform magnetic field, two charged particles rotate in a circle at the same speed. The mass of the second particle is 4 times the mass of the first, the charge of the second particle is twice the charge of the first. How many times the radius of the circle along which the second particle moves is greater than the radius of the first particle?
What is a polarizer.
3. a device that converts natural light into polarized light.
What is polarimetry?
1. transformation of natural light into polarized;
4. rotation of the plane of oscillations of polarized light.
It's called accommodation. . .
1. adaptation of the eye to vision in the dark;
2. adaptation of the eye to a clear vision of objects at various distances;
3. adaptation of the eye to the perception of different shades of the same color;
4. the reciprocal of the threshold brightness.
152. Refractive media of the eye:
1) cornea, anterior chamber fluid, lens, vitreous body;
2) pupil, cornea, anterior chamber fluid, lens, vitreous body;
3) air-cornea, cornea - lens, lens - visual cells.
What is a wave?
1. any process that is more or less exactly repeated at regular intervals;
2. the process of propagation of any oscillations in the medium;
3. change in time shift according to the law of sine or cosine.
What is a polarizer.
1. a device that measures the concentration of sucrose;
2. a device that rotates the plane of oscillations of the light vector;
3. a device that converts natural light into polarized light.
What is polarimetry?
1. transformation of natural light into polarized;
2. a device for determining the concentration of a solution of a substance;
3. method for determining the concentration of optically active substances;
4. rotation of the plane of oscillations of polarized light.
180. Sensors are used for:
1. electrical signal measurements;
2. conversion of biomedical information into an electrical signal;
3. voltage measurement;
4. electromagnetic influence to the object.
181. electrodes are used only to pick up an electrical signal:
182. electrodes are used for:
1. primary amplification of the electrical signal;
2. conversion of the measured value into an electrical signal;
3. electromagnetic impact on the object;
4. removal of biopotentials.
183. Generator sensors include:
1. inductive;
2. piezoelectric;
3. induction;
4. rheostatic.
Match the correct sequence of image formation of an object in a microscope when visually viewed at the distance of best vision: 1) Eyepiece.2) Subject.3) Imaginary image.4) Real image.5) Light source.6) Objective
190. Indicate the correct statement:
1) Laser radiation is coherent, and that is why it is widely used in medicine.
2) As light propagates in a medium with an inverse population, its intensity increases.
3) Lasers create a large radiation power, since their radiation is monochromatic.
4) If an excited particle spontaneously passes to the lower level, then an induced emission of a photon occurs.
1. Only 1, 2 and 3
2. All - 1,2,3 and 4
3. Only 1 and 2
4. Only 1
5. Only 2
192. Electromagnetic wave is emitted... .
1. a charge that moves with acceleration
2. uniformly moving charge
3. resting charge
4. electric shock
5. other reasons
Which of the following conditions lead to electromagnetic waves: 1) Change in time of the magnetic field. 2) The presence of motionless charged particles. 3) The presence of conductors with direct current. 4) The presence of an electrostatic field. 5) Change in time of the electric field.
What is the angle between the main sections of the polarizer and analyzer if the intensity of natural light passing through the polarizer and analyzer has decreased by 4 times? Considering the transparency coefficients of the polarizer and analyzer to be 1, indicate the correct answer.
2. 45 degrees
It is known that the phenomenon of rotation of the plane of polarization consists in the rotation of the plane of oscillations of a light wave by an angle when it passes a distance d in the optical active substance. What is the relationship between the angle of rotation and d for solid optically active bodies?
Match the types of luminescence with the methods of excitation: 1. a - ultraviolet radiation; 2. b - electron beam; 3. c - electric field; 4. d - cathodoluminescence; 5. e - photoluminescence; 6. e - electroluminescence
hell bg ve
18. Properties of laser radiation: a. a wide range; b. monochromatic radiation; in. high beam directivity; d. strong beam divergence; e. coherent radiation;
What is recombination?
1. interaction of an ionizing particle with an atom;
2. transformation of an atom into an ion;
3. interaction of an ion with electrons with the formation of an atom by them;
4. interaction of a particle with an antiparticle;
5. change in the combination of atoms in a molecule.
36. Specify correct statements:
1) An ion is an electrically charged particle formed by the loss or gain of electrons by atoms, molecules, radicals.
2) Ions can have a positive or negative charge that is a multiple of the charge of an electron.
3) The properties of an ion and an atom are the same.
4) Ions can be in a free state or in the composition of molecules.
37. Indicate the correct statements:
1) Ionization - the formation of ions and free electrons from atoms, molecules.
2) Ionization - the transformation of atoms, molecules into ions.
3) Ionization - the transformation of ions into atoms, molecules.
4) Ionization energy - the energy received by an electron in an atom, sufficient to overcome the binding energy with the nucleus and its departure from the atom.
38. Indicate the correct statements:
1) Recombination - the formation of an atom from an ion and an electron.
2) Recombination - the formation of two gamma quanta from an electron and a positron.
3) Annihilation - the interaction of an ion with an electron to form an atom.
4) Annihilation - the transformation of particles and antiparticles as a result of interaction into electromagnetic radiation.
5) Annihilation - the transformation of matter from one form to another, one of the types of interconversion of particles.
48. Indicate the type of ionizing radiation, the quality factor of which has highest value:
1. beta radiation;
2. gamma radiation;
3. x-rays;
4. alpha radiation;
5. neutron flux.
The degree of oxidation of the patient's blood plasma was studied by luminescence. Plasma containing, among other components, blood lipid oxidation products capable of luminescence was used. For a certain time interval, the mixture, having absorbed 100 quanta of light with a wavelength of 410 nm, highlighted 15 quanta of radiation with a wavelength of 550 nm. What is the luminescence quantum yield of this blood plasma?
Which of listed properties refer to thermal radiation: 1-electromagnetic nature of radiation, 2-radiation can be in equilibrium with the radiating body, 3-continuous frequency spectrum, 4-discrete frequency spectrum.
1. Only 1, 2 and 3
2. All - 1,2,3 and 4
3. Only 1 and 2
4. Only 1
5. Only 2
What formula is used to calculate the probability of the opposite event if the probability P(A) of the event A is known?
A. P(Aav) = 1 + P(A);
B. P(Aav) = P(A) P(Aav A);
C. P(Aav) = 1 - P(A).
Which formula is correct?
A. P(ABC) = P(A)P(B/A)P(BC);
B. P (ABC) \u003d P (A) P (B) P (C);
C. P (ABC) \u003d P (A / B) P (B / A) P (B / C).
43. The probability of occurrence of at least one of the events A1, A2, ..., An, independent of each other, is equal to
A. 1 - (P(A1) P(A2)P ... P(An));
B. 1 – (Р(А1) Р(А2/А1)Р ·… · Р(Аn));
S. 1 - (P(Aav1) P(Aav2)P ... P(Aavn)).
The device has three independently installed trouble signaling devices. The probability that in case of an accident the first will work is 0.9, the second - 0.7, the third - 0.8. Find the probability that none of the alarms go off during an accident.
62. Nikolai and Leonid perform test. Nikolai has a 70% probability of error in calculations, and 30% for Leonid. Find the probability that Leonid makes a mistake and Nikolay does not.
63. The music school is recruiting students. The probability of not being enrolled during the test of musical ear is 40%, and the sense of rhythm is 10%. What is the probability of a positive test?
64. Each of the three shooters shoots at the target once, and the probability of hitting 1 shooter is 80%, the second - 70%, the third - 60%. Find the probability that only the second shooter hits the target.
65. There are fruits in the basket, among which 30% are bananas and 60% are apples. What is the probability that a fruit chosen at random is a banana or an apple?
The district doctor received 35 patients during the week, of which five patients were diagnosed with stomach ulcers. Determine the relative frequency of appearance at the reception of a patient with stomach disease.
76. Events A and B are opposite, if P (A) \u003d 0.4, then P (B) \u003d ...
D. There is no correct answer.
77. If events A and B are incompatible and P (A) \u003d 0.2 and P (B) \u003d 0.05, then P (A + B) \u003d ...
78. If P(B/A) = P(B), then events A and B:
A. reliable;
B. opposite;
C. dependent;
D. there is no correct answer
79. The conditional probability of an event A under the condition is written as:
Vibrations and waves
In the harmonic oscillation equation, the value under the cosine sign is called
A. Amplitude
B. cyclic frequency
C. initial phase
E. offset from the equilibrium position
fluctuations called such processes in which the system repeatedly passes through the equilibrium position with a greater or lesser frequency.
Oscillation classification:
a) by nature (mechanical, electromagnetic, fluctuations in concentration, temperature, etc.);
b) in form (simple = harmonic; complex, which are the sum of simple harmonic vibrations);
in) according to the degree of periodicity = periodic (characteristics of the system are repeated after a strictly defined period of time (period)) and aperiodic;
G) in relation to time (undamped = constant amplitude; damped = decreasing amplitude);
G) energy – free (single input of energy into the system from outside = single external impact); forced (multiple (periodic) supply of energy to the system from the outside = periodic external influence); self-oscillations (undamped oscillations arising due to the system's ability to regulate the flow of energy from a constant source).
Conditions for the occurrence of oscillations.
a) The presence of an oscillatory system (a pendulum on a suspension, a spring pendulum, an oscillatory circuit, etc.);
b) The presence of an external source of energy that is able to bring the system out of equilibrium at least once;
c) The emergence of a quasi-elastic restoring force in the system (i.e., a force proportional to the displacement);
d) Presence of inertia (inertial element) in the system.
As an illustrative example, consider the movement of a 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 α
there is a tangential component of gravity F=-
mg
sinα.
The minus sign in this formula means that the tangential component is directed in the direction opposite to the pendulum deflection. She is a restoring force. At small angles α (of the order of 15-20 o), this force is proportional to the displacement of the pendulum, i.e. is quasi-elastic, and the oscillations of the pendulum are harmonic.
When the pendulum is deflected, it rises to a certain height, i.e. he is given a certain amount of potential energy ( E sweat = mgh). When the pendulum moves to the equilibrium position, the transition of potential energy into kinetic energy occurs. At the moment when the pendulum passes the equilibrium position, the potential energy is equal to zero, and the kinetic energy is maximum. Due to the presence of mass m(mass is a physical quantity that determines the inertial and gravitational properties of matter) the pendulum passes the equilibrium position and deviates in the opposite direction. In the absence of friction in the system, the pendulum will continue to oscillate indefinitely.
The harmonic oscillation equation has the form:
x(t) = x m cos (ω 0 t +φ 0 ),
where X- displacement of the body from the equilibrium position;
x m (BUT) is the oscillation amplitude, that is, the maximum displacement modulus,
ω 0 - cyclic (or circular) frequency of oscillations,
t- time.
The value under the cosine sign φ = ω 0 t + φ 0 called phase harmonic vibration. Phase determines the offset at a given time t. The phase is expressed in angular units (radians).
At t= 0 φ = φ 0 , That's why φ 0 called initial phase.
The period of time after which certain states of the oscillatory system are repeated is called period of oscillation T.
The physical quantity reciprocal to the period of oscillation is called oscillation frequency: 
. Oscillation frequency ν
shows how many oscillations are made per unit of time. Frequency unit - hertz (Hz) - unicycle per second.
Oscillation frequency ν
related to cyclic frequency ω
and oscillation period T ratios: 
.
That is, the circular frequency is the number of complete oscillations that occur in 2π units of time.
Graphically, harmonic oscillations can be represented as a dependence X from t and the method of vector diagrams.
The method of vector diagrams allows you to visualize all the parameters included in the equation of harmonic oscillations. Indeed, if the amplitude vector BUT placed at an angle φ to the axis X, then its projection onto the axis X will be equal to: x = Acos(φ ) . Injection φ and eat initial phase. If the vector BUT put into rotation with angular velocityω 0 , equal to the circular frequency of oscillations, then the projection of the end of the vector will move along the axis X and take values ranging from -A before +A, and the coordinate of this projection will change over time according to the law: x(t) = BUTcos(ω 0 t+ φ) . The time it takes for the amplitude vector to make one complete revolution is equal to the period T harmonic vibrations. The number of revolutions of the vector per second is equal to the oscillation frequency ν .
The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.
So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) makes on vertical screen(Fig. 1) harmonic oscillatory motion.
The displacement from the equilibrium position during harmonic vibrations is described by the equation (it is called the kinematic law harmonic movement) of the form:
where x - displacement - a value characterizing the position of the oscillating point at time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given time; A - oscillation amplitude - the maximum displacement of the body from the equilibrium position; T - oscillation period - the time of one complete oscillation; those. the shortest amount of time after which values are repeated physical quantities characterizing the oscillation; - initial phase;
The phase of the oscillation at time t. The oscillation phase is an argument of a periodic function, which, for a given oscillation amplitude, determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.
If at the initial moment of time the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law
If the oscillating point at is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law
The value V, the reciprocal of the period and equal to the number of complete oscillations performed in 1 s, is called the oscillation frequency:
If in time t the body makes N complete oscillations, then
the value 
, showing how many oscillations the body makes in s, is called cyclic (circular) frequency.
The kinematic law of harmonic motion can be written as:
Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).
Figure 2, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case .
Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:
where is the amplitude of the velocity projection on the x-axis.
This formula shows that during harmonic oscillations, the projection of the body velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by (Fig. 2, b).
To find out the dependence of acceleration, we find the time derivative of the velocity projection:
where is the amplitude of the acceleration projection on the x-axis.
For harmonic oscillations, the acceleration projection leads the phase shift by k (Fig. 2, c).
Changes in time according to a sinusoidal law:
where X- the value of the fluctuating quantity at the moment of time t, BUT- 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 BUT, ω and φ - permanent.
For mechanical vibrations fluctuating value X are, in particular, displacement and speed, for electrical oscillations - voltage and current strength.
Harmonic vibrations occupy a special place among all types of vibrations, since this is the only type of vibration whose shape is not distorted when passing through any homogeneous medium, i.e., waves propagating from a source of harmonic vibrations 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 Wk 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 — maximum speed body (in equilibrium position), x m = BUT- 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.
Have mathematical expression. Their properties are characterized by a set trigonometric equations, the complexity of which is determined by the complexity of the oscillatory process itself, the properties of the system and the environment in which they occur, i.e., external factors affecting the oscillatory process.
For example, in mechanics, a harmonic oscillation is a movement that is characterized by:
Rectilinear character;
unevenness;
The movement of a physical body that occurs along a sinusoidal or cosine trajectory, and depends on time.
Based on these properties, we can bring the equation of harmonic oscillations, which has the form:
x \u003d A cos ωt or the form x \u003d A sin ωt, where x is the value of the coordinate, A is the value of the oscillation amplitude, ω is the coefficient.
Such an equation of harmonic oscillations is the main one for all harmonic oscillations that are considered in kinematics and mechanics.
The indicator ωt, which in this formula is under the sign trigonometric function, is called a phase, and it determines the location of an oscillating material point at a given specific moment in time with a given amplitude. When considering cyclical fluctuations, this indicator is equal to 2l, it shows the amount within the time cycle and is denoted by w. In this case, the equation of harmonic oscillations contains it as an indicator of the magnitude of the cyclic (circular) frequency.
The equation of harmonic oscillations we are considering, as already noted, can take various forms, depending on a number of factors. For example, here is an option. To consider free harmonic oscillations, one should take into account the fact that they are all characterized by damping. This phenomenon manifests itself in different ways: the stopping of a moving body, the cessation of radiation in electrical systems. The simplest example showing a decrease in the vibrational potential is its conversion into thermal energy.
The equation under consideration is: d²s / dt² + 2β x ds / dt + ω²s \u003d 0. In this formula: s is the value of the oscillating quantity that characterizes the properties of a particular system, β is a constant showing the attenuation coefficient, ω is the cyclic frequency.
The use of such a formula allows one to approach the description of oscillatory processes in linear systems from a unified point of view, as well as to design and simulate oscillatory processes at the scientific and experimental level.
For example, it is known that final stage their manifestations already cease to be harmonic, that is, the categories of frequency and period for them become simply meaningless and are not reflected in the formula.
The classic way to study harmonic oscillations is In its simplest form, it represents a system that is described by such differential equation harmonic oscillations: ds/dt + ω²s = 0. But the variety of oscillatory processes naturally leads to the fact that there is a large number of oscillators. We list their main types:
A spring oscillator is an ordinary load with a certain mass m, which is suspended on an elastic spring. It performs harmonic type, which are described by the formula F = - kx.
Physical oscillator (pendulum) - a rigid body that oscillates around a static axis under the influence of a certain force;
- (almost never occurs in nature). It is an ideal model of a system that includes an oscillating physical body with a certain mass, which is suspended on a rigid weightless thread.
