Physics oscillatory motion formula. Mechanical vibrations. Energy conversion in oscillatory systems

4.2. Concepts and definitions of the section “oscillations and waves”

Equation harmonic vibrations and his solution:

, x=Acos(ω 0t+α ) ,

A– amplitude of oscillations;

α – initial phase of oscillations.

Oscillation period material point oscillating under the action of elastic force:

Where m– mass of a material point;

k– stiffness coefficient.

Period of oscillation of a mathematical pendulum:

Where l– length of the pendulum;

g= 9.8 m/s 2 – free fall acceleration.

The amplitude of vibrations obtained by adding two equally directed harmonic vibrations:

Where A 1 and A 2 – amplitudes of the vibration components;

φ 1 and φ 2 are the initial phases of the components of the oscillations.

The initial phase of oscillations obtained by adding two equally directed harmonic oscillations:

.

Equation damped oscillations and his solution:

, ,

– frequency of damped oscillations,

here ω 0 is the natural frequency of oscillations.

Logarithmic damping decrement:

where β is the attenuation coefficient;

– period of damped oscillations.

Quality factor of the oscillatory system:

where θ is the logarithmic attenuation decrement

The equation of forced oscillations and its steady-state solution:

, x=A cos (ω t-φ ),

Where F 0 – amplitude value of force;

– amplitude of damped oscillations;

φ= – initial phase.

Resonance frequency fluctuations:

,

where ω 0 – natural cyclic frequency of oscillations;

β is the attenuation coefficient.

Damped electromagnetic oscillations in a circuit consisting of a capacitanceC, inductanceLand resistanceR:

,

Where q– charge on the capacitor;

q m– amplitude value of the charge on the capacitor;

β = R/2L– attenuation coefficient,

Here R– circuit resistance;

L– coil inductance;

– cyclic frequency of oscillations;

here ω 0 – natural frequency of oscillations;

α – initial phase of oscillations.

Period of electromagnetic oscillations:

,

Where WITH– capacitor capacity;

L– coil inductance;

R– circuit resistance.

If the circuit resistance is small, what ( R/2L) 2 <<1/L.C., then the period of oscillation:

Wavelength:

Where v – wave propagation speed;

T– period of oscillation.

Plane wave equation:

ξ = A cos (ω t-kx),

Where A– amplitude;

ω – cyclic frequency;

– wave number.

Spherical wave equation:

,

Where A– amplitude;

ω – cyclic frequency;

k– wave number;

r– distance from the center of the wave to the considered point in the medium.

? Free harmonic oscillations in the circuit

An ideal circuit is an electrical circuit consisting of a capacitor connected in series with a capacitance WITH and inductors L. According to the harmonic law, the voltage on the capacitor plates and the current in the inductor will change.

? Harmonic oscillator. Spring, physical and mathematical pendulums, their periods of oscillation

Harmonic oscillator is any physical system that oscillates. Classical oscillators - spring, physical and mathematical pendulums. Spring pendulum - mass mass m, suspended on an absolutely elastic spring and performing harmonic oscillations under the action of an elastic force. T= . A physical pendulum is a rigid body of arbitrary shape that oscillates under the influence of gravity around a horizontal axis that does not pass through its center of gravity. T= . A mathematical pendulum is an isolated system consisting of a material point with a mass m, suspended on an inextensible weightless thread of length L, and oscillating under the influence of gravity. T= .

? Free undamped mechanical vibrations (equation, speed, acceleration, energy). Graphic representation of harmonic vibrations.

Oscillations are called free if they occur due to the initially imparted energy in the subsequent absence of external influences on the oscillatory system. The value changes according to the law of sine or cosine. , S- displacement from the equilibrium position, A– amplitude, w 0 - cyclic frequency, – initial phase of oscillations. Speed, acceleration. Full energy - E= . Graphically - using a sine or cosine wave.

? The concept of oscillatory processes. Harmonic oscillations and their characteristics. Period, amplitude, frequency and phase of oscillations. Graphic representation of harmonic vibrations.

Periodic processes that repeat over time are called oscillatory. Periodic oscillations, in which the coordinate of a body changes over time according to the law of sine or cosine, are called harmonic. Period is the time of one oscillation. Amplitude is the maximum displacement of a point from its equilibrium position. Frequency is the number of complete oscillations per unit of time. Phase is a quantity under the sine or cosine sign. Equation: , Here S- a quantity characterizing the state of an oscillating system - cyclic frequency. Graphically - using a sine or cosine wave.

? Damped oscillations. Differential equation of these oscillations. Logarithmic damping decrement, relaxation time, quality factor.

Oscillations whose amplitude decreases over time, for example, due to friction. Equation: , Here S- a quantity characterizing the state of an oscillating system, - cyclic frequency, - damping coefficient. Logarithmic damping decrement, where N– the number of oscillations completed during the amplitude decrease in N once. Relaxation time t - during which the amplitude decreases by e times. Quality factor Q= .

? Undamped forced oscillations. Differential equation of these oscillations. What is resonance? Amplitude and phase of forced oscillations.

If the loss of oscillation energy, leading to their damping, is fully compensated, undamped oscillations are established. Equation: . Here the right side is the external influence changing according to the harmonic law. If the natural frequency of oscillations of the system coincides with the external one, resonance occurs - a sharp increase in the amplitude of the system. Amplitude , .

? Describe the addition of vibrations of the same direction and the same frequency, mutually perpendicular vibrations. What are beats?

The amplitude of the resulting oscillation resulting from the addition of two harmonic oscillations of the same direction and the same frequency is here A– amplitudes, j – initial phases. Initial phase of the resulting oscillation . Mutually perpendicular oscillations - trajectory equation , Here A And IN amplitudes of added oscillations, j-phase difference.

? Describe relaxation oscillations; self-oscillations.

Relaxation - self-oscillations, sharply different in shape from harmonic ones, due to significant energy dissipation in self-oscillating systems (friction in mechanical systems). Self-oscillations are undamped oscillations supported by external energy sources in the absence of an external variable force. The difference from forced ones is that the frequency and amplitude of self-oscillations are determined by the properties of the oscillatory system itself. They differ from free oscillations - they differ in the independence of the amplitude from time and from the initial short-term influence that excites the oscillation process. An example of a self-oscillating system is a clock.

? Waves (basic concepts). Longitudinal and transverse waves. Standing wave. Wavelength, its relationship with period and frequency.

The process of propagation of vibrations in space is called a wave. The direction in which a wave transfers vibrational energy is the direction in which the wave moves. Longitudinal - vibration of medium particles occurs in the direction of wave propagation. Transverse - vibrations of particles of the medium occur perpendicular to the direction of propagation of the wave. A standing wave is formed by the superposition of two traveling waves propagating towards each other with the same frequencies and amplitudes, and in the case of transverse waves, the same polarization. Wavelength is the distance a wave travels in one period. ( wavelength, v- wave speed, T- period of oscillation)

? The principle of superposition (overlay) of waves. Group velocity and its relationship with phase velocity.

The principle of superposition - when several waves propagate in a linear medium, each one propagates as if there were no other waves, and the resulting displacement of a particle of the medium at any time is equal to the geometric sum of the displacements that the particles receive while participating in each of the constituent wave processes. Group velocity is the speed of movement of a group of waves that form a localized wave packet in space at each moment of time. The speed of movement of the wave phase is the phase speed. In a non-dispersed environment they coincide.

? Electromagnetic wave and its properties. Energy of electromagnetic waves.

An electromagnetic wave is an electromagnetic oscillation propagating in space. Experimentally obtained by Hertz in 1880. Properties - can propagate in media and vacuum, in vacuum equal to c, in media less, transverse, E And B mutually perpendicular and perpendicular to the direction of propagation. The intensity increases with increasing acceleration of the radiating charged particle; under certain conditions, typical wave properties appear - diffraction, etc. Volumetric energy density .

Optics

Basic formulas of optics

Speed ​​of light in the medium:

Where c– speed of light in vacuum;

n– refractive index of the medium.

Optical light wave path length:

L = ns,

Where s– geometric path length of a light wave in a medium with a refractive index n.

Optical path difference between two light waves:

∆ = L 1 – L 2 .

Dependence of the phase difference on the optical difference in the path of light waves:

where λ is the light wavelength.

Condition for maximum light amplification during interference:

∆ = kλ ( = 0, 1, 2, …) .

Condition for maximum light attenuation:

Optical difference in the path of light waves that occurs when monochromatic light is reflected from a thin film:

∆ = 2d ,

Where d– film thickness;

n– refractive index of the film;

I i– angle of refraction of light in the film.

Radius of light Newton's rings in reflected light:

r k = , (k = 1, 2, 3, …),

Where k– ring number;

R– radius of curvature.

Radius of Newton's dark rings in reflected light:

r k = .

The angle φ of deflection of the rays, corresponding to the maximum (light band) during diffraction by one slit, is determined from the condition

a sinφ = (k = 0, 1, 2, 3, …),

Where a– slot width;

k– serial number of the maximum.

Cornerφdeviation of the rays, corresponding to the maximum (light band) during diffraction of light on a diffraction grating, is determined from the condition

d sinφ = (k = 0, 1, 2, 3, …),

Where d– period of the diffraction grating.

Diffraction grating resolution:

R= = kN,

where ∆λ is the smallest difference in wavelengths of two adjacent spectral lines (λ and λ+∆λ), at which these lines can be seen separately in the spectrum obtained by this grating;

N– total number of grating slits.

Wulf–Bragg formula:

2d sin θ = κ λ,

where θ is the grazing angle (the angle between the direction of a parallel X-ray beam incident on the crystal and the atomic plane in the crystal);

d is the distance between the atomic planes of the crystal.

Brewster's Law:

tan ε B=n 21 ,

where ε B– angle of incidence at which the beam reflected from the dielectric is completely polarized;

n 21 – relative refractive index of the second medium relative to the first.

Malus's Law:

I = I 0 cos 2 α ,

Where I 0 – intensity of plane-polarized light incident on the analyzer;

I– intensity of this light after the analyzer;

α is the angle between the direction of oscillations of the electric vector of light incident on the analyzer and the transmittance plane of the analyzer (if the oscillations of the electric vector of the incident light coincide with this plane, then the analyzer transmits this light without attenuation).

Angle of rotation of the plane of polarization of monochromatic light when passing through an optically active substance:

a) φ = αd(in solids),

Where α – rotation constant;

d– length of the path traveled by light in an optically active substance;

b) φ = [α]pd(in solutions),

Where [α] – specific rotation;

p– mass concentration of an optically active substance in solution.

Light pressure at normal incidence on a surface:

,

Where Her– energy illumination (irradiance);

ω – volumetric radiation energy density;

ρ – reflection coefficient.

4.2. Concepts and definitions of the “optics” section

? Wave interference. Coherence. Maximum and minimum conditions.

Interference is the mutual strengthening or weakening of coherent waves when they are superimposed (coherent - having the same length and a constant phase difference at the point of their superposition).

Maximum ;

minimum .

Here D is the optical path difference, l is the wavelength.

? Huygens-Fresnel principle. The phenomenon of diffraction. Slit diffraction, diffraction grating.

The Huygens-Fresnel principle - every point in space that a propagating wave has reached at a given moment in time becomes a source of elementary coherent waves. Diffraction is the bending of waves around obstacles, if the size of the obstacle is comparable to the wavelength, the deviation of light from rectilinear propagation. Slit diffraction is in parallel rays. A plane wave falls on an obstacle; the diffraction pattern is observed on a screen located in the focal plane of a collecting lens installed in the path of light passing through the obstacle. The screen produces a “diffraction image” of a distant light source. A diffraction grating is a system of parallel slits of equal width, lying in the same plane, separated by opaque spaces of equal width. Used to split light into a spectrum and measure wavelengths.

? Light dispersion (normal and abnormal). Bouguer's law. The meaning of absorption coefficient.

Dispersion of light - dependence of the absolute refractive index of a substance n on the frequency ν (or wavelength λ) of light incident on the substance (). The speed of light in a vacuum does not depend on frequency, so there is no dispersion in a vacuum. Normal dispersion of light - if the refractive index increases monotonically with increasing frequency (decreases with increasing wavelength). Anomalous dispersion - if the refractive index decreases monotonically with increasing frequency (increases with increasing wavelength). The consequence of dispersion is the decomposition of white light into a spectrum when it is refracted in a substance. The absorption of light in a substance is described by Bouguer's law

I 0 and I– intensity of a plane monochromatic light wave at the input and output of a layer of absorbing substance of thickness X, a is the absorption coefficient, depends on the wavelength, and is different for different substances.

? What is wave polarization called? Obtaining polarized waves. Malus's law.

Polarization consists in acquiring a preferential orientation of the direction of oscillations in transverse waves. Orderliness in the orientation of the vectors of electric and magnetic field strengths of an electromagnetic wave in a plane perpendicular to the direction of propagation of the light beam. E , B - perpendicular. Natural light can be converted into polarized light using polarizers. Malus's law ( I 0 – passed through the analyzer, I– passed through a polarizer).

? Corpuscular-wave dualism. De Broglie's hypothesis.

Historically, two theories of light have been put forward: corpuscular - luminous bodies emit corpuscular particles (evidence - black body radiation, photoelectric effect) and wave - a luminous body causes elastic vibrations in the environment, propagating like sound waves in the air (evidence - phenomena of interference, diffraction, polarization of light). Broglie's hypothesis - particle-wave properties are inherent not only to photons, but also to particles that have a rest mass - electrons, protons, neutrons, atoms, molecules. ? Photo effect. Einstein's equation.

Photoelectric effect is the phenomenon of interaction of light with matter, as a result of which the energy of photons is transferred to the electrons of the substance. Equation: (photon energy is spent on the work function of the electron and imparting kinetic energy to the electron)

Any vibrations represent movement with variable acceleration. Deflection, speed and acceleration in this case are functions of time. Any fluctuations are characterized by periodicity, i.e. the movement is repeated after time has elapsed T, called the duration or period of oscillation. Oscillations occur in cases where energy is imparted to a system capable of oscillating.
It is necessary to distinguish:

Undamped oscillations

Undamped oscillations occur with constant amplitude Ym. It is assumed that in this case the supplied energy is conserved. Approximately such conditions occur at low energy losses and short observation time. To obtain truly undamped oscillations, it is necessary to regularly replenish the lost energy.

Damped oscillations

Damped oscillations gradually reduce their amplitude Ym. Without replenishment of energy, any vibrations die out.

Important Vibration Characteristics

Harmonic oscillations occur according to the law:

x = A cos(ω t + φ 0),

Where x– displacement of the particle from the equilibrium position, A– amplitude of oscillations, ω – circular frequency, φ 0 – initial phase, t- time.

Oscillation period T = .

Speed ​​of oscillating particle:

υ = = – Aω sin(ω t + φ 0),

acceleration a = = –Aω 2 cos (ω t + φ 0).

Kinetic energy of a particle undergoing oscillatory motion: E k = =
sin 2 (ω t+ φ 0).

Potential energy:

E n=
cos 2 (ω t + φ 0).

Periods of pendulum oscillations

– spring T =
,

Where m– mass of cargo, k– spring stiffness coefficient,

– mathematical T = ,

Where l– suspension length, g– free fall acceleration,

– physical T =
,

Where I– moment of inertia of the pendulum relative to the axis passing through the suspension point, m– mass of the pendulum, l– distance from the suspension point to the center of mass.

The reduced length of a physical pendulum is found from the condition: l np = ,

The designations are the same as for a physical pendulum.

When two harmonic oscillations of the same frequency and one direction are added, a harmonic oscillation of the same frequency with amplitude is obtained:

A = A 1 2 + A 2 2 + 2A 1 A 2 cos(φ 2 – φ 1)

and initial phase: φ = arctan
.

Where A 1 , A 2 – amplitudes, φ 1, φ 2 – initial phases of folded oscillations.

The trajectory of the resulting movement when adding mutually perpendicular oscillations of the same frequency:

+ – cos (φ 2 – φ 1) = sin 2 (φ 2 – φ 1).

Damped oscillations occur according to the law:

x = A 0 e - β t cos(ω t + φ 0),

where β is the damping coefficient, the meaning of the remaining parameters is the same as for harmonic oscillations, A 0 – initial amplitude. At a moment in time t vibration amplitude:

A = A 0 e - β t .

The logarithmic damping decrement is called:

λ = log
= β T,

Where T– oscillation period: T = .

The quality factor of an oscillatory system is called:

The equation of a plane traveling wave has the form:

y = y 0 cos ω( t ± ),

Where at– displacement of the oscillating quantity from the equilibrium position, at 0 – amplitude, ω – angular frequency, t- time, X– coordinate along which the wave propagates, υ – speed of wave propagation.

The “+” sign corresponds to a wave propagating against the axis X, the “–” sign corresponds to a wave propagating along the axis X.

The wavelength is called its spatial period:

λ = υ T,

Where υ – wave propagation speed, T– period of propagating oscillations.

The wave equation can be written:

y = y 0 cos 2π (+).

A standing wave is described by the equation:

y = (2y 0cos ) cos ω t.

The amplitude of the standing wave is enclosed in parentheses. Points with maximum amplitude are called antinodes,

x n = n ,

points with zero amplitude - nodes,

x y = ( n + ) .

Examples of problem solving

Problem 20

The amplitude of harmonic oscillations is 50 mm, the period is 4 s and the initial phase . a) Write down the equation of this oscillation; b) find the displacement of the oscillating point from the equilibrium position at t=0 and at t= 1.5 s; c) draw a graph of this movement.

Solution

The oscillation equation is written as x = a cos( t+  0).

According to the condition, the period of oscillation is known. Through it we can express the circular frequency  = . The remaining parameters are known:

A) x= 0.05cos( t + ).

b) Offset x at t= 0.

x 1 = 0.05 cos = 0.05 = 0.0355 m.

At t= 1.5 s

x 2 = 0.05 cos( 1,5 + )= 0.05 cos  = – 0.05 m.

V ) graph of a function x=0.05cos ( t + ) looks like this:

Let's determine the position of several points. Known X 1 (0) and X 2 (1.5), as well as the oscillation period. So, through  t= 4 s value X repeats, and after  t = 2 s changes sign. Between the maximum and minimum in the middle is 0.

Problem 21

The point undergoes a harmonic oscillation. The oscillation period is 2 s, the amplitude is 50 mm, the initial phase is zero. Find the speed of the point at the moment of time when its displacement from the equilibrium position is 25 mm.

Solution

1 way. We write down the equation of point oscillation:

x= 0.05 cos t, because  = =.

Finding the speed at the moment of time t:

υ = = – 0,05 cos t.

We find the moment in time when the displacement is 0.025 m:

0.025 = 0.05 cos t 1 ,

hence cos  t 1 = ,  t 1 = . We substitute this value into the expression for speed:

υ = – 0.05  sin = – 0.05  = 0.136 m/s.

Method 2. Total energy of oscillatory motion:

E =
,

Where A– amplitude,  – circular frequency, m – particle mass.

At each moment of time it consists of the potential and kinetic energy of the point

E k = , E n = , But k = m 2, which means E n =
.

Let's write down the law of conservation of energy:

= +
,

from here we get: a 2  2 = υ 2 +  2 x 2 ,

υ = 
= 
= 0.136 m/s.

Problem 22

Amplitude of harmonic oscillations of a material point A= 2 cm, total energy E= 3∙10 -7 J. At what displacement from the equilibrium position does the force act on the oscillating point F = 2.25∙10 -5 N?

Solution

The total energy of a point performing harmonic oscillations is equal to: E =
. (13)

The modulus of elastic force is expressed through the displacement of points from the equilibrium position x as follows:

F = k x (14)

Formula (13) includes mass m and circular frequency , and in (14) – the stiffness coefficient k. But the circular frequency is related to m And k:

 2 = ,

from here k = m 2 and F = m 2 x. Having expressed m 2 from relation (13) we obtain: m 2 = , F = x.

From where we get the expression for the displacement x: x = .

Substituting numeric values ​​gives:

x =
= 1.5∙10 -2 m = 1.5 cm.

Problem 23

The point participates in two oscillations with the same periods and initial phases. Oscillation amplitudes A 1 = 3 cm and A 2 = 4 cm. Find the amplitude of the resulting vibration if: 1) the vibrations occur in one direction; 2) the oscillations are mutually perpendicular.

Solution

If oscillations occur in one direction, then the amplitude of the resulting oscillation is determined as:

Where A 1 and A 2 – amplitudes of added oscillations,  1 and  2 – initial phases. According to the condition, the initial phases are the same, which means  2 –  1 = 0, and cos 0 = 1.

Hence:

A =
=
= A 1 +A 2 = 7 cm.

If the oscillations are mutually perpendicular, then the equation of the resulting motion will be:

cos( 2 –  1) = sin 2 ( 2 –  1).

Since by condition  2 –  1 = 0, cos 0 = 1, sin 0 = 0, the equation will be written as:
=0,

or
=0,

or
.

The resulting relationship between x And at can be depicted on a graph. The graph shows that the result will be a oscillation of a point on a straight line MN. The amplitude of this oscillation is determined as: A =
= 5 cm.

Problem 24

Period of damped oscillations T=4 s, logarithmic damping decrement  = 1.6, initial phase is zero. Point displacement at t = equal to 4.5 cm. 1) Write the equation of this vibration; 2) Construct a graph of this movement for two periods.

Solution

The equation of damped oscillations with zero initial phase has the form:

x = A 0 e -  t cos2 .

There are not enough initial amplitude values ​​to substitute numerical values A 0 and attenuation coefficient .

The attenuation coefficient can be determined from the relation for the logarithmic attenuation decrement:

 = T.

Thus  = = = 0.4 s -1 .

The initial amplitude can be determined by substituting the second condition:

4.5 cm = A 0
cos 2 = A 0
cos = A 0
.

From here we find:

A 0 = 4,5∙

(cm) = 7.75 cm.

The final equation of motion is:

x = 0,0775
cost.

Problem 25

What is the logarithmic damping decrement of a mathematical pendulum, if for t = 1 min amplitude of oscillations decreased by half? Pendulum length l = 1 m.

Solution

The logarithmic damping decrement can be found from the relation: =  T,

where  is the attenuation coefficient, T– period of oscillation. Natural circular frequency of a mathematical pendulum:

 0 =
= 3.13 s -1 .

The oscillation damping coefficient can be determined from the condition: A 0 = A 0 e -  t ,

t= ln2 = 0.693,

 =
= 0.0116c -1 .

Since <<  0 , то в формуле  =
can be neglected compared to  0 and the oscillation period can be determined by the formula: T = = 2c.

We substitute  and T into the expression for the logarithmic damping decrement and we get:

 = T= 0.0116 s -1 ∙ 2 s = 0.0232.

Problem 26

The equation of undamped oscillations is given in the form x= 4 sin600  t cm.

Find the displacement from the equilibrium position of a point located at a distance l= 75 cm from the vibration source, through t= 0.01 s after the start of oscillation. Oscillation propagation speed υ = 300 m/s.

Solution

Let us write down the equation of a wave propagating from a given source: x= 0.04 sin 600 ( t– ).

We find the phase of the wave at a given time in a given place:

t– = 0,01 –= 0,0075 ,

600 ∙ 0.0075 = 4.5,

sin 4.5 = sin = 1.

Therefore, the point displacement x= 0.04 m, i.e. at a distance l =75 cm from the source at time t= 0.01 s maximum point displacement.

References

Volkenshtein V.S.. Collection of problems for the general course of physics. – St. Petersburg: SpetsLit, 2001.

Savelyev I.V.. Collection of questions and problems in general physics. – M.: Nauka, 1998.

Harmonic oscillations are oscillations performed according to the laws of sine and cosine. The following figure shows a graph of changes in the coordinates of a point over time according to the cosine law.

picture

Oscillation amplitude

The amplitude of a harmonic vibration is the greatest value of the displacement of a body from its equilibrium position. The amplitude can take on different values. It will depend on how much we displace the body at the initial moment of time from the equilibrium position.

The amplitude is determined by the initial conditions, that is, the energy imparted to the body at the initial moment of time. Since sine and cosine can take values ​​in the range from -1 to 1, the equation must contain a factor Xm, expressing the amplitude of the oscillations. Equation of motion for harmonic vibrations:

x = Xm*cos(ω0*t).

Oscillation period

The period of oscillation is the time it takes to complete one complete oscillation. The period of oscillation is designated by the letter T. The units of measurement of the period correspond to the units of time. That is, in SI these are seconds.

Oscillation frequency is the number of oscillations performed per unit of time. The oscillation frequency is designated by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.

ν = 1/T.

Frequency units are in SI 1/sec. This unit of measurement is called Hertz. The number of oscillations in a time of 2*pi seconds will be equal to:

ω0 = 2*pi* ν = 2*pi/T.

Oscillation frequency

This quantity is called the cyclic frequency of oscillations. In some literature the name circular frequency appears. The natural frequency of an oscillatory system is the frequency of free oscillations.

The frequency of natural oscillations is calculated using the formula:

The frequency of natural vibrations depends on the properties of the material and the mass of the load. The greater the spring stiffness, the greater the frequency of its own vibrations. The greater the mass of the load, the lower the frequency of natural oscillations.

These two conclusions are obvious. The stiffer the spring, the greater the acceleration it will impart to the body when the system is thrown out of balance. The greater the mass of a body, the slower the speed of this body will change.

Free oscillation period:

T = 2*pi/ ω0 = 2*pi*√(m/k)

It is noteworthy that at small angles of deflection the period of oscillation of the body on the spring and the period of oscillation of the pendulum will not depend on the amplitude of the oscillations.

Let's write down the formulas for the period and frequency of free oscillations for a mathematical pendulum.

then the period will be equal

T = 2*pi*√(l/g).

This formula will be valid only for small deflection angles. From the formula we see that the period of oscillation increases with increasing length of the pendulum thread. The longer the length, the slower the body will vibrate.

The period of oscillation does not depend at all on the mass of the load. But it depends on the acceleration of free fall. As g decreases, the oscillation period will increase. This property is widely used in practice. For example, to measure the exact value of free acceleration.