Mathematical pendulum called material point, suspended on a weightless and inextensible thread attached to the suspension and located in the field of gravity (or other force).
We investigate the oscillations of a mathematical pendulum in inertial system reference relative to which the point of its suspension is at rest or moves uniformly in a straight line. We will neglect the force of air resistance (ideal mathematical pendulum). Initially, the pendulum is at rest in the equilibrium position C. In this case, the force of gravity \(\vec F\) acting on it and the elastic force \(\vec F_(ynp)\) of the thread are mutually compensated.
Let us remove the pendulum from the equilibrium position (by deflecting it, for example, to position A) and release it without initial speed(Fig. 13.11). In this case, the forces \(\vec F\) and \(\vec F_(ynp)\) do not balance each other. The tangential component of gravity \(\vec F_\tau\), acting on the pendulum, tells it tangential acceleration\(\vec a_\tau\) (component full acceleration, directed along the tangent to the trajectory of the mathematical pendulum), and the pendulum begins to move towards the equilibrium position with a speed increasing in absolute value. The tangential component of gravity \(\vec F_\tau\) is thus a restoring force. The normal component \(\vec F_n\) of the gravity force is directed along the thread against the elastic force \(\vec F_(ynp)\). The resultant of the forces \(\vec F_n\) and \(\vec F_(ynp)\) imparts the normal acceleration \(~a_n\) to the pendulum, which changes the direction of the velocity vector, and the pendulum moves along an arc ABCD.
The closer the pendulum comes to the equilibrium position C, the smaller the value of the tangential component \(~F_\tau = F \sin \alpha\) becomes. In the equilibrium position it is zero, and the speed reaches maximum value, and the pendulum moves further by inertia, rising upward in an arc. In this case, the component \(\vec F_\tau\) is directed against the speed. With increasing angle of deflection a, the modulus of force \(\vec F_\tau\) increases, and the modulus of velocity decreases, and at point D the speed of the pendulum becomes equal to zero. The pendulum stops for a moment and then begins to move in the opposite direction to the equilibrium position. Having passed it again by inertia, the pendulum, slowing down its movement, will reach point A (there is no friction), i.e. will complete a complete swing. After this, the movement of the pendulum will be repeated in the sequence already described.
Let us obtain an equation describing the free oscillations of a mathematical pendulum.
Let the pendulum in at the moment time is at point B. Its displacement S from the equilibrium position at this moment is equal to the length of the arc SV (i.e. S = |SV|). Let us denote the length of the suspension thread l, and the mass of the pendulum is m.
From Figure 13.11 it is clear that \(~F_\tau = F \sin \alpha\), where \(\alpha =\frac(S)(l).\) At small angles \(~(\alpha<10^\circ)\) отклонения маятника \(\sin \alpha \approx \alpha,\) поэтому
\(F_\tau = -F\frac(S)(l) = -mg\frac(S)(l).\)
The minus sign is placed in this formula because the tangential component of gravity is directed towards the equilibrium position, and the displacement is counted from the equilibrium position.
According to Newton's second law \(m \vec a = m \vec g + F_(ynp).\) Let's project the vector quantities of this equation onto the direction of the tangent to the trajectory of the mathematical pendulum
\(~F_\tau = ma_\tau .\)
From these equations we get
\(a_\tau = -\frac(g)(l)S\) - dynamic equation of motion of a mathematical pendulum. The tangential acceleration of a mathematical pendulum is proportional to its displacement and directed towards the equilibrium position. This equation can be written as\. Comparing it with Eq. harmonic vibrations\(~a_x + \omega^2x = 0\) (see § 13.3), we can conclude that the mathematical pendulum performs harmonic oscillations. And since the considered oscillations of the pendulum occurred under the influence only internal forces, then these were free oscillations of the pendulum. Hence, free oscillations of a mathematical pendulum with small deviations are harmonic.
Let us denote \(\frac(g)(l) = \omega^2.\) From where \(\omega = \sqrt \frac(g)(l)\) is the cyclic frequency of the pendulum.
The period of oscillation of the pendulum is \(T = \frac(2 \pi)(\omega).\) Therefore,
\(T = 2 \pi \sqrt( \frac(l)(g) )\)
This expression is called Huygens' formula. It determines the period of free oscillations of a mathematical pendulum. It follows from the formula that at small angles of deviation from the equilibrium position, the period of oscillation of a mathematical pendulum: 1) does not depend on its mass and amplitude of oscillations; 2) proportional to the square root of the length of the pendulum and inversely proportional to the square root of the acceleration free fall. This is consistent with the experimental laws of small oscillations of a mathematical pendulum, which were discovered by G. Galileo.
We emphasize that this formula can be used to calculate the period if two conditions are simultaneously met: 1) the pendulum oscillations must be small; 2) the suspension point of the pendulum must be at rest or move uniformly in a straight line relative to the inertial frame of reference in which it is located.
If the suspension point of a mathematical pendulum moves with acceleration \(\vec a\), then the tension force of the thread changes, which leads to a change in the restoring force, and, consequently, the frequency and period of oscillations. As calculations show, the period of oscillation of the pendulum in this case can be calculated using the formula
\(T = 2 \pi \sqrt( \frac(l)(g") )\)
where \(~g"\) is the "effective" acceleration of the pendulum in a non-inertial frame of reference. It is equal to the geometric sum of the acceleration of gravity \(\vec g\) and the vector opposite to the vector \(\vec a\), i.e. it can be calculated using the formula
\(\vec g" = \vec g + (- \vec a).\)
Literature
Aksenovich L. A. Physics in secondary school: Theory. Assignments. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyakhavanne, 2004. - P. 374-376.
Imagine a certain mechanical system that consists of a certain material point (body) that hangs on an inextensible weightless thread (the mass of the thread is negligible compared to the mass of the body). This mechanical system is a pendulum or oscillator, as it is also called. However, there may be other types of such devices. Why is a mathematical pendulum or oscillator interesting for us? The fact is that with its help you can gain insight into many interesting natural phenomena in physics.
Oscillations of a mathematical pendulum
The formula for the period of oscillation of a mathematical pendulum was first discovered by the Dutch scientist Huygens back in the 17th century. Being a contemporary of Isaac Newton, Huygens was very fascinated by such pendulums, so fascinated that he even invented a special clock with pendulum mechanisms, and these clocks were one of the most accurate for that time.
Huygens pendulum clock.
The appearance of such an invention was of great benefit to physics, especially in the field of physical experiments, where the accurate measurement of time is a very important factor.
But let's return to the pendulum, so, the basis of the work of the pendulum is its oscillations, which can be expressed by a formula, more precisely by the following differential equation:
x + w2 sin x = 0
Where x (t) is an unknown function (this is the angle of deviation from the lower equilibrium position at moment t, expressed in radians); w is a positive constant, which is determined from the parameters of the pendulum (w = √ g/L, where g is the acceleration of free fall, and L is the length of the mathematical pendulum (suspension).
In addition to the oscillations themselves, the pendulum can also be in an equilibrium position, while the force of gravity acting on it will be balanced by the tension force of the thread. An ordinary flat pendulum resting on an inextensible thread is a system with two degrees of freedom. But if, for example, the thread is replaced by a rod, then our pendulum will become a system with only one degree of freedom, since its movements will be two-dimensional, not three-dimensional.
But if our pendulum still remains on the string and at the same time makes intense oscillations up and down, then the mechanical system acquires a stable position called “upside down”; it is also called the Kapitsa pendulum.
Properties of a pendulum
The pendulum has a number of interesting properties, confirmed by physical laws. Thus, the period of oscillation of any pendulum depends on factors such as its size, body shape, and the distance between the center of gravity and the point of suspension. Therefore, determining the period of a pendulum is not an easy task. But the period of a mathematical pendulum can be calculated exactly using the formula given below.
During observations of pendulums, the following patterns were derived:
- If different loads with different weights are suspended from the pendulum, but at the same time maintaining the same length of the pendulum, then the period of its oscillation will be the same regardless of the mass of the load.
- If, when starting oscillations, the pendulum is deflected at not very large, but still different angles, then it will begin to oscillate in the same period, but with different amplitudes. Consequently, the period of oscillation of such a pendulum does not depend on the amplitude of the oscillation; this phenomenon was called isochronism, which can be translated from ancient Greek as “chronos” - time, “iso” - equal, that is, “equal in time”.
Period of a mathematical pendulum
The period of a pendulum is an indicator that represents the period of the actual oscillations of the pendulum, their duration. The formula for the period of a mathematical pendulum can be written as follows.
Where L is the length of the thread of a mathematical pendulum, g is the acceleration of gravity, and π is the number Pi, a mathematical constant.
The period of small oscillations of a mathematical pendulum does not depend in any way on the mass of the pendulum and the amplitude of the oscillation; in this situation, it moves like a mathematical pendulum with a given length.
Practical application of a mathematical pendulum
Now we get to the most interesting thing, why we need a mathematical pendulum and what its application is in practice in life. First of all, the acceleration of a mathematical pendulum is used for geological exploration, with its help they search for minerals. How does this happen? The fact is that the acceleration of gravity changes with geographic latitude, since the density of the crust in different places on our planet is far from the same and where rocks with higher density occur, the acceleration will be slightly greater. This means that simply by counting the number of oscillations of a pendulum, you can find ore or coal in the bowels of the Earth, since they have a higher density than other loose rocks.
Also, the mathematical pendulum was used by many outstanding scientists of the past, starting from antiquity, in particular Archimedes, Aristotle, Plato, Plutarch. So Archimedes even used a mathematical pendulum in all his calculations, and some people even believed that the pendulum could influence people’s destinies and tried to make predictions about the future with its help.
Mathematical pendulum, video
And finally, an educational video on the topic of our article.
Oscillatory motion- periodic or almost periodic movement of a body, the coordinate, speed and acceleration of which at equal intervals of time take on approximately the same values.
Mechanical vibrations occur when, when a body is removed from an equilibrium position, a force appears that tends to return the body back.
Displacement x is the deviation of the body from the equilibrium position.
Amplitude A is the module of the maximum displacement of the body.
Oscillation period T - time of one oscillation:
Oscillation frequency
The number of oscillations performed by a body per unit of time: During oscillations, the speed and acceleration periodically change. In the equilibrium position, the speed is maximum and the acceleration is zero. At the points of maximum displacement, the acceleration reaches a maximum and the speed becomes zero.
HARMONIC VIBRATION SCHEDULE
Harmonic vibrations that occur according to the law of sine or cosine are called:
where x(t) is the displacement of the system at time t, A is the amplitude, ω is the cyclic frequency of oscillations.
If you plot the deviation of the body from the equilibrium position along the vertical axis, and time along the horizontal axis, you will get a graph of oscillation x = x(t) - the dependence of the body’s displacement on time. For free harmonic oscillations, it is a sine wave or cosine wave. The figure shows graphs of the dependence of displacement x, projections of velocity V x and acceleration a x on time.
As can be seen from the graphs, at maximum displacement x, the speed V of the oscillating body is zero, the acceleration a, and therefore the force acting on the body, is maximum and directed opposite to the displacement. In the equilibrium position, the displacement and acceleration become zero, and the speed is maximum. The acceleration projection always has the opposite sign to the displacement.
ENERGY OF VIBRATIONAL MOTION
The total mechanical energy of an oscillating body is equal to the sum of its kinetic and potential energies and, in the absence of friction, remains constant:
At the moment when the displacement reaches a maximum x = A, the speed, and with it the kinetic energy, goes to zero.
In this case, the total energy is equal to the potential energy:
The total mechanical energy of an oscillating body is proportional to the square of the amplitude of its oscillations.
When the system passes the equilibrium position, the displacement and potential energy are zero: x = 0, E p = 0. Therefore, the total energy is equal to the kinetic energy:
The total mechanical energy of an oscillating body is proportional to the square of its speed in the equilibrium position. Hence:
MATHEMATICAL PENDULUM
1. Math pendulum is a material point suspended on a weightless inextensible thread.
In the equilibrium position, the force of gravity is compensated by the tension of the thread. If the pendulum is deflected and released, then the forces will cease to compensate each other, and a resultant force will arise directed towards the equilibrium position. Newton's second law:
For small oscillations, when the displacement x is much less than l, the material point will move almost along the horizontal x axis. Then from the triangle MAB we get:
Because sin a = x/l, then the projection of the resulting force R onto the x axis is equal to
The minus sign shows that the force R is always directed opposite the displacement x.
2. So, during oscillations of a mathematical pendulum, as well as during oscillations spring pendulum, the restoring force is proportional to the displacement and is directed in the opposite direction.
Let's compare the expressions for the restoring force of mathematical and spring pendulums:
It can be seen that mg/l is an analogue of k. Replacing k with mg/l in the formula for the period of a spring pendulum
we obtain the formula for the period of a mathematical pendulum:
The period of small oscillations of a mathematical pendulum does not depend on the amplitude.
A mathematical pendulum is used to measure time and determine the acceleration of gravity at a given location on the earth's surface.
Free oscillations of a mathematical pendulum at small angles of deflection are harmonic. They happen thanks to acting force gravity and tension force of the thread, as well as the inertia of the load. The resultant of these forces is the restoring force.
Example. Determine the acceleration due to gravity on a planet where a pendulum 6.25 m long has a period of free oscillation of 3.14 s.
The period of oscillation of a mathematical pendulum depends on the length of the thread and the acceleration of gravity:
By squaring both sides of the equality, we get:
Answer: the acceleration of gravity is 25 m/s 2 .
Problems and tests on the topic "Topic 4. "Mechanics. Oscillations and waves."
- Transverse and longitudinal waves. Wavelength
Lessons: 3 Assignments: 9 Tests: 1
- Sound waves. Speed of sound - Mechanical vibrations and waves. Sound 9th grade
In technology and the world around us we often have to deal with periodic(or almost periodic) processes that repeat at regular intervals. Such processes are called oscillatory.
Oscillations are one of the most common processes in nature and technology. The wings of insects and birds in flight, high-rise buildings and high-voltage wires under the influence of the wind, the pendulum of a wound clock and a car on springs while driving, the river level throughout the year and the temperature of the human body during illness, sound is fluctuations in air density and pressure, radio waves - periodic changes in the strengths of electric and magnetic fields, visible light is also electromagnetic vibrations, only with slightly different wavelengths and frequencies, earthquakes are soil vibrations, the pulse is periodic contractions of the human heart muscle, etc.
Oscillations can be mechanical, electromagnetic, chemical, thermodynamic and various others. Despite such diversity, they all have much in common.
Oscillatory phenomena of various physical nature obey general laws. For example, current oscillations in an electrical circuit and oscillations of a mathematical pendulum can be described by the same equations. The generality of vibrational patterns allows us to consider oscillatory processes of different natures from a single point of view. A sign of oscillatory motion is its periodicity.
Mechanical vibrations –Thismovements that are repeated exactly or approximately at regular intervals.
Examples of simple oscillatory systems can serve as a load on a spring (spring pendulum) or a ball on a string (mathematical pendulum).
During mechanical vibrations, kinetic and potential energies change periodically.
At maximum deviation body from its equilibrium position, its speed, and therefore kinetic energy goes to zero. In this position potential energy oscillating body reaches maximum value. For a load on a spring, potential energy is the energy of elastic deformation of the spring. For a mathematical pendulum, this is energy in the Earth’s gravitational field.
When a body, in its movement, passes through equilibrium position, its speed is maximum. The body overshoots the equilibrium position according to the law of inertia. At this moment it has maximum kinetic and minimum potential energy. An increase in kinetic energy occurs due to a decrease in potential energy.
With further movement, potential energy begins to increase due to a 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 mechanical vibrations remains unchanged.
For spring load:
At the position of maximum deflection, the total energy of the pendulum is equal to the potential energy of the deformed spring:
When passing through the equilibrium position, the total energy is equal to the kinetic energy of the load:
For small oscillations of a mathematical pendulum:
At the position of maximum deviation, the total energy of the pendulum is equal to the potential energy of the body raised to a height h:
When passing through the equilibrium position, the total energy is equal to the kinetic energy of the body:
Here h m– the maximum height of the pendulum in the Earth’s gravitational field, x m and υ m = ω 0 x m– maximum values of the pendulum’s deviation from the equilibrium position and its speed.
Harmonic oscillations and their characteristics. Equation of harmonic vibration.
The simplest type of oscillatory process are simple harmonic vibrations, which are described by the equation
x = x m cos(ω t + φ 0).
Here x– displacement of the body from the equilibrium position,
x m– amplitude of oscillations, that is, the maximum displacement from the equilibrium position,
ω – cyclic or circular frequency hesitation,
t- time.
Characteristics of oscillatory motion.
Offset x – deviation of an oscillating point from its equilibrium position. The unit of measurement is 1 meter.
Oscillation amplitude A – maximum deviation of an oscillating point from its equilibrium position. The unit of measurement is 1 meter.
Oscillation periodT– the minimum time interval during which one complete oscillation occurs is called. The unit of measurement is 1 second.
T=t/N
where t is the time of oscillations, N is the number of oscillations completed during this time.
From the graph of harmonic oscillations, one can determine the period and amplitude of oscillations:
Oscillation frequency ν – a physical quantity equal to the number of oscillations per unit of time.
ν=N/t
Frequency is the reciprocal of the oscillation period:
Frequency oscillations ν shows how many oscillations occur in 1 s. The unit of frequency is hertz(Hz).
Cyclic frequency ω– number of oscillations in 2π seconds.
Oscillation frequency ν is related to cyclic frequency ω and oscillation period T ratios:
Phase harmonic process - a quantity under the sine or cosine sign in the equation of harmonic oscillations φ = ω t + φ 0 . At t= 0 φ = φ 0 , therefore φ 0 called initial phase.
Harmonic graph represents a sine or cosine wave.
In all three cases for blue curves φ 0 = 0:
only greater amplitude(x" m > x m);
the red curve is different from the blue one only meaning period(T" = T / 2);
the red curve is different from the blue one only meaning initial phase(glad).
At oscillatory movement body along a straight line (axis OX) the velocity vector is always directed along this straight line. The speed of movement of the body is determined by the expression
In mathematics, the procedure for finding the limit of the ratio Δх/Δt at Δ t→ 0 is called calculating the derivative of the function x(t) by time t and is denoted as x"(t).The speed is equal to the derivative of the function x( t) by time t.
For the harmonic law of motion x = x m cos(ω t+ φ 0) calculating the derivative leads to the following result:
υ X =x"(t)= ω x m sin (ω t + φ 0)
Acceleration is determined in a similar way a x bodies during harmonic vibrations. Acceleration a is equal to the derivative of the function υ( t) by time t, or the second derivative of the function x(t). Calculations give:
and x =υ x "(t) =x""(t)= -ω 2 x m cos(ω t+ φ 0)=-ω 2 x
The minus sign in this expression means that the acceleration a(t) always has a sign, opposite sign offsets x(t), and, therefore, according to Newton’s second law, the force causing the body to perform harmonic oscillations is always directed towards the equilibrium position ( x = 0).
The figure shows graphs of the coordinates, speed and acceleration of a body performing harmonic oscillations.
Graphs of coordinates x(t), velocity υ(t) and acceleration a(t) of a body performing harmonic oscillations.
Spring pendulum.
Spring pendulumis a load of some mass m attached to a spring of stiffness k, the second end of which is fixedly fixed.
Natural frequencyω 0 free oscillations of the load on the spring is found by the formula:
Period T harmonic vibrations of the load on the spring is equal to
This means that the period of oscillation of a spring pendulum depends on the mass of the load and the stiffness of the spring.
Physical properties of the oscillatory system determine only the natural frequency of oscillations ω 0 and the period T . Parameters of the oscillation process such as amplitude x m And initial phaseφ 0 are determined by the way in which the system was brought out of equilibrium at the initial moment of time.
Mathematical pendulum.
Mathematical pendulumcalled a small body 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 plumb, the force of gravity is balanced by the tension force of the thread N. When the pendulum deviates from the equilibrium position by a certain angle φ, a tangential component of the force of gravity appears F τ = – mg sin φ. The minus sign in this formula means that the tangential component is directed in the direction opposite to the deflection of the pendulum.
Mathematical pendulum.φ – angular deviation of the pendulum from the equilibrium position,
x= lφ – displacement of the pendulum along the arc
The natural frequency of small oscillations of a mathematical pendulum is expressed by the formula:
Period of oscillation of a mathematical pendulum:
This means that the period of oscillation of a mathematical pendulum depends on the length of the thread and on the acceleration of free fall of the area where the pendulum is installed.
Free and forced vibrations.
Mechanical vibrations, like oscillatory processes of any other physical nature, can be free And forced.
Free vibrations –These are oscillations that occur in a system under the influence of internal forces, after the system has been removed from a stable equilibrium position.
Oscillations of a weight on a spring or oscillations of a pendulum are free oscillations.
In order for free vibrations to occur along harmonic law, it is necessary that the force tending to return the body to the equilibrium position be proportional to the displacement of the body from the equilibrium position and directed in the direction opposite to the displacement.
IN real conditions any oscillatory system is under the influence of friction forces (resistance). Moreover, part mechanical energy turns into internal energy thermal movement of atoms and molecules, and vibrations become fading.
Fading called oscillations whose amplitude decreases with time.
To prevent the oscillations from fading, it is necessary to provide the system with additional energy, i.e. influence the oscillatory system with a periodic force (for example, to rock a swing).
Oscillations occurring under the influence of an external periodically changing force are calledforced.
An external force does positive work and provides an energy flow to the oscillatory system. It does not allow vibrations to die out, despite the action of friction forces.
A periodic external force can vary over time according to various laws. Special Interest represents the case when an external force, varying according to a harmonic law with a frequency ω, acts on an oscillatory system capable of performing its own 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 external force .
Phenomenon sharp increase amplitudes forced oscillations when the frequency of natural oscillations coincides with the frequency of the external driving force, it is calledresonance.
Amplitude dependence x m forced oscillations from the frequency ω of the driving force is called resonant characteristic or resonance curve.
Resonance curves at various levels attenuation:
1 – oscillatory system without friction; at resonance, the amplitude x m of forced oscillations increases indefinitely;
2, 3, 4 – real resonance curves for oscillatory systems with different friction.
In the absence of friction, the amplitude of forced oscillations during resonance should increase without limit. In real conditions, the amplitude of steady-state forced oscillations is determined by the condition: the work of the external force during the oscillation period must be equal to the loss of mechanical energy during the same time due to friction. The less friction, the greater the amplitude of forced oscillations during resonance.
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 arises, for example, due to the rotation of an unbalanced motor.
(lat. amplitude- magnitude) is the greatest deviation of an oscillating body from its equilibrium position.
For a pendulum, this is the maximum distance by which the ball moves away from its equilibrium position (figure below). For oscillations with small amplitudes, such a distance can be taken as the length of the arc 01 or 02, and the lengths of these segments.
The amplitude of oscillations is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve (see figure below).
Oscillation period.
Oscillation period- This smallest gap the time after which the oscillating system returns again to the same state in which it was at the initial moment of time, chosen arbitrarily.
In other words, the oscillation period ( T) is the time during which one complete oscillation occurs. For example, in the figure below, this is the time it takes for the pendulum bob to move from the rightmost point through the equilibrium point ABOUT to the far left point and back through the point ABOUT again to the far right.
For full period oscillations, thus the body travels a path equal to four amplitudes. The period of oscillation is measured in time units - seconds, minutes, etc. The period of oscillation can be determined by famous graphic artist vibrations (see figure below).
The concept of “oscillation period”, strictly speaking, is valid only when the values of the oscillating quantity are exactly repeated after a certain period of time, i.e. for harmonic oscillations. However, this concept also applies to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency- this is the number of oscillations performed per unit of time, for example, in 1 s.
The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, this means that every second there is one oscillation. The frequency and period of oscillations are related by the relations:
In the theory of oscillations they also use the concept cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:
.
Cyclic frequency is the number of oscillations performed per 2π seconds
