mechanical system that consists of a material point (the body), hanging on the weightless inextensible filament (its mass is negligible compared to the weight of the body) in a uniform gravitational field, called the mathematical pendulum (another name - the oscillator).There are other types of devices.Instead of a filament can be used weightless rod.Pendulum can clearly reveal the essence of many interesting phenomena.At low amplitude fluctuations of its motion is called harmonic.
Understanding the mechanical system
Formula period of oscillation of the pendulum was bred Dutch scientist Huygens (1629-1695 gg.).This contemporary of Isaac Newton was very fond of the mechanical system.In 1656 he created the first clock with a pendulum mechanism.They measured the time with extreme precision for those times.This invention was a major step in the development of physical experiments and practical activities.
If the pendulum is in its equilibrium position (hanging vertically), the force of gravity is balanced by the force of the thread tension.Flat pendulum on a non-stretchable yarn is a system with two degrees of freedom with a link.If you change only one component of the change characteristics of all its parts.Thus, if a string is replaced by a rod, then given mechanical system is only one degree of freedom.What were the properties of mathematical pendulum?In this simple system under the influence of a periodic perturbation there is chaos.In the case where the point of suspension does not move, and oscillates the pendulum appears at a new position of equilibrium.If rapid fluctuations up and down the mechanical system becomes stable position "upside down."It also has its name.It is called the Kapitza pendulum.
Properties
pendulum Pendulum has very interesting properties.All of them are supported by well-known physical laws.The period of oscillation of the pendulum any other depends of various factors such as the size and shape of the body, the distance between the point of suspension and the center of gravity, weight distribution with respect to this point.That is why the definition of the period of the hanging body is quite challenging.It is much easier to calculate the period of a simple pendulum, the formula of which is given below.As a result of observations of such mechanical systems can be set such laws:
• If, while maintaining the same length of the pendulum, suspended various loads, the period of oscillation received the same, although their weight will vary greatly.Therefore, the period of such a pendulum is independent of the load mass.
• If the system starts to deflect the pendulum is not too big, but different angles, it will fluctuate with the same period but in different amplitudes.As long as the deviation from the center of balance is not too large fluctuations in their form are close enough harmonic.The period of the pendulum does not depend on the vibrational amplitude.This property of the mechanical system is called isochronism (in Greek "chronos" - time "Izosov" - equal).
period of a simple pendulum
This figure represents a period of natural oscillations.Despite the complicated wording, the process is very simple.If the length of the thread of a simple pendulum L, and the gravitational acceleration g, then this value is:
T = 2π√L / g
small period of natural oscillations in no way independent of the mass of the pendulum and the amplitude of oscillation.In this case, the pendulum moves as a mathematical length from here.
Fluctuations mathematical pendulum
Pendulum oscillates, which can be described by a simple differential equation:
x + ω2 sin x = 0,
where x (t) - unknown function (this is the angle of deviation from the lower equilibrium positiontime t, expressed in radians);ω - a positive constant, which is determined by the parameters of the pendulum (ω = √g / L, where g - is the acceleration due to gravity, and L - length of a simple pendulum (suspension).
equation of small oscillations near the equilibrium position (harmonic equation) is as follows:
x + ω2 sin x = 0
vibrational motion of the pendulum
Pendulum, which makes small oscillations, moving sinusoid. The differential equation of the second order meets all the requirements and parameters of such a movement. To determine the path you need to set the speed and coordinates,which later determined the independent constants:
x = A sin (θ0 + ωt),
where θ0 - the initial phase, A - amplitude of oscillation, ω - angular frequency, which is determined from the equation of motion.
Pendulum (the formula for largeamplitudes)
This mechanical system, make their vibrations with a significant amplitude is subject to more complex traffic laws.For such a pendulum they are calculated according to the formula:
sin x / 2 = u * sn (ωt / u),
where sn - Jacobi sine, which for u & lt;1 is a periodic function, and for small u it coincides with the simple trigonometric sine.U values determined by the following expression:
u = (ε + ω2) / 2ω2,
where ε = E / mL2 (mL2 - energy of the pendulum).
Determining the oscillation period of a nonlinear pendulum is performed by the formula:
T = 2π / Ω,
where Ω = π / 2 * ω / 2K (u), K - elliptic integral, π - 3,14.
pendulum movement on separatrix
called separatrix trajectory of the dynamic system, in which a two-dimensional phase space.Pendulum moves on noncyclic.In an infinitely distant point in time he falls from the uppermost position in the direction of zero velocity and then gradually gaining it.He eventually stopped, returning to its original position.
If the amplitude of oscillation of the pendulum approaches the number π , this suggests that the motion in the phase plane is close to the separatrix.In this case, under the influence of small periodic driving force mechanical system exhibits chaotic behavior.
In the event of a simple pendulum from the equilibrium position with an angle φ occurs tangential gravity Fτ = -mg sin φ."Minus" sign means that the tangential component is directed to the opposite side of the pendulum.In designating by x pendulum displacement along the arc of a circle with radius L of its angular displacement is equal to φ = x / L.Isaac Newton's Second Law, designed for projections of the vector acceleration and give the desired value:
mg τ = Fτ = -mg sin x / L
Based on this ratio, it is clear that the pendulum is a nonlinear system, because the forcewhich tends to return it to a position of equilibrium is not always proportional to the displacement of x, and sin x / L.
Only when the mathematical pendulum carries out small vibrations, it is the harmonic oscillator.In other words, it becomes a mechanical system capable of performing harmonic oscillations.This approximation is valid for almost angles 15-20 °.Pendulum with large amplitudes is not harmonious.
Newton's law for small oscillations of a pendulum
If the mechanical system performs small oscillations, the 2nd law of Newton will look like this:
mg τ = Fτ = -m * g / L * x.
On this basis, we can conclude that the tangential acceleration of a simple pendulum is proportional to its displacement with the sign "minus".This is a condition whereby the system becomes a harmonic oscillator.Module proportionality factor between the displacement and the acceleration is equal to the square of the angular frequency:
ω02 = g / L;ω0 = √ g / L.
This formula reflects the natural frequency of small oscillations of this type of pendulum.On this basis,
T = 2π / ω0 = 2π√ g / L.
Calculations based on the law of conservation of energy
Properties of oscillatory motion of the pendulum can be described with the help of the law of conservation of energy.It should be borne in mind that the potential energy of the pendulum in a gravitational field is equal to:
E = mgΔh = mgL (1 - cos α) = mgL2sin2 α / 2
full mechanical kinetic energy equal to or maximum potential: Epmax = Ekmsx = E
After you have written the law of conservation of energy, taking the derivative of the left and right sides of the equation:
Ep + Ek = const
Since the derivative of the constant values equal to 0, then (Ep + Ek) '= 0. The derivative is equal to the sum ofsum derivatives:
Ep '= (mg / L * x2 / 2)' = mg / 2L * 2x * x '= mg / L * v + Ek' = (mv2 / 2) = m / 2 (v2) '= m / 2 * 2v * v '= mv * α,
thus:
Mg / L * xv + mva = v (mg / L * x + m α) = 0.
From the last formula we find:α = - g / L * x.
Practical application of mathematical pendulum
acceleration due to gravity varies with latitude, because the density of the Earth's crust on the planet is not the same.Where rock occur with higher density, it will be somewhat higher.Acceleration of mathematical pendulum is often used for exploration.In seeking the help of a variety of minerals.Simply counting the number of oscillations of a pendulum, can be found in the bowels of the earth coal or ore.This is due to the fact that these resources have a density and mass greater than lying beneath loose rocks.
mathematical pendulum used by such prominent scholars as Socrates, Aristotle, Plato, Plutarch, Archimedes.Many of them believed that the mechanical system can affect the fate and life of man.Archimedes used a mathematical pendulum at his calculations.Nowadays, many psychics and occultists use this mechanical system for the implementation of its prophecies, or the search for missing people.
famous French astronomer and scientist K. Flammarion for their research also used the mathematical pendulum.He claimed that with his help he was able to predict the discovery of a new planet, the appearance of the Tunguska meteorite, and other important events.During the Second World War in Germany (Berlin) is a specialized institute of the pendulum.Today, such research engaged the Munich Institute of Parapsychology.His work with the pendulum the staff of this institution called "radiesteziey."
