A mechanical system that consists of a material point (body) hanging on an inextensible weightless thread (its mass is negligible compared to the weight of the body) in a uniform gravity field is called a mathematical pendulum (another name is an oscillator). There are other types of this device. Instead of a thread, a weightless rod can be used. A mathematical pendulum can clearly reveal the essence of many interesting phenomena. With a small amplitude of oscillation, its movement is called harmonic.
Mechanical system overview
The formula for the oscillation period of this pendulum was derived by the Dutch scientist Huygens (1629-1695). This contemporary of I. Newton was very fond of this mechanical system. In 1656 he created the first pendulum clock. They measured time with exceptionalfor those times accuracy. This invention has become a major milestone in the development of physical experiments and practical activities.
If the pendulum is in equilibrium (hanging vertically), then the force of gravity will be balanced by the force of the thread tension. A flat pendulum on an inextensible thread is a system with two degrees of freedom with a connection. When you change just one component, the characteristics of all its parts change. So, if the thread is replaced by a rod, then this mechanical system will have only 1 degree of freedom. What are the properties of a mathematical pendulum? In this simplest system, chaos arises under the influence of a periodic perturbation. In the case when the suspension point does not move, but oscillates, the pendulum has a new equilibrium position. With rapid up and down oscillations, this mechanical system acquires a stable upside down position. She also has her own name. It is called Kapitza's pendulum.
Pendulum properties
Mathematical pendulum has very interesting properties. All of them are confirmed by known physical laws. The period of oscillation of any other pendulum depends on various circumstances, such as the size and shape of the body, the distance between the point of suspension and the center of gravity, the distribution of mass relative to this point. That is why determining the period of a hanging body is a rather difficult task. It is much easier to calculate the period of a mathematical pendulum, the formula of which will be given below. As a result of observations of similarmechanical systems can establish the following patterns:
• If, while maintaining the same length of the pendulum, we hang different weights, then the period of their oscillations will be the same, although their masses will vary greatly. Therefore, the period of such a pendulum does not depend on the mass of the load.
• When starting the system, if the pendulum is deflected by not too large, but different angles, it will begin to oscillate with the same period, but with different amplitudes. As long as the deviations from the center of equilibrium are not too large, the oscillations in their form will be close enough to harmonic ones. The period of such a pendulum does not depend on the oscillation amplitude in any way. This property of this mechanical system is called isochronism (translated from the Greek "chronos" - time, "isos" - equal).
Period of the mathematical pendulum
This indicator represents the period of natural oscillations. Despite the complex wording, the process itself is very simple. If the length of the thread of a mathematical pendulum is L, and the acceleration of gravity is g, then this value is:
T=2π√L/g
The period of small natural oscillations in no way depends on the mass of the pendulum and the amplitude of oscillations. In this case, the pendulum moves like a mathematical pendulum with a reduced length.
Swings of the mathematical pendulum
A mathematical pendulum oscillates, which can be described by a simple differential equation:
x + ω2 sin x=0, where x (t) is an unknown function (this is the angle of deviation from the lowerequilibrium position at time t, expressed in radians); ω is a positive constant, which is determined from the parameters of the pendulum (ω=√g/L, where g is the free fall acceleration and L is the length of the mathematical pendulum (suspension).
The equation of small fluctuations near the equilibrium position (harmonic equation) looks like this:
x + ω2 sin x=0
Oscillatory movements of the pendulum
A mathematical pendulum that makes small oscillations moves along a sinusoid. The second-order differential equation meets all the requirements and parameters of such a motion. To determine the trajectory, you must specify the speed and coordinate, from which independent constants are then determined:
x=A sin (θ0 + ωt), where θ0 is the initial phase, A is the oscillation amplitude, ω is the cyclic frequency determined from the equation of motion.
Mathematical pendulum (formulas for large amplitudes)
This mechanical system, which makes its oscillations with a significant amplitude, obeys more complex laws of motion. For such a pendulum, they are calculated by the formula:
sin x/2=usn(ωt/u), where sn is the Jacobi sine, which for u < 1 is a periodic function, and for small u it coincides with a simple trigonometric sine. The value of u is determined by the following expression:
u=(ε + ω2)/2ω2, where ε=E/mL2 (mL2 is the energy of the pendulum).
Determining the oscillation period of a non-linear pendulumcarried out according to the formula:
T=2π/Ω, where Ω=π/2ω/2K(u), K is the elliptic integral, π - 3, 14.
Movement of the pendulum along the separatrix
A separatrix is a trajectory of a dynamical system with a two-dimensional phase space. The mathematical pendulum moves along it non-periodically. At an infinitely distant moment of time, it falls from the extreme upper position to the side with zero velocity, then gradually picks it up. It eventually stops, returning to its original position.
If the amplitude of the pendulum's oscillations approaches the number π, this indicates that the motion on the phase plane is approaching the separatrix. In this case, under the action of a small driving periodic force, the mechanical system exhibits chaotic behavior.
When the mathematical pendulum deviates from the equilibrium position with a certain angle φ, a tangential force of gravity Fτ=–mg sin φ arises. The minus sign means that this tangential component is directed in the opposite direction from the pendulum deflection. When the displacement of the pendulum along the arc of a circle with radius L is denoted by x, its angular displacement is equal to φ=x/L. The second law of Isaac Newton, designed for projections of the acceleration vector and force, will give the desired value:
mg τ=Fτ=–mg sin x/L
Based on this ratio, it is clear that this pendulum is a non-linear system, since the force that seeks to returnit to the equilibrium position, is always proportional not to the displacement x, but to sin x/L.
Only when the mathematical pendulum makes small oscillations, it is a harmonic oscillator. In other words, it becomes a mechanical system capable of performing harmonic vibrations. This approximation is practically valid for angles of 15–20°. Pendulum oscillations with large amplitudes are not harmonic.
Newton's law for small oscillations of a pendulum
If this mechanical system performs small vibrations, Newton's 2nd law will look like this:
mg τ=Fτ=–m g/L x.
Based on this, we can conclude that the tangential acceleration of the mathematical pendulum is proportional to its displacement with a minus sign. This is the condition due to which the system becomes a harmonic oscillator. The modulus of the proportional gain between displacement and acceleration is equal to the square of the circular frequency:
ω02=g/L; ω0=√ g/L.
This formula reflects the natural frequency of small oscillations of this type of pendulum. Based on this, T=2π/ ω0=2π√ g/L.
Calculations based on the law of conservation of energy
The properties of the oscillatory movements of the pendulum can also be described using the law of conservation of energy. In this case, it should be taken into account that the potential energy of the pendulum in the gravitational field is:
E=mg∆h=mgL(1 – cos α)=mgL2sin2 α/2
Total mechanical energyequals kinetic or maximum potential: Epmax=Ekmsx=E
After the law of conservation of energy is written, take the derivative of the right and left sides of the equation:
Ep + Ek=const
Since the derivative of constant values is 0, then (Ep + Ek)'=0. The derivative of the sum is equal to the sum of the derivatives:
Ep'=(mg/Lx2/2)'=mg/2L2xx'=mg/Lv + Ek'=(mv2/2)=m/2(v2)'=m/22vv'=mv α, hence:
Mg/Lxv + mva=v (mg/Lx + m α)=0.
Based on the last formula, we find: α=- g/Lx.
Practical application of the mathematical pendulum
The acceleration of free fall varies with geographic latitude, since the density of the earth's crust throughout the planet is not the same. Where rocks with a higher density occur, it will be somewhat higher. The acceleration of a mathematical pendulum is often used for geological exploration. It is used to search for various minerals. Simply by counting the number of swings of the pendulum, you can find coal or ore in the bowels of the Earth. This is due to the fact that such fossils have a density and mass greater than the loose rocks underlying them.
The mathematical pendulum was used by such prominent scientists as Socrates, Aristotle, Plato, Plutarch, Archimedes. Many of them believed that this mechanical system could influence the fate and life of a person. Archimedes used a mathematical pendulum in his calculations. Nowadays, many occultists and psychicsuse this mechanical system to fulfill their prophecies or search for missing people.
The famous French astronomer and naturalist K. Flammarion also used a mathematical pendulum for his research. 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) a specialized Pendulum Institute worked. Today, the Munich Institute of Parapsychology is engaged in similar research. The employees of this institution call their work with the pendulum “radiesthesia.”
