A mechanical system that consists of a material point (body) hanging on an inextensible weightless thread (its mass is negligible compared to the body weight) 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. The mathematical pendulum can clearly reveal the essence of many interesting phenomena. With a small amplitude of the oscillation, its motion is called harmonic.
General information about the mechanical system
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 interested in this mechanical system. In 1656, he created the first watch with a pendulum mechanism. They measured time with accuracy that was exceptional for those times. This invention has become a critical step 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 tension of the thread. A flat pendulum on an inextensible string is a system with two degrees of freedom with a connection. When changing only one component, the characteristics of all its parts change. So, if the thread is replaced with a rod, then this mechanical system will have only 1 degree of freedom. What properties does a mathematical pendulum have? In this simplest system, chaos occurs under the influence of periodic perturbation. In the case when the suspension point does not move, but vibrates, a new equilibrium position appears at the pendulum. With rapid up and down vibrations, this mechanical system acquires a stable upside down position. She has her own name. It is called the Kapitsa pendulum.
Pendulum Properties
The mathematical pendulum has very interesting properties. All of them are confirmed by well-known physical laws. The oscillation period of any other pendulum depends on various circumstances, such as the size and shape of the body, the distance between the suspension point 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 similar mechanical systems, the following patterns can be established:
• If, while maintaining the same length of the pendulum, suspend different loads, then the period of their oscillations will be the same, although their masses will vary greatly. Consequently, the period of such a pendulum does not depend on the mass of the cargo.
• If, when starting up the system, the pendulum is rejected by not too large but different angles, then 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 quite close to harmonic. The period of such a pendulum in no way depends on the vibrational amplitude. This property of this mechanical system is called isochronism (translated from Greek as “chronos” - time, “isos” - equal).
Mathematical Pendulum Period
This indicator represents a period of natural oscillations. Despite the complicated formulation, the process itself is very simple. If the length of the thread of the mathematical pendulum L, and the acceleration of gravity g, then this value is equal to:
T = 2π√L / g
The period of small natural vibrations does not in any way depend on the mass of the pendulum and the amplitude of the oscillations. In this case, the pendulum moves as a mathematical one with a reduced length.
Oscillations of a mathematical pendulum
The 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 lower equilibrium 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 acceleration of gravity, and L is the length of the mathematical pendulum (suspension).
The equation of small oscillations near the equilibrium position (harmonic equation) looks like this:
x + ω2 sin x = 0
Pendulum oscillations
The mathematical pendulum, which performs 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, it is necessary to set the speed and coordinate, from which independent constants are then determined:
x = A sin (θ 0 + ωt),
where θ 0 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 vibrates with considerable amplitude, obeys more complex laws of motion. For such a pendulum, they are calculated by the formula:
sin x / 2 = u * sn (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).
The period of oscillation of a nonlinear pendulum is determined by the formula:
T = 2π / Ω,
where Ω = π / 2 * ω / 2K (u), K is the elliptic integral, π is 3.14.
The motion of the pendulum along the separatrix
Separatrix is the trajectory of a dynamical system in which a two-dimensional phase space. The mathematical pendulum moves along it non-periodically. At an infinitely distant moment of time, he falls from the extreme upper position to the side at zero speed, then gradually picks it up. In the end, he stops, returning to its original position.
If the amplitude of the oscillations of the pendulum approaches the number π , this indicates that the motion on the phase plane approaches the separatrix. In this case, under the action of a small coercive periodic force, the mechanical system exhibits chaotic behavior.
When the mathematical pendulum deviates from the equilibrium position with a certain angle φ, a tangent of gravity appears Fτ = –mg sin φ. The minus sign means that this tangent component is directed in the direction opposite to the deviation of the pendulum. When denoting by x the displacement of the pendulum along an arc of a circle with radius L, its angular displacement is equal to φ = x / L. The second law of Isaac Newton, intended for projections of the vector of acceleration and force, will give the desired value:
mg τ = Fτ = –mg sin x / L
Based on this relation, it is clear that this pendulum is a nonlinear system, since the force that tends to return it to the equilibrium position is always proportional not to the displacement x, but to sin x / L.
Only when the mathematical pendulum performs small oscillations, is it a harmonic oscillator. In other words, it becomes a mechanical system capable of performing harmonic vibrations. Such an approximation is practically valid for angles of 15–20 °. Oscillations of a pendulum with large amplitudes are not harmonic.
Newton's law for small pendulum vibrations
If a given 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 a 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 proportionality coefficient 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.
Energy conservation law calculations
The properties of the oscillatory movements of the pendulum can also be described using the law of conservation of energy. It should be borne in mind that the potential energy of the pendulum in the gravitational field is:
E = mg∆h = mgL (1 - cos α) = mgL2sin2 α / 2
The total mechanical energy is equal to the 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 the constant values is 0, then (Ep + Ek) '= 0. The derivative of the sum is the sum of the 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,
hence:
Mg / L * xv + mva = v (mg / L * x + m α) = 0.
Based on the last formula, we find: α = - g / L * x.
The practical application of the mathematical pendulum
The acceleration of gravity varies with geographical 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 slightly higher. Acceleration of a mathematical pendulum is often used for geological exploration. In it, they are looking for various minerals. Just by calculating the number of oscillations 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 lying beneath them.
Such outstanding scientists as Socrates, Aristotle, Plato, Plutarch, Archimedes used the mathematical pendulum. 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 psychics use this mechanical system to fulfill their prophecies or search for missing people.
The famous French astronomer and natural scientist 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, a specialized pendulum Institute worked in Germany (Berlin). Today, the Munich Institute of Parapsychology is engaged in such studies. Employees of this institution call their work with a pendulum "radio esthesia."