Mathematical pendulum: period, acceleration and formulas

A mechanical system that consists of a material point (body) hanging on an inextensible weightless thread (its mass is negligibly small compared to the body weight) in a uniform gravitational 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 visually reveal the essence of many interesting phenomena. With a small amplitude of vibration, 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 I. Newton was very fond of this mechanical system. In 1656, he created the first clock with a pendulum mechanism. They measured time with an accuracy that was exceptional for those times. This invention has become an important stage in the development of physical experiments and practical activities.

If the pendulum is in the equilibrium position (hanging vertically), then the force of gravity will be balanced by the tension of the thread. A flat pendulum on an inextensible thread is a system with two degrees of freedom with a link. When changing only 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 periodic perturbation. In the case when the suspension point does not move but oscillates, a new equilibrium position appears at the pendulum. With rapid oscillations up and down this mechanical system acquires a stable position "upside down". It has its own name. It is called Kapitza's pendulum.

Properties of the pendulum

The 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 a given point. That is why determining the period of the hanging body is quite a challenge. 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 on similar mechanical systems, one can establish such regularities:

• If, keeping 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, at the system start-up, the pendulum is rejected by not too large, but different angles, it will oscillate with the same period, but in different amplitudes. While the deviations from the center of equilibrium are not too great, the fluctuations in their form will be fairly close to harmonic. The period of such a pendulum does not depend on the vibrational amplitude. This property of this mechanical system is called isochronism (in translation from the Greek "chronos" - time, "isos" - equal).

Period of the mathematical pendulum

This indicator is a period of natural oscillations. Despite the complicated formulation, the process itself is very simple. If the length of a string of a mathematical pendulum L, and the acceleration of gravity g, then this value is equal to:

T = 2π√L / g

The period of small natural oscillations does not depend in any measure on the mass of the pendulum and the amplitude of the oscillations. In this case, the pendulum moves as a mathematical pendulum with the given length.

Fluctuation of a 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 deviation angle from the lower equilibrium position at time t, expressed in radians); Ω is a positive constant, which is determined from the pendulum parameters (ω = √g / L, where g is the acceleration due to gravity, and L is the length of the pendulum.

The equation of small oscillations near the equilibrium position (harmonic equation) looks like this:

X + ω2 sin x = 0

Oscillatory motion of the pendulum

A mathematical pendulum, which 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 (θ ₀ + ωt),

Where θ ₀ is the initial phase, A is the amplitude of the oscillation, ω is the cyclic frequency determined from the equation of motion.

Mathematical pendulum (formulas for large amplitudes)

This mechanical system, which oscillates with a significant 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 an elliptic integral, and π is 3.14.

Movement of the pendulum along the separatrix

A separatrix is the trajectory of a dynamical system with a two-dimensional phase space. The mathematical pendulum moves along it non-periodically. At an infinitely far point in time, he falls from the extreme upper position to the side at zero speed, then gradually picks it up. Eventually, it stops, returning to its original position.

If the amplitude of the oscillations of the pendulum approaches π , this indicates that the motion on the phase plane approaches the separatrix. In this case, under the influence of a small forcing periodic force, the mechanical system exhibits chaotic behavior.

When the mathematical pendulum deviates from the equilibrium position with some angle φ, a gravitational tangent arises Fτ = -mg sin φ. The minus sign means that this tangential component is directed in the opposite direction from the deviation of the pendulum. If we denote the displacement of the pendulum through x 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 the projections of the vector of acceleration and force, will give the desired value:

Mg τ = Fτ = -mg sin x / L

Proceeding from 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 sin x / L.

Only when the mathematical pendulum carries out small oscillations, it is a harmonic oscillator. In other words, it becomes a mechanical system capable of performing harmonic oscillations. This approximation is practically valid for angles of 15-20 °. The oscillations of a pendulum with large amplitudes are not harmonic.

Newton's law for small oscillations of a pendulum

If this mechanical system performs small oscillations, Newton's 2nd law will look like this:

Mg τ = Fτ = -m * g / L * x.

Proceeding from this, we can conclude that the tangential acceleration of a mathematical pendulum is proportional to its displacement with the minus sign. This is the condition by which the system becomes a harmonic oscillator. The modulus of proportionality 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 by means of the law of conservation of energy. It should be borne in mind that the potential energy of the pendulum in the gravitational field is equal to:

E = mgΔh = mgL (1 - cos α) = mgL2sin2 α / 2

The total mechanical energy equals the kinetic or maximum potential energy: Epmax = Ekmsx = E

After the law of conservation of energy is recorded, take the derivative of the right and left parts of the equation:

Ep + Ek = const

Since the derivative of the constants is 0, then (Ep + Ek) '= 0. The derivative of the sum is equal to 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,

Consequently:

Mg / L * xv + mva = v (mg / L * x + m α) = 0.

Proceeding from the last formula, we find: α = - g / L * x.

Practical application of the mathematical pendulum

Acceleration of free fall varies with geographical latitude, since the density of the earth's crust throughout the entire planet is not the same. Where there are rocks with greater density, it will be slightly higher. Acceleration of a mathematical pendulum is often used for geological prospecting. It is used to search for various minerals. Just calculating the number of oscillations of the pendulum, you can find in the bowels of the earth coal or ore. This is due to the fact that such fossils have a density and mass greater than the loose rock beneath them.

The mathematical pendulum was used by such outstanding scientists as Socrates, Aristotle, Plato, Plutarch, Archimedes. Many of them believed that this mechanical system can 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 carry out their prophecies or search for missing people.

Famous French astronomer and natural scientist K. Flammarion also used a mathematical pendulum for his research. He claimed that with his help he managed 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 operated in Germany (Berlin). These days, the Munich Institute of Parapsychology is engaged in similar studies. The employees of this institution call their work with a pendulum "radeesthesia".