Definition
Mathematical pendulum- this is a special case of a physical pendulum, the mass of which is at one point.
Usually, a small ball (material point), having a large mass, suspended on a long inextensible thread (suspension) is considered to be a mathematical pendulum. This is an idealized system that oscillates under the influence of gravity. Only for angles of the order of 50-100 is the mathematical pendulum a harmonic oscillator, that is, it performs harmonic oscillations.
Studying the swing of a chandelier on a long chain, Galileo studied the properties of a mathematical pendulum. He realized that the oscillation period of a given system does not depend on the amplitude at small deflection angles.
The formula for the oscillation period of a mathematical pendulum
Let the suspension point of the pendulum be fixed. A load suspended from a pendulum thread moves along an arc of a circle (Fig.1(a)) with acceleration, and some restoring force ($\overline(F)$) acts on it. This force changes as the load moves. As a result, the calculation of motion becomes complex. Let's introduce some simplifications. Let the pendulum oscillate not in a plane, but describe a cone (Fig. 1 (b)). The load in this case moves in a circle. The period of oscillations of interest to us will coincide with the period of the conical movement of the load. The period of revolution of a conical pendulum around the circumference is equal to the time that the weight spends on one turn around the circumference:
where $L$ is the circumference; $v$ - the speed of the cargo movement. If the angles of deviation of the thread from the vertical are small (small oscillation amplitudes), then it is assumed that the restoring force ($F_1$) is directed along the radius of the circle that the load describes. Then this force is equal to the centripetal force:
Consider similar triangles: AOB and DBC (Fig. 1 (b)).
We equate the right parts of expressions (2) and (3), we express the speed of movement of the load:
\[\frac(mv^2)(R)=mg\frac(R)(l)\ \to v=R\sqrt(\frac(g)(l))\left(4\right).\]
We substitute the resulting speed into formula (1), we have:
\ \
From formula (5) we see that the period of a mathematical pendulum depends only on the length of its suspension (the distance from the suspension point to the center of gravity of the load) and the free fall acceleration. Formula (5) for the period of a mathematical pendulum is called the Huygens formula; it is fulfilled when the suspension point of the pendulum does not move.
Using the dependence of the oscillation period of a mathematical pendulum on the free fall acceleration, the value of this acceleration is determined. To do this, measure the length of the pendulum, considering a large number of oscillations, find the period $T$, then calculate the acceleration of free fall.
Examples of problems with a solution
Example 1
Exercise. As you know, the magnitude of the acceleration of free fall depends on latitude. What is the acceleration of free fall at the latitude of Moscow if the period of oscillation of a mathematical pendulum of length $l=2.485\cdot (10)^(-1)$m is T=1 c?\textit()
Solution. As a basis for solving the problem, we take the formula for the period of a mathematical pendulum:
Let us express from (1.1) the free fall acceleration:
Let's calculate the desired acceleration:
Answer.$g=9.81\frac(m)(s^2)$
Example 2
Exercise. What will be the period of oscillation of a mathematical pendulum if the point of its suspension moves vertically downward 1) at a constant speed? 2) with acceleration $a$? The length of the thread of this pendulum is $l.$
Solution. Let's make a drawing.
1) The period of a mathematical pendulum whose suspension point moves uniformly is equal to the period of a pendulum with a fixed suspension point:
2) The acceleration of the pendulum's suspension point can be considered as the appearance of an additional force equal to $F=ma$, which is directed against the acceleration. That is, if the acceleration is directed upwards, then the additional force is directed downwards, which means that it is added to the force of gravity ($mg$). If the suspension point moves with downward acceleration, then the additional force is subtracted from the force of gravity.
The period of a mathematical pendulum that oscillates and for which the suspension point moves with acceleration, we find as:
Answer. 1) $T_1=2\pi \sqrt(\frac(l)(g))$; 2) $T_1=2\pi \sqrt(\frac(l)(g-a))$
The period of oscillation of a physical pendulum depends on many circumstances: on the size and shape of the body, on the distance between the center of gravity and the point of suspension, and on the distribution of body mass relative to this point; therefore, calculating the period of a suspended body is a rather difficult task. The situation is simpler for the mathematical pendulum. From observations of such pendulums, the following simple laws can be established.
1. If, while maintaining the same length of the pendulum (the distance from the point of suspension to the center of gravity of the load), different loads are suspended, then the oscillation period will be the same, although the masses of the loads differ greatly. The period of a mathematical pendulum does not depend on the mass of the load.
2. If, when starting the pendulum, it is deflected to different (but not too large) angles, then it will oscillate with the same period, although with different amplitudes. As long as the amplitudes are not too large, the oscillations are close enough in their form to harmonic (§ 5) and the period of the mathematical pendulum does not depend on the amplitude of the oscillations. This property is called isochronism (from the Greek words "isos" - equal, "chronos" - time).
This fact was first established in 1655 by Galileo allegedly under the following circumstances. Galileo observed in the Pisa Cathedral the swinging of a chandelier on a long chain, which was pushed when ignited. During the course of the service, the amplitude of the swings gradually faded (§ 11), i.e., the amplitude of the oscillations decreased, but the period remained the same. Galileo used his own pulse as an indicator of time.
We now derive a formula for the period of oscillation of a mathematical pendulum.
Rice. 16. Oscillations of a pendulum in a plane (a) and movement along a cone (b)
When the pendulum swings, the load moves accelerated along an arc (Fig. 16, a) under the action of a restoring force, which changes during movement. The calculation of the motion of a body under the action of a non-constant force is rather complicated. Therefore, for simplicity, we will proceed as follows.
Let us make the pendulum not oscillate in one plane, but describe the cone so that the load moves in a circle (Fig. 16, b). This movement can be obtained by adding two independent vibrations: one still in the plane of the drawing and the other in the perpendicular plane. Obviously, the periods of both of these plane oscillations are the same, since any oscillation plane is no different from any other. Consequently, the period of the complex movement - the rotation of the pendulum along the cone - will be the same as the period of the swing of the water plane. This conclusion can be easily illustrated by direct experience, taking two identical pendulums and telling one of them to swing in a plane, and the other to rotate along a cone.
But the period of revolution of the "conical" pendulum is equal to the length of the circle described by the load, divided by the speed:
If the angle of deviation from the vertical is small (small amplitudes), then we can assume that the restoring force is directed along the radius of the circle, i.e., equal to the centripetal force:
On the other hand, it follows from the similarity of triangles that . Since , then from here
Equating both expressions to each other, we get for the velocity of circulation
Finally, substituting this into the period expression, we find
So, the period of a mathematical pendulum depends only on the acceleration of free fall and on the length of the pendulum, i.e., the distance from the point of suspension to the center of gravity of the load. From the obtained formula it follows that the period of the pendulum does not depend on its mass and on the amplitude (provided that it is sufficiently small). In other words, we obtained by calculation those basic laws that were established earlier from observations.
But our theoretical derivation gives us more: it allows us to establish a quantitative relationship between the period of the pendulum, its length and the acceleration of free fall. The period of a mathematical pendulum is proportional to the square root of the ratio of the length of the pendulum to the acceleration due to gravity. The coefficient of proportionality is .
A very accurate way of determining this acceleration is based on the dependence of the period of the pendulum on the acceleration of free fall. By measuring the length of the pendulum and determining the period from a large number of oscillations, we can calculate using the formula obtained. This method is widely used in practice.
It is known (see Volume I, §53) that the acceleration of free fall depends on the geographical latitude of the place (at the pole, and at the equator). Observations on the swing period of a certain reference pendulum make it possible to study the distribution of free fall acceleration over latitude. This method is so accurate that even more subtle differences in meaning on the earth's surface can be detected with its help. It turns out that even on the same parallel, the values \u200b\u200bare different at different points on the earth's surface. These anomalies in the distribution of gravitational acceleration are associated with the uneven density of the earth's crust. They are used to study the distribution of density, in particular, to detect the occurrence of any minerals in the thickness of the earth's crust. Extensive gravimetric changes, which made it possible to judge the occurrence of dense masses, were carried out in the USSR in the region of the so-called Kursk magnetic anomaly (see Volume II, § 130) under the guidance of the Soviet physicist Pyotr Petrovich Lazarev. In combination with data on the anomaly of the earth's magnetic field, these gravimetric data made it possible to establish the distribution of the occurrence of iron masses, which determine the Kursk magnetic and gravitational anomalies.
A mechanical system, which 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.
General information about the mechanical system
The formula for the period of oscillation 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 exceptional accuracy for those times. This invention became the most important stage in the development of physical experiments and practical activities.
If the pendulum is in the equilibrium position (hanging vertically), then it 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 the pendulum of Kapitsa.
pendulum properties
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 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 regularities can be established:
If, while maintaining the same length of the pendulum, different weights are suspended, then the period of their oscillations will turn out to be the same, although their masses will differ greatly. Therefore, the period of such a pendulum does not depend on the mass of the load.
If, when starting the system, the pendulum is deflected 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 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).
The 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 free fall acceleration is g, then this value is equal to:
The period of the small ones is in no way dependent on the mass of the pendulum and the amplitude of the oscillations. In this case, the pendulum moves like a mathematical pendulum with a reduced length.
Oscillations 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 angle of deviation from the lower equilibrium position at time t, expressed in radians); ω is a positive constant that is determined from the parameters of the pendulum (ω = √g/L, where g is the gravitational acceleration 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
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 \u003d 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 makes its oscillations with a significant amplitude, is subject to 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 Jacobian sine, which for u< 1 является периодической функцией, а при малых u он совпадает с простым тригонометрическим синусом. Значение u определяют следующим выражением:
u = (ε + ω2)/2ω2,
where ε = E/mL2 (mL2 is the energy of the pendulum).
The oscillation period of a non-linear pendulum is determined by the formula:
where Ω = π/2 * ω/2K(u), K is the elliptic integral, π - 3,14.
The movement of the pendulum along the separatrix
A separatrix is a trajectory of a dynamical system that has 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 oscillation 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 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, which is for projections and force, will give the desired value:
mg τ = Fτ = -mg sinx/L
Based on this relationship, it can be seen that this pendulum is a non-linear system, since the force that tends to return it to its equilibrium position is always proportional not to the displacement x, but to sin x/L.
Only when the mathematical pendulum makes small oscillations is it 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 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 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 factor 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 a pendulum can also be described using the law of conservation of energy. In this case, it should be taken into account that the pendulum in the field of gravity is equal to:
E = mg∆h = mgL(1 - cos α) = mgL2sin2 α/2
Total equals kinetic or maximum potential: Epmax = Ekmsx = E
After the law of conservation of energy is written, the derivative of the right and left sides of the equation is taken:
Since the derivative of 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,
hence:
Mg/L*xv + mva = v (mg/L*x + mα) = 0.
Based on the last formula, we find: α = - g/L*x.
Practical application of the mathematical pendulum
Acceleration varies with geographic latitude, since the density of the earth's crust is not the same throughout the planet. 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 psychics use this mechanical system to fulfill their prophecies or search for missing people.
The famous French astronomer and naturalist C. 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”.
Mathematical pendulum called a material point suspended on a weightless and inextensible thread attached to a suspension and located in the field of gravity (or other force).
We study the oscillations of a mathematical pendulum in an inertial frame of 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 (an ideal mathematical pendulum). Initially, the pendulum is at rest in the equilibrium position C. In this case, the force of gravity acting on it and the force of elasticity F?ynp of the thread are mutually compensated.
We bring the pendulum out of the equilibrium position (deflecting it, for example, to position A) and let it go without initial velocity (Fig. 1). In this case, the forces and do not balance each other. The tangential component of gravity, acting on the pendulum, gives it a tangential acceleration a?? (the component of the total acceleration directed along the tangent to the trajectory of the mathematical pendulum), and the pendulum begins to move towards the equilibrium position with an increasing speed in absolute value. The tangential component of gravity is thus the restoring force. The normal component of gravity is directed along the thread against the elastic force. The resultant force and tells the pendulum normal acceleration, which changes the direction of the velocity vector, and the pendulum moves along the arc ABCD.
The closer the pendulum approaches the equilibrium position C, the smaller the value of the tangential component becomes. In the equilibrium position, it is equal to zero, and the speed reaches its maximum value, and the pendulum moves further by inertia, rising upward along the arc. In this case, the component is directed against the speed. With an increase in the angle of deflection a, the modulus of force 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 again passed it by inertia, the pendulum, slowing down, will reach point A (no friction), i.e. makes a full swing. After that, the movement of the pendulum will be repeated in the sequence already described.
We obtain an equation describing the free oscillations of a mathematical pendulum.
Let the pendulum at a given moment of time be at point B. Its displacement S from the equilibrium position at this moment is equal to the length of the arc CB (i.e. S = |CB|). Let us denote the length of the suspension thread as l, and the mass of the pendulum as m.
Figure 1 shows that , where . At small angles () pendulum deflection, therefore
The minus sign in this formula is put 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. We project the vector quantities of this equation onto the direction of the tangent to the trajectory of the mathematical pendulum
From these equations we get
Dynamic equation of motion of a mathematical pendulum. The tangential acceleration of a mathematical pendulum is proportional to its displacement and is directed towards the equilibrium position. This equation can be written as
Comparing it with the equation of harmonic oscillations 
, we can conclude that the mathematical pendulum makes harmonic oscillations. And since the considered oscillations of the pendulum occurred under the action of only internal forces, these were free oscillations of the pendulum. Consequently, free oscillations of a mathematical pendulum with small deviations are harmonic.
Denote
Cyclic frequency of pendulum oscillations.
The period of oscillation of the pendulum. Hence,
This expression is called the Huygens formula. It determines the period of free oscillations of the mathematical pendulum. It follows from the formula that at small angles of deviation from the equilibrium position, the oscillation period of the mathematical pendulum:
- does not depend on its mass and amplitude of oscillations;
- proportional to the square root of the length of the pendulum and inversely proportional to the square root of the free fall acceleration.
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 when two conditions are met simultaneously:
- pendulum oscillations should be small;
- the suspension point of the pendulum must be at rest or move uniformly rectilinearly relative to the inertial reference frame in which it is located.
If the suspension point of a mathematical pendulum moves with acceleration, then the tension force of the thread changes, which leads to a change in the restoring force, and, consequently, the frequency and period of oscillation. As calculations show, the period of oscillation of the pendulum in this case can be calculated by the formula
where is the "effective" acceleration of the pendulum in a non-inertial frame of reference. It is equal to the geometric sum of the gravitational acceleration and the vector opposite to the vector , i.e. it can be calculated using the formula
Definition
Mathematical pendulum- this is an oscillatory system, which is a special case of a physical pendulum, the entire mass of which is concentrated at one point, the center of mass of the pendulum.
Usually a mathematical pendulum is represented as a ball suspended on a long weightless and inextensible thread. This is an idealized system that performs harmonic oscillations under the influence of gravity. A good approximation to a mathematical pendulum is a massive small ball that oscillates on a thin long thread.
Galileo was the first to study the properties of a mathematical pendulum, considering the swing of a chandelier on a long chain. He obtained that the period of oscillation of a mathematical pendulum does not depend on the amplitude. If, when the pendulum is launched, it is deflected at different small angles, then its oscillations will occur with the same period, but with different amplitudes. This property is called isochronism.
The equation of motion of a mathematical pendulum
The mathematical pendulum is a classic example of a harmonic oscillator. It performs harmonic oscillations, which are described by the differential equation:
\[\ddot(\varphi )+(\omega )^2_0\varphi =0\ \left(1\right),\]
where $\varphi $ is the angle of deviation of the thread (suspension) from the equilibrium position.
The solution to equation (1) is the function $\varphi (t):$
\[\varphi (t)=(\varphi )_0(\cos \left((\omega )_0t+\alpha \right)\left(2\right),\ )\]
where $\alpha $ - initial phase of oscillations; $(\varphi )_0$ - oscillation amplitude; $(\omega )_0$ - cyclic frequency.
The oscillation of a harmonic oscillator is an important example of periodic motion. The oscillator serves as a model in many problems of classical and quantum mechanics.
Cyclic frequency and period of oscillation of a mathematical pendulum
The cyclic frequency of a mathematical pendulum depends only on the length of its suspension:
\[\ (\omega )_0=\sqrt(\frac(g)(l))\left(3\right).\]
The oscillation period of the mathematical pendulum ($T$) in this case is equal to:
Expression (4) shows that the period of a mathematical pendulum depends only on the length of its suspension (the distance from the suspension point to the center of gravity of the load) and the free fall acceleration.
Energy equation for a mathematical pendulum
When considering vibrations of mechanical systems with one degree of freedom, it is often taken as the initial not Newton's equation of motion, but the energy equation. Since it is easier to compose, and it is an equation of the first order in time. Let us assume that there is no friction in the system. The law of conservation of energy for a mathematical pendulum making free oscillations (small oscillations) can be written as:
where $E_k$ is the kinetic energy of the pendulum; $E_p$ - potential energy of the pendulum; $v$ - the speed of the pendulum; $x$ - linear displacement of the pendulum weight from the equilibrium position along the arc of a circle of radius $l$, while the angle - displacement is related to $x$ as:
\[\varphi =\frac(x)(l)\left(6\right).\]
The maximum value of the potential energy of a mathematical pendulum is:
Maximum value of kinetic energy:
where $h_m$ is the maximum lifting height of the pendulum; $x_m$ - maximum deviation of the pendulum from the equilibrium position; $v_m=(\omega )_0x_m$ - maximum speed.
Examples of problems with a solution
Example 1
Exercise. What is the maximum height of the ball of a mathematical pendulum if its speed of movement when passing through the equilibrium position was $v$?
Solution. Let's make a drawing.
Let the potential energy of the ball be zero in its equilibrium position (point 0). At this point, the speed of the ball is maximum and equal to $v$ by the condition of the problem. At the point of maximum lifting of the ball above the equilibrium position (point A), the speed of the ball is zero, the potential energy is maximum. Let us write down the law of conservation of energy for the considered two positions of the ball:
\[\frac(mv^2)(2)=mgh\ \left(1.1\right).\]
From equation (1.1) we find the desired height:
Answer.$h=\frac(v^2)(2g)$
Example 2
Exercise. What is the acceleration of gravity if a mathematical pendulum of length $l=1\ m$ oscillates with a period equal to $T=2\ s$? Consider the oscillations of the mathematical pendulum to be small.\textit()
Solution. As a basis for solving the problem, we take the formula for calculating the period of small oscillations:
Let's express the acceleration from it:
Let's calculate the acceleration of gravity:
Answer.$g=9.87\ \frac(m)(s^2)$