Mathematical pendulum called material point suspended on a weightless and inextensible thread attached to the 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 \(\vec F\) and the elastic force \(\vec F_(ynp)\) of the thread acting on it are mutually compensated.

Let's bring the pendulum out of the equilibrium position (deflecting it, for example, to position A) and let it go without initial speed (Fig. 13.11). In this case, the forces \(\vec F\) and \(\vec F_(ynp)\) do not balance each other. The tangential component of gravity \(\vec F_\tau\), acting on the pendulum, gives it a tangential acceleration \(\vec a_\tau\) (component of the total acceleration directed along the tangent to the trajectory of the mathematical pendulum), and the pendulum begins to move to the equilibrium position with increasing modulus of speed. The tangential component of gravity \(\vec F_\tau\) is thus the restoring force. The normal component \(\vec F_n\) of gravity is directed along the thread against the elastic force \(\vec F_(ynp)\). The resultant of the forces \(\vec F_n\) and \(\vec F_(ynp)\) gives the pendulum a normal acceleration \(~a_n\), which changes the direction of the velocity vector, and the pendulum moves along an arc ABCD.

The closer the pendulum approaches the equilibrium position C, the smaller the value of the tangential component \(~F_\tau = F \sin \alpha\) 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 \(\vec F_\tau\) is directed against the speed. With an increase in the deflection angle a, the modulus of force \(\vec F_\tau\) increases, and the modulus of velocity decreases, and at point D the pendulum's velocity 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|). Denote the length of the suspension thread l, and the mass of the pendulum - m.

Figure 13.11 shows that \(~F_\tau = F \sin \alpha\), where \(\alpha =\frac(S)(l).\) At small angles \(~(\alpha<10^\circ)\) отклонения маятника \(\sin \alpha \approx \alpha,\) поэтому

\(F_\tau = -F\frac(S)(l) = -mg\frac(S)(l).\)

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 \(m \vec a = m \vec g + F_(ynp).\) We project the vector quantities of this equation onto the direction of the tangent to the trajectory of the mathematical pendulum

\(~F_\tau = ma_\tau .\)

From these equations we get

\(a_\tau = -\frac(g)(l)S\) - 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 in the form \. Comparing it with the equation of harmonic oscillations \(~a_x + \omega^2x = 0\) (see § 13.3), we can conclude that the mathematical pendulum performs 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. Hence, free oscillations of a mathematical pendulum with small deviations are harmonic.

Denote \(\frac(g)(l) = \omega^2.\) Whence \(\omega = \sqrt \frac(g)(l)\) is the cyclic frequency of the pendulum.

The period of oscillation of the pendulum \(T = \frac(2 \pi)(\omega).\) Therefore,

\(T = 2 \pi \sqrt( \frac(l)(g) )\)

This expression is called 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 a mathematical pendulum: 1) does not depend on its mass and oscillation amplitude; 2) is proportional to the square root of the length of the pendulum and inversely proportional to the square root of the gravitational 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 if two conditions are met simultaneously: 1) the oscillations of the pendulum must be small; 2) the suspension point of the pendulum must be at rest or move uniformly rectilinearly relative to the inertial frame of reference in which it is located.

If the suspension point of a mathematical pendulum moves with acceleration \(\vec a\), 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

\(T = 2 \pi \sqrt( \frac(l)(g") )\)

where \(~g"\) is the "effective" acceleration of the pendulum in a non-inertial reference frame. It is equal to the geometric sum of the free fall acceleration \(\vec g\) and the vector opposite to the vector \(\vec a\), i.e. it can be calculated using the formula

\(\vec g" = \vec g + (- \vec a).\)

Literature

Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - S. 374-376.

Pendulum Foucault- a pendulum, which is used to experimentally demonstrate the daily rotation of the Earth.

The Foucault pendulum is a massive weight suspended on a wire or thread, the upper end of which is reinforced (for example, with a cardan joint) so that it allows the pendulum to swing in any vertical plane. If the Foucault pendulum is deflected from the vertical and released without initial velocity, then the forces of gravity and tension of the thread acting on the pendulum's weight will lie all the time in the plane of the pendulum's swings and will not be able to cause its rotation with respect to the stars (to the inertial frame of reference associated with the stars) . An observer who is on the Earth and rotates with it (i.e., located in a non-inertial frame of reference) will see that the swing plane of the Foucault pendulum slowly rotates relative to the earth's surface in the direction opposite to the direction of the Earth's rotation. This confirms the fact of the daily rotation of the Earth.

At the North or South Pole, the swing plane of the Foucault pendulum will rotate 360° per sidereal day (15 o per sidereal hour). At a point on the earth's surface, the geographical latitude of which is equal to φ, the horizon plane rotates around the vertical with an angular velocity of ω 1 = ω sinφ (ω is the Earth's angular velocity module) and the swing plane of the pendulum rotates with the same angular velocity. Therefore, the apparent angular velocity of rotation of the oscillation plane of the Foucault pendulum at latitude φ, expressed in degrees per sidereal hour, has the value rotates). In the Southern Hemisphere, the rotation of the rocking plane will be observed in the direction opposite to that observed in the Northern Hemisphere. The refined calculation gives the value

ω m = 15 o sinφ

Where A- the amplitude of oscillations of the pendulum weight, l- thread length. The additional term, which reduces the angular velocity, the less, the more l. Therefore, to demonstrate the experience, it is advisable to use the Foucault pendulum with the largest possible length of the thread (several tens of meters).

Story

For the first time this device was designed by the French scientist Jean Bernard Leon Foucault.

This device was a five-kilogram brass ball suspended from the ceiling on a two-meter steel wire.

Foucault's first experience was in the basement of his own house. January 8, 1851. This was recorded in the scientist's scientific diary.

February 3, 1851 Jean Foucault demonstrated his pendulum at the Paris Observatory to academicians who received letters like this: "I invite you to follow the rotation of the Earth."

The first public demonstration of the experience took place at the initiative of Louis Bonaparte in the Paris Panthéon in April of that year. A metal ball was suspended under the dome of the Pantheon. weighing 28 kg with a point fixed on it on a steel wire 1.4 mm in diameter and 67 m long. pendulum allowed him to freely oscillate in all directions. Under the attachment point was made a circular fence with a diameter of 6 meters, along the edge of the fence a sandy path was poured in such a way that the pendulum in its movement could draw marks on the sand when crossing it. To avoid a lateral push when starting the pendulum, he was taken aside and tied with a rope, after which the rope burned out. The oscillation period was 16 seconds.

The experiment was a great success and caused a wide response in the scientific and public circles of France and other countries of the world. Only in 1851 were other pendulums created on the model of the first, and Foucault's experiments were carried out at the Paris Observatory, in the Cathedral of Reims, in the church of St. Ignatius in Rome, in Liverpool, in Oxford, Dublin, in Rio de Janeiro, in city ​​of Colombo in Ceylon, New York.

In all these experiments, the dimensions of the ball and the length of the pendulum were different, but they all confirmed the conclusionsJean Bernard Leon Foucault.

Elements of the pendulum, which was demonstrated in the Pantheon, are now kept in the Paris Museum of Arts and Crafts. And Foucault's pendulums are now in many parts of the world: in polytechnic and natural history museums, scientific observatories, planetariums, university laboratories and libraries.

There are three Foucault pendulums in Ukraine. One is kept at the National Technical University of Ukraine “KPI named after I. Igor Sikorsky", the second - at the Kharkiv National University. V.N. Karazin, the third - at the Kharkiv Planetarium.

Mathematical pendulum.

A mathematical pendulum is a material point suspended on an inextensible weightless thread, oscillating in one vertical plane under the action of gravity.

Such a pendulum can be considered a heavy ball of mass m, suspended on a thin thread, the length l of which is much larger than the size of the ball. If it is deflected by an angle α (Fig. 7.3.) from the vertical line, then under the influence of the force F - one of the components of the weight P, it will oscillate. The other component , directed along the thread, is not taken into account, because balanced by the tension in the string. At small displacement angles and, then the x-coordinate can be counted in the horizontal direction. From Fig. 7.3 it can be seen that the weight component perpendicular to the thread is equal to

The moment of force relative to the point O: , and the moment of inertia:
M=FL .
Moment of inertia J in this case
Angular acceleration:

Taking into account these values, we have:

(7.8)

His decision
,

where and

(7.9)

As you can see, the period of oscillation of a mathematical pendulum depends on its length and the acceleration of gravity and does not depend on the amplitude of the oscillations.

physical pendulum.

A physical pendulum is a rigid body fixed on a fixed horizontal axis (suspension axis) that does not pass through the center of gravity and oscillates about this axis under the action of gravity. Unlike a mathematical pendulum, the mass of such a body cannot be considered as a point mass.

At small deflection angles α (Fig. 7.4), the physical pendulum also performs harmonic oscillations. We will assume that the weight of the physical pendulum is applied to its center of gravity at point C. The force that returns the pendulum to the equilibrium position, in this case, will be the component of gravity - the force F.

The minus sign on the right side means that the force F is directed towards decreasing the angle α. Taking into account the smallness of the angle α

To derive the law of motion of mathematical and physical pendulums, we use the basic equation for the dynamics of rotational motion

Moment of force: cannot be determined explicitly. Taking into account all the quantities included in the original differential equation of the oscillations of a physical pendulum, it has the form

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:

1. does not depend on its mass and amplitude of oscillations;
2. 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:

1. pendulum oscillations should be small;
2. 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