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).
Exploring fluctuations mathematical pendulum in inertial system 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 \(\vec F_(ynp)\) informs the pendulum 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 frame of reference. It is equal to the geometric sum of the gravitational 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 φ, the horizon plane rotates around the vertical with an angular velocity of ω 1 = ω sinφ (ω is the Earth's angular velocity modulus) and the pendulum swing plane rotates with the same angular velocity. Therefore, the apparent angular velocity of rotation of the plane of swing 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 sand 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.
A mathematical pendulum is a model of an ordinary pendulum. A mathematical pendulum is a material point that is suspended on a long weightless and inextensible thread.
Bring the ball out of equilibrium and release it. There are two forces acting on the ball: gravity and tension in the string. When the pendulum moves, the force of air friction will still act on it. But we will consider it very small.
Let us decompose the force of gravity into two components: the force directed along the thread, and the force directed perpendicular to the tangent to the trajectory of the ball.
These two forces add up to gravity. The elastic forces of the thread and the component of gravity Fn impart centripetal acceleration to the ball. The work of these forces will be equal to zero, and therefore they will only change the direction of the velocity vector. At any point in time, it will be tangent to the arc of the circle.
Under the action of the gravity component Fτ, the ball will move along the arc of a circle with a speed increasing in absolute value. The value of this force always changes in absolute value; when passing through the equilibrium position, it is equal to zero.
Dynamics of oscillatory motion
The equation of motion of a body oscillating under the action of an elastic force.
General equation of motion:
Oscillations in the system occur under the action of an elastic force, which, according to Hooke's law, is directly proportional to the displacement of the load
Then the equation of motion of the ball will take the following form:
Divide this equation by m, we get the following formula:
And since the mass and coefficient of elasticity are constant values, then the ratio (-k / m) will also be constant. We have obtained an equation that describes the vibrations of a body under the action of an elastic force.
The projection of the acceleration of the body will be directly proportional to its coordinate, taken with the opposite sign.
The equation of motion of a mathematical pendulum
The equation of motion of a mathematical pendulum is described by the following formula:
This equation has the same form as the equation for the movement of a load on a spring. Consequently, the oscillations of the pendulum and the movement of the ball on the spring occur in the same way.
The displacement of the ball on the spring and the displacement of the pendulum body from the equilibrium position change with time according to the same laws.
The pendulums shown in fig. 2, are extended bodies of various shapes and sizes, oscillating around a suspension or support point. Such systems are called physical pendulums. In a state of equilibrium, when the center of gravity is on the vertical below the point of suspension (or support), the force of gravity is balanced (through the elastic forces of the deformed pendulum) by the reaction of the support. When deviating from the equilibrium position, gravity and elastic forces determine at each moment of time the angular acceleration of the pendulum, i.e., determine the nature of its movement (oscillation). We will now consider the dynamics of oscillations in more detail using the simplest example of the so-called mathematical pendulum, which is a small weight suspended on a long thin thread.
In a mathematical pendulum, we can neglect the mass of the thread and the deformation of the weight, i.e., we can assume that the mass of the pendulum is concentrated in the weight, and the elastic forces are concentrated in the thread, which is considered inextensible. Let us now look under the influence of what forces our pendulum oscillates after it is brought out of equilibrium in some way (by push, deflection).
When the pendulum is at rest in the equilibrium position, the force of gravity acting on its weight and directed vertically downwards is balanced by the tension in the thread. In the deflected position (Fig. 15), gravity acts at an angle to the tension force directed along the thread. We decompose the force of gravity into two components: in the direction of the thread () and perpendicular to it (). When the pendulum oscillates, the tension force of the thread slightly exceeds the component - by the value of the centripetal force, which causes the load to move in an arc. The component is always directed towards the equilibrium position; she seems to be striving to restore this position. Therefore, it is often called the restoring force. The modulus is greater, the more the pendulum is deflected.
Rice. 15. The restoring force when the pendulum deviates from the equilibrium position
So, as soon as the pendulum, during its oscillations, begins to deviate from the equilibrium position, say, to the right, a force appears that slows down its movement the more, the farther it is deflected. Ultimately, this force will stop him and drag him back to the equilibrium position. However, as we approach this position, the force will become less and less and in the equilibrium position itself will turn to zero. Thus, the pendulum passes through the equilibrium position by inertia. As soon as it begins to deviate to the left, a force will again appear, growing with an increase in the deviation, but now directed to the right. The movement to the left will again slow down, then the pendulum will stop for a moment, after which the accelerated movement to the right will begin, etc.
What happens to the energy of a pendulum as it swings?
Twice during the period - at the largest deviations to the left and to the right - the pendulum stops, that is, at these moments the speed is zero, which means that the kinetic energy is also zero. But it is precisely at these moments that the center of gravity of the pendulum is raised to the greatest height and, consequently, the potential energy is greatest. On the contrary, at the moments of passage through the equilibrium position, the potential energy is the smallest, and the speed and kinetic energy reach the maximum value.
We assume that the forces of friction of the pendulum on the air and the friction at the point of suspension can be neglected. Then, according to the law of conservation of energy, this maximum kinetic energy is exactly equal to the excess of potential energy in the position of greatest deviation over the potential energy in the equilibrium position.
So, when the pendulum oscillates, a periodic transition of kinetic energy into potential energy and vice versa occurs, and the period of this process is half as long as the period of oscillation of the pendulum itself. However, the total energy of the pendulum (the sum of potential and kinetic energies) is constant all the time. It is equal to the energy that was imparted to the pendulum at the start, no matter whether it is in the form of potential energy (initial deflection) or in the form of kinetic energy (initial push).
This is the case for all vibrations in the absence of friction or any other processes that take energy from the oscillating system or impart energy to it. That is why the amplitude remains unchanged and is determined by the initial deviation or the force of the push.
We get the same changes in the restoring force and the same transition of energy if, instead of hanging the ball on a thread, we make it roll in a vertical plane in a spherical cup or in a trough curved around the circumference. In this case, the role of the thread tension will be assumed by the pressure of the walls of the cup or trough (again, we neglect the friction of the ball against the walls and air).
