(lat. amplitude- magnitude) - this is the largest deviation of the oscillating body from the equilibrium position.
For a pendulum, this is the maximum distance that the ball moves from its equilibrium position (figure below). For oscillations with small amplitudes, this distance can be taken as the length of the arc 01 or 02, as well as the lengths of these segments.
The oscillation amplitude is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve, (see figure below).
Oscillation period.
Oscillation period- this is the smallest period of time after which the system, making oscillations, again returns to the same state in which it was at the initial moment of time, chosen arbitrarily.
In other words, the oscillation period ( T) is the time for which one complete oscillation takes place. For example, in the figure below, this is the time it takes for the weight of the pendulum to move from the rightmost point through the equilibrium point O to the leftmost point and back through the point O again to the far right.
For a full period of oscillation, therefore, the body travels a path equal to four amplitudes. The oscillation period is measured in units of time - seconds, minutes, etc. The oscillation period can be determined from the well-known oscillation graph, (see figure below).
The concept of “oscillation period”, strictly speaking, is valid only when the values of the oscillating quantity are exactly repeated after a certain period of time, that is, for harmonic oscillations. However, this concept is also applied to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency is the number of oscillations per unit of time, for example, in 1 s.
The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, then this means that one oscillation is made for every second. The frequency and period of oscillations are related by the relations:
In the theory of oscillations, the concept is also used cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:
.
Cyclic frequency is the number of oscillations per 2π seconds.
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. Consequently,
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
What is the period of oscillation? What is this quantity, what physical meaning does it have and how to calculate it? In this article, we will deal with these issues, consider various formulas by which the period of oscillations can be calculated, and also find out what relationship exists between such physical quantities as the period and frequency of oscillations of a body / system.
Definition and physical meaning
The period of oscillation is such a period of time in which the body or system makes one oscillation (necessarily complete). In parallel, we can note the parameter at which the oscillation can be considered complete. The role of such a condition is the return of the body to its original state (to the original coordinate). The analogy with the period of a function is very well drawn. Incidentally, it is a mistake to think that it takes place exclusively in ordinary and higher mathematics. As you know, these two sciences are inextricably linked. And the period of functions can be encountered not only when solving trigonometric equations, but also in various branches of physics, namely, we are talking about mechanics, optics and others. When transferring the oscillation period from mathematics to physics, it must be understood as simply a physical quantity (and not a function), which has a direct dependence on the passing time.
What are the fluctuations?
Oscillations are divided into harmonic and anharmonic, as well as periodic and non-periodic. It would be logical to assume that in the case of harmonic oscillations, they occur according to some harmonic function. It can be either sine or cosine. In this case, the coefficients of compression-stretching and increase-decrease may also turn out to be in the case. Also, vibrations are damped. That is, when a certain force acts on the system, which gradually “slows down” the oscillations themselves. In this case, the period becomes shorter, while the frequency of oscillations invariably increases. The simplest experiment using a pendulum demonstrates such a physical axiom very well. It can be spring type, as well as mathematical. It doesn't matter. By the way, the oscillation period in such systems will be determined by different formulas. But more on that later. Now let's give examples.
Experience with pendulums
You can take any pendulum first, there will be no difference. The laws of physics are the laws of physics, that they are respected in any case. But for some reason, the mathematical pendulum is more to my liking. If someone does not know what it is: it is a ball on an inextensible thread that is attached to a horizontal bar attached to the legs (or the elements that play their role - to keep the system in balance). The ball is best taken from metal, so that the experience is clearer.
So, if you take such a system out of balance, apply some force to the ball (in other words, push it), then the ball will begin to swing on the thread, following a certain trajectory. Over time, you can notice that the trajectory along which the ball passes is reduced. At the same time, the ball begins to scurry back and forth faster and faster. This indicates that the oscillation frequency is increasing. But the time it takes for the ball to return to its original position decreases. But the time of one complete oscillation, as we found out earlier, is called a period. If one value decreases and the other increases, then they speak of inverse proportionality. So we got to the first moment, on the basis of which formulas are built to determine the period of oscillations. If we take a spring pendulum for testing, then the law will be observed there in a slightly different form. In order for it to be most clearly represented, we set the system in motion in a vertical plane. To make it clearer, it was first worth saying what a spring pendulum is. From the name it is clear that a spring must be present in its design. And indeed it is. Again, we have a horizontal plane on supports, to which a spring of a certain length and stiffness is suspended. To it, in turn, a weight is suspended. It can be a cylinder, a cube or another figure. It may even be some third-party item. In any case, when the system is taken out of equilibrium, it will begin to perform damped oscillations. The increase in frequency is most clearly seen in the vertical plane, without any deviation. On this experience, you can finish.
So, in their course, we found out that the period and frequency of oscillations are two physical quantities that have an inverse relationship.
Designation of quantities and dimensions
Usually, the oscillation period is denoted by the Latin letter T. Much less often, it can be denoted differently. The frequency is denoted by the letter µ (“Mu”). As we said at the very beginning, a period is nothing more than the time during which a complete oscillation occurs in the system. Then the dimension of the period will be a second. And since the period and frequency are inversely proportional, the frequency dimension will be unit divided by a second. In the record of tasks, everything will look like this: T (s), µ (1/s).
Formula for a mathematical pendulum. Task #1
As in the case with the experiments, I decided first of all to deal with the mathematical pendulum. We will not go into the derivation of the formula in detail, since such a task was not originally set. Yes, and the conclusion itself is cumbersome. But let's get acquainted with the formulas themselves, find out what kind of quantities they include. So, the formula for the period of oscillation for a mathematical pendulum is as follows:
Where l is the length of the thread, n \u003d 3.14, and g is the acceleration of gravity (9.8 m / s ^ 2). The formula should not cause any difficulties. Therefore, without additional questions, we will immediately proceed to solving the problem of determining the period of oscillation of a mathematical pendulum. A metal ball weighing 10 grams is suspended from an inextensible thread 20 centimeters long. Calculate the period of oscillation of the system, taking it for a mathematical pendulum. The solution is very simple. As in all problems in physics, it is necessary to simplify it as much as possible by discarding unnecessary words. They are included in the context in order to confuse the decisive one, but in fact they have absolutely no weight. In most cases, of course. Here it is possible to exclude the moment with “inextensible thread”. This phrase should not lead to a stupor. And since we have a mathematical pendulum, we should not be interested in the mass of the load. That is, the words about 10 grams are also simply designed to confuse the student. But we know that there is no mass in the formula, so with a clear conscience we can proceed to the solution. So, we take the formula and simply substitute the values \u200b\u200binto it, since it is necessary to determine the period of the system. Since no additional conditions were specified, we will round the values to the 3rd decimal place, as is customary. Multiplying and dividing the values, we get that the period of oscillation is 0.886 seconds. Problem solved.
Formula for a spring pendulum. Task #2
Pendulum formulas have a common part, namely 2p. This value is present in two formulas at once, but they differ in the root expression. If in the problem concerning the period of a spring pendulum, the mass of the load is indicated, then it is impossible to avoid calculations with its use, as was the case with the mathematical pendulum. But you should not be afraid. This is how the period formula for a spring pendulum looks like:
In it, m is the mass of the load suspended from the spring, k is the coefficient of spring stiffness. In the problem, the value of the coefficient can be given. But if in the formula of a mathematical pendulum you don’t really clear up - after all, 2 out of 4 values are constants - then a 3rd parameter is added here, which can change. And at the output we have 3 variables: the period (frequency) of oscillations, the coefficient of spring stiffness, the mass of the suspended load. The task can be oriented towards finding any of these parameters. Searching for a period again would be too easy, so we'll change the condition a bit. Find the stiffness of the spring if the full swing time is 4 seconds and the weight of the spring pendulum is 200 grams.
To solve any physical problem, it would be good to first make a drawing and write formulas. They are half the battle here. Having written the formula, it is necessary to express the stiffness coefficient. It is under our root, so we square both sides of the equation. To get rid of the fraction, multiply the parts by k. Now let's leave only the coefficient on the left side of the equation, that is, we divide the parts by T^2. In principle, the problem could be a little more complicated by setting not a period in numbers, but a frequency. In any case, when calculating and rounding (we agreed to round up to the 3rd decimal place), it turns out that k = 0.157 N/m.
The period of free oscillations. Free period formula
The formula for the period of free oscillations is understood to mean those formulas that we examined in the two previously given problems. They also make up an equation of free oscillations, but there we are already talking about displacements and coordinates, and this question belongs to another article.
1) Before taking on a task, write down the formula that is associated with it.
2) The simplest tasks do not require drawings, but in exceptional cases they will need to be done.
3) Try to get rid of roots and denominators if possible. An equation written in a line that does not have a denominator is much more convenient and easier to solve.
In technology and the world around us, we often have to deal with periodical(or almost periodic) processes that repeat at regular intervals. Such processes are called oscillatory.
Vibrations are one of the most common processes in nature and technology. Wings of insects and birds in flight, high-rise buildings and high-voltage wires under the action of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness, sound is fluctuations in air density and pressure, radio waves - periodic changes in the strength of electric and magnetic fields, visible light is also electromagnetic oscillations, only with a slightly different wavelength and frequency, earthquakes - soil vibrations, pulse beats - periodic contractions of the human heart muscle, etc.
Vibrations are mechanical, electromagnetic, chemical, thermodynamic and various others. Despite this diversity, they all have much in common.
Oscillatory phenomena of various physical nature are subject to general laws. For example, current oscillations in an electrical circuit and oscillations of a mathematical pendulum can be described by the same equations. The commonality of oscillatory regularities makes it possible to consider oscillatory processes of various nature from a single point of view. A sign of oscillatory motion is its periodicity.
Mechanical vibrations -this ismovements that repeat exactly or approximately at regular intervals.
Examples of simple oscillatory systems are a weight on a spring (spring pendulum) or a ball on a thread (mathematical pendulum).
During mechanical vibrations, the kinetic and potential energies change periodically.
At maximum deviation body from the equilibrium position, its speed, and consequently, and kinetic energy goes to zero. In this position potential energy oscillating body reaches the maximum value. For a load on a spring, the potential energy is the energy of the elastic deformation of the spring. For a mathematical pendulum, this is the energy in the Earth's gravitational field.
When a body in its motion passes through equilibrium position, its speed is maximum. The body skips the equilibrium position according to the law of inertia. At this moment it has maximum kinetic and minimum potential energy. An increase in kinetic energy occurs at the expense of a decrease in potential energy.
With further movement, the potential energy begins to increase due to the decrease in kinetic energy, etc.
Thus, with harmonic vibrations, there is a periodic transformation of kinetic energy into potential energy and vice versa.
If there is no friction in the oscillatory system, then the total mechanical energy during mechanical vibrations remains unchanged.
For spring load:
In the position of maximum deflection, the total energy of the pendulum is equal to the potential energy of the deformed spring:
When passing through the equilibrium position, the total energy is equal to the kinetic energy of the load:
For small oscillations of a mathematical pendulum:
In the position of maximum deviation, the total energy of the pendulum is equal to the potential energy of the body raised to a height h:
When passing through the equilibrium position, the total energy is equal to the kinetic energy of the body:
Here h m is the maximum lifting height of the pendulum in the Earth's gravitational field, x m and υ m = ω 0 x m are the maximum deviations of the pendulum from the equilibrium position and its velocity.
Harmonic oscillations and their characteristics. Equation of harmonic oscillation.
The simplest type of oscillatory process are simple harmonic vibrations, which are described by the equation
x = x m cos(ω t + φ 0).
Here x- displacement of the body from the equilibrium position,
x m- the amplitude of oscillations, that is, the maximum displacement from the equilibrium position,
ω – cyclic or circular frequency hesitation,
t- time.
Characteristics of oscillatory motion.
Offset x - deviation of the oscillating point from the equilibrium position. The unit of measurement is 1 meter.
Oscillation amplitude A - the maximum deviation of the oscillating point from the equilibrium position. The unit of measurement is 1 meter.
Oscillation periodT- the minimum time interval for which one complete oscillation occurs is called. The unit of measurement is 1 second.
T=t/N
where t is the oscillation time, N is the number of oscillations made during this time.
According to the graph of harmonic oscillations, you can determine the period and amplitude of oscillations:
Oscillation frequency ν – a physical quantity equal to the number of oscillations per unit of time.
ν=N/t
Frequency is the reciprocal of the oscillation period:
Frequency oscillations ν shows how many oscillations occur in 1 s. The unit of frequency is hertz(Hz).
Cyclic frequency ω is the number of oscillations in 2π seconds.
The oscillation frequency ν is related to cyclic frequency ω and oscillation period T ratios:
Phase harmonic process - a value that is under the sign of sine or cosine in the equation of harmonic oscillations φ = ω t + φ 0 . At t= 0 φ = φ 0 , therefore φ 0 called initial phase.
Graph of harmonic oscillations is a sine wave or a cosine wave.
In all three cases for the blue curves φ 0 = 0:
only greater amplitude(x" m > x m);
the red curve is different from the blue one only meaning period(T" = T / 2);
the red curve is different from the blue one only meaning initial phase(glad).
When the body oscillates along a straight line (axis OX) the velocity vector is always directed along this straight line. The speed of the body is determined by the expression
In mathematics, the procedure for finding the limit of the ratio Δx / Δt at Δ t→ 0 is called the calculation of the derivative of the function x(t) by time t and is denoted as x"(t).The speed is equal to the derivative of the function x( t) by time t.
For the harmonic law of motion x = x m cos(ω t+ φ 0) the calculation of the derivative leads to the following result:
υ X =x"(t)= ω x m sin(ω t + φ 0)
Acceleration is defined in a similar way a x bodies under harmonic vibrations. Acceleration a is equal to the derivative of the function υ( t) by time t, or the second derivative of the function x(t). The calculations give:
a x \u003d υ x "(t) =x""(t)= -ω 2 x m cos(ω t+ φ 0)=-ω 2 x
The minus sign in this expression means that the acceleration a(t) always has the opposite sign of the offset x(t), and, therefore, according to Newton's second law, the force that causes the body to perform harmonic oscillations is always directed towards the equilibrium position ( x = 0).
The figure shows graphs of the coordinates, velocity and acceleration of a body that performs harmonic oscillations.
Graphs of coordinate x(t), velocity υ(t) and acceleration a(t) of a body performing harmonic oscillations.
Spring pendulum.
Spring pendulumcall a load of some mass m, attached to a spring of stiffness k, the second end of which is fixed motionless.
natural frequencyω 0 free vibrations of the load on the spring is found by the formula:
Period T harmonic vibrations of the load on the spring is equal to
This means that the period of oscillation of a spring pendulum depends on the mass of the load and on the stiffness of the spring.
Physical properties of the oscillatory system determine only the natural oscillation frequency ω 0 and the period T . Such parameters of the oscillation process as amplitude x m and the initial phase φ 0 , are determined by the way in which the system was brought out of equilibrium at the initial moment of time.
Mathematical pendulum.
Mathematical pendulumcalled a body of small size, suspended on a thin inextensible thread, the mass of which is negligible compared to the mass of the body.
In the equilibrium position, when the pendulum hangs on a plumb line, the gravity force is balanced by the thread tension force N. When the pendulum deviates from the equilibrium position by a certain angle φ, a tangential component of the gravity force appears F τ = – mg sin phi. The minus sign in this formula means that the tangential component is directed in the direction opposite to the pendulum deflection.
Mathematical pendulum.φ - angular deviation of the pendulum from the equilibrium position,
x= lφ – displacement of the pendulum along the arc
The natural frequency of small oscillations of a mathematical pendulum is expressed by the formula:
Oscillation period of a mathematical pendulum:
This means that the period of oscillation of a mathematical pendulum depends on the length of the thread and on the acceleration of free fall of the area where the pendulum is installed.
Free and forced vibrations.
Mechanical oscillations, like oscillatory processes of any other physical nature, can be free and forced.
Free vibrations -These are oscillations that occur in the system under the action of internal forces, after the system has been brought out of a position of stable equilibrium.
The oscillations of a weight on a spring or the oscillations of a pendulum are free oscillations.
In order for free oscillations to occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position is proportional to the displacement of the body from the equilibrium position and is directed in the direction opposite to the displacement.
In real conditions, any oscillatory system is under the influence of friction forces (resistance). In this case, part of the mechanical energy is converted into the internal energy of the thermal motion of atoms and molecules, and the vibrations become fading.
Decaying called vibrations, the amplitude of which decreases with time.
In order for the oscillations not to damp, it is necessary to impart additional energy to the system, i.e. act on the oscillatory system with a periodic force (for example, to swing a swing).
Oscillations that occur under the influence of an external periodically changing force are calledforced.
The external force performs positive work and provides an influx of energy to the oscillatory system. It does not allow oscillations to fade, despite the action of friction forces.
A periodic external force can vary in time according to various laws. Of particular interest is the case when an external force, changing according to a harmonic law with a frequency ω, acts on an oscillatory system capable of performing natural oscillations at a certain frequency ω 0 .
If free vibrations occur at a frequency ω 0 , which is determined by the parameters of the system, then steady forced oscillations always occur on frequency ω of the external force .
The phenomenon of a sharp increase in the amplitude of forced oscillations when the frequency of natural oscillations coincides with the frequency of the external driving force is calledresonance.
Amplitude dependence x m forced oscillations from the frequency ω of the driving force is called resonant characteristic or resonance curve.
Resonance curves at various damping levels:
1 - oscillatory system without friction; at resonance, the amplitude x m of forced oscillations increases indefinitely;
2, 3, 4 - real resonance curves for oscillatory systems with different friction.
In the absence of friction, the amplitude of forced oscillations at resonance should increase indefinitely. In real conditions, the amplitude of steady-state forced oscillations is determined by the condition: the work of an external force during the period of oscillations must be equal to the loss of mechanical energy over the same time due to friction. The less friction, the greater the amplitude of forced oscillations at resonance.
The phenomenon of resonance can cause the destruction of bridges, buildings and other structures, if the natural frequencies of their oscillations coincide with the frequency of a periodically acting force, which has arisen, for example, due to the rotation of an unbalanced motor.
oscillatory motion- periodic or almost periodic movement of a body, the coordinate, velocity and acceleration of which at regular intervals take approximately the same values.
Mechanical oscillations occur when, when a body is taken out of equilibrium, a force appears that tends to bring the body back.
Displacement x - deviation of the body from the equilibrium position.
Amplitude A - the module of the maximum displacement of the body.
Oscillation period T - time of one oscillation:
Oscillation frequency
The number of oscillations made by the body per unit time: During oscillations, the speed and acceleration change periodically. In the equilibrium position, the speed is maximum, the acceleration is zero. At the points of maximum displacement, the acceleration reaches its maximum, and the velocity vanishes.
GRAPH OF HARMONIC OSCILLATIONS
Harmonic oscillations occurring according to the law of sine or cosine are called:
where x(t) is the displacement of the system at time t, A is the amplitude, ω is the cyclic oscillation frequency.
If the deviation of the body from the equilibrium position is plotted along the vertical axis, and time is plotted along the horizontal axis, then we get a graph of the oscillation x = x(t) - the dependence of the body's displacement on time. With free harmonic oscillations, it is a sinusoid or a cosine wave. The figure shows graphs of displacement x, velocity projections V x and acceleration a x versus time.
As can be seen from the graphs, at the maximum displacement x, the speed V of the oscillating body is zero, the acceleration a, and hence the force acting on the body, are maximum and directed opposite to the displacement. In the equilibrium position, the displacement and acceleration vanish, the speed is maximum. The acceleration projection always has the opposite sign of the displacement.
ENERGY OF VIBRATIONAL MOVEMENT
The total mechanical energy of an oscillating body is equal to the sum of its kinetic and potential energies and, in the absence of friction, remains constant:
At the moment when the displacement reaches its maximum x = A, the speed, and with it the kinetic energy, vanishes.
In this case, the total energy is equal to the potential energy:
The total mechanical energy of an oscillating body is proportional to the square of the amplitude of its oscillations.
When the system passes the equilibrium position, the displacement and potential energy are equal to zero: x \u003d 0, E p \u003d 0. Therefore, the total energy is equal to the kinetic:
The total mechanical energy of an oscillating body is proportional to the square of its velocity in the equilibrium position. Consequently:
MATHEMATICAL PENDULUM
1. Mathematical pendulum is a material point suspended on a weightless inextensible thread.
In the equilibrium position, the force of gravity is compensated by the tension of the thread. If the pendulum is deflected and released, then the forces and will cease to compensate each other, and there will be a resultant force directed to the equilibrium position. Newton's second law:
For small fluctuations, when the displacement x is much less than l, the material point will move almost along the horizontal x axis. Then from the triangle MAB we get:
Because sin a \u003d x / l, then the projection of the resulting force R on the x-axis is equal to
The minus sign indicates that the force R is always directed against the displacement x.
2. So, during oscillations of a mathematical pendulum, as well as during oscillations of a spring pendulum, the restoring force is proportional to the displacement and is directed in the opposite direction.
Let's compare the expressions for the restoring force of the mathematical and spring pendulums:
It can be seen that mg/l is an analogue of k. Replacing k with mg/l in the formula for the period of a spring pendulum
we get the formula for the period of a mathematical pendulum:
The period of small oscillations of a mathematical pendulum does not depend on the amplitude.
A mathematical pendulum is used to measure time, to determine the acceleration of free fall at a given location on the earth's surface.
Free oscillations of a mathematical pendulum at small deflection angles are harmonic. They occur due to the resultant force of gravity and the tension of the thread, as well as the inertia of the load. The resultant of these forces is the restoring force.
Example. Determine the free fall acceleration on a planet where a pendulum 6.25 m long has a period of free oscillation of 3.14 s.
The period of oscillation of a mathematical pendulum depends on the length of the thread and the acceleration of free fall:
By squaring both sides of the equation, we get:
Answer: free fall acceleration is 25 m/s 2 .
Tasks and tests on the topic "Topic 4. "Mechanics. Vibrations and waves.
- Transverse and longitudinal waves. Wavelength
Lessons: 3 Assignments: 9 Tests: 1
- Sound waves. Sound speed - Mechanical oscillations and waves. Sound grade 9
