Displacement of a horizontal spring pendulum with a mass of 10 g. Free vibrations

A spring pendulum is an oscillatory system consisting of a material point of mass m and a spring. Let's consider a horizontal spring pendulum (Fig. 1, a). It consists of a massive body, drilled in the middle and placed on a horizontal rod, along which it can slide without friction (an ideal oscillating system). The rod is fixed between two vertical supports.

A weightless spring is attached to the body at one end. Its other end is fixed to a support, which in the simplest case is at rest relative to the inertial reference frame in which the pendulum oscillates. At the beginning, the spring is not deformed, and the body is in the equilibrium position C. If, by stretching or compressing the spring, the body is taken out of the equilibrium position, then an elastic force will begin to act on it from the side of the deformed spring, always directed towards the equilibrium position.

Let us compress the spring, moving the body to position A, and release it. Under the influence of elastic force, it will move faster. In this case, in position A the maximum elastic force acts on the body, since here the absolute elongation x m of the spring is greatest. Therefore, in this position the acceleration is maximum. As the body moves toward the equilibrium position, the absolute elongation of the spring decreases, and consequently, the acceleration imparted by the elastic force decreases. But since the acceleration during a given movement is co-directed with the speed, the speed of the pendulum increases and in the equilibrium position it will be maximum.

Having reached the equilibrium position C, the body will not stop (although in this position the spring is not deformed and the elastic force is zero), but having speed, it will move further by inertia, stretching the spring. The elastic force that arises is now directed against the movement of the body and slows it down. At point D, the speed of the body will be equal to zero, and the acceleration will be maximum, the body will stop for a moment, after which, under the influence of the elastic force, it will begin to move in the opposite direction, to the equilibrium position. Having passed it again by inertia, the body, compressing the spring and slowing down the movement, will reach point A (since there is no friction), i.e. will complete a complete swing. After this, the body movement will be repeated in the described sequence. So, the reasons for the free oscillations of a spring pendulum are the action of the elastic force that occurs when the spring is deformed and the inertia of the body.

According to Hooke's law, F x = -kx. According to Newton's second law, F x = ma x. Therefore, ma x = -kx. From here

Dynamic equation of motion of a spring pendulum.

We see that the acceleration is directly proportional to the mixing and is directed oppositely to it. Comparing the resulting equation with the equation of harmonic vibrations , we see that the spring pendulum performs harmonic oscillations with a cyclic frequency

Any periodically repeating movement is called oscillatory. Therefore, the dependences of the coordinates and speed of a body on time during oscillations are described by periodic functions of time. In the school physics course, vibrations are considered in which the dependencies and velocities of the body are trigonometric functions , or a combination thereof, where is a certain number. Such oscillations are called harmonic (functions And often called harmonic functions). To solve problems on oscillations included in the program of the unified state exam in physics, you need to know the definitions of the main characteristics of oscillatory motion: amplitude, period, frequency, circular (or cyclic) frequency and phase of oscillations. Let us give these definitions and connect the listed quantities with the parameters of the dependence of the body coordinates on time, which in the case of harmonic oscillations can always be represented in the form

where , and are some numbers.

The amplitude of oscillations is the maximum deviation of an oscillating body from its equilibrium position. Since the maximum and minimum values ​​of the cosine in (11.1) are equal to ±1, the amplitude of oscillations of the body oscillating (11.1) is equal to . The period of oscillation is the minimum time after which the movement of a body is repeated. For dependence (11.1), the period can be set from the following considerations. Cosine is a periodic function with period. Therefore, the movement is completely repeated through such a value that . From here we get

The circular (or cyclic) frequency of oscillations is the number of oscillations performed per unit of time. From formula (11.3) we conclude that the circular frequency is the quantity from formula (11.1).

The oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinate on time. From formula (11.1) we see that the phase of oscillations of the body, the movement of which is described by dependence (11.1), is equal to . The value of the oscillation phase at time = 0 is called the initial phase. For dependence (11.1), the initial phase of oscillations is equal to . Obviously, the initial phase of oscillations depends on the choice of the time reference point (moment = 0), which is always conditional. By changing the origin of time, the initial phase of oscillations can always be “made” equal to zero, and the sine in formula (11.1) can be “turned” into a cosine or vice versa.

The Unified State Exam program also includes knowledge of formulas for the frequency of oscillations of spring and mathematical pendulums. A spring pendulum is usually called a body that can oscillate on a smooth horizontal surface under the action of a spring, the second end of which is fixed (left figure). A mathematical pendulum is a massive body, the dimensions of which can be neglected, oscillating on a long, weightless and inextensible thread (right figure). The name of this system, “mathematical pendulum,” is due to the fact that it represents an abstract mathematical model of real ( physical) pendulum. It is necessary to remember the formulas for the period (or frequency) of oscillations of spring and mathematical pendulums. For a spring pendulum

where is the length of the thread, is the acceleration of gravity. Let's consider the application of these definitions and laws using the example of problem solving.

To find the cyclic frequency of oscillations of the load in task 11.1.1 Let's first find the period of oscillation, and then use formula (11.2). Since 10 m 28 s is 628 s, and during this time the load oscillates 100 times, the period of oscillation of the load is 6.28 s. Therefore, the cyclic frequency of oscillations is 1 s -1 (answer 2 ). IN problem 11.1.2 the load made 60 oscillations in 600 s, so the oscillation frequency is 0.1 s -1 (answer 1 ).

To understand how far the load will travel in 2.5 periods ( problem 11.1.3), let's follow his movement. After a period, the load will return back to the point of maximum deflection, completing a complete oscillation. Therefore, during this time, the load will travel a distance equal to four amplitudes: to the equilibrium position - one amplitude, from the equilibrium position to the point of maximum deviation in the other direction - the second, back to the equilibrium position - the third, from the equilibrium position to the starting point - the fourth. During the second period, the load will again go through four amplitudes, and during the remaining half of the period - two amplitudes. Therefore, the distance traveled is equal to ten amplitudes (answer 4 ).

The amount of movement of the body is the distance from the starting point to the ending point. Over 2.5 periods in task 11.1.4 the body will have time to complete two full and half a full oscillation, i.e. will be at the maximum deviation, but on the other side of the equilibrium position. Therefore, the magnitude of the displacement is equal to two amplitudes (answer 3 ).

By definition, the oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinates of an oscillating body on time. Therefore the correct answer is problem 11.1.5 - 3 .

A period is the time of complete oscillation. This means that the return of a body back to the same point from which the body began to move does not mean that a period has passed: the body must return to the same point with the same speed. For example, a body, having started oscillations from an equilibrium position, will have time to deviate by a maximum amount in one direction, return back, deviate by a maximum in the other direction, and return back again. Therefore, during the period the body will have time to deviate by the maximum amount from the equilibrium position twice and return back. Consequently, the passage from the equilibrium position to the point of maximum deviation ( problem 11.1.6) the body spends a quarter of the period (answer 3 ).

Harmonic oscillations are those in which the dependence of the coordinates of the oscillating body on time is described by a trigonometric (sine or cosine) function of time. IN task 11.1.7 these are the functions and , despite the fact that the parameters included in them are designated as 2 and 2 . The function is a trigonometric function of the square of time. Therefore, vibrations of only quantities and are harmonic (answer 4 ).

During harmonic vibrations, the speed of the body changes according to the law , where is the amplitude of the speed oscillations (the time reference point is chosen so that the initial phase of the oscillations is equal to zero). From here we find the dependence of the kinetic energy of the body on time
(problem 11.1.8). Using further the well-known trigonometric formula, we obtain

From this formula it follows that the kinetic energy of a body changes during harmonic oscillations also according to the harmonic law, but with double the frequency (answer 2 ).

Behind the relationship between the kinetic energy of the load and the potential energy of the spring ( problem 11.1.9) is easy to follow from the following considerations. When the body is deflected by the maximum amount from the equilibrium position, the speed of the body is zero, and, therefore, the potential energy of the spring is greater than the kinetic energy of the load. On the contrary, when the body passes through the equilibrium position, the potential energy of the spring is zero, and therefore the kinetic energy is greater than the potential energy. Therefore, between the passage of the equilibrium position and the maximum deflection, the kinetic and potential energy are compared once. And since during a period the body passes four times from the equilibrium position to the maximum deflection or back, then during the period the kinetic energy of the load and the potential energy of the spring are compared with each other four times (answer 2 ).

Amplitude of speed fluctuations ( task 11.1.10) is easiest to find using the law of conservation of energy. At the point of maximum deflection, the energy of the oscillatory system is equal to the potential energy of the spring , where is the spring stiffness coefficient, is the vibration amplitude. When passing through the equilibrium position, the energy of the body is equal to the kinetic energy , where is the mass of the body, is the speed of the body when passing through the equilibrium position, which is the maximum speed of the body during the oscillation process and, therefore, represents the amplitude of the speed oscillations. Equating these energies, we find

(answer 4 ).

From formula (11.5) we conclude ( problem 11.2.2), that its period does not depend on the mass of a mathematical pendulum, and with an increase in length by 4 times, the period of oscillations increases by 2 times (answer 1 ).

A clock is an oscillatory process that is used to measure intervals of time ( problem 11.2.3). The words “clock is in a hurry” mean that the period of this process is less than what it should be. Therefore, to clarify the progress of these clocks, it is necessary to increase the period of the process. According to formula (11.5), to increase the period of oscillation of a mathematical pendulum, it is necessary to increase its length (answer 3 ).

To find the amplitude of oscillations in problem 11.2.4, it is necessary to represent the dependence of the body coordinates on time in the form of a single trigonometric function. For the function given in the condition, this can be done by introducing an additional angle. Multiplying and dividing this function by and using the formula for adding trigonometric functions, we get

where is the angle such that . From this formula it follows that the amplitude of body oscillations is (answer 4 ).

A spring pendulum is a material point with mass attached to an absolutely elastic weightless spring with a stiffness . There are two simplest cases: horizontal (Fig. 15, A) and vertical (Fig. 15, b) pendulums.

A) Horizontal pendulum(Fig. 15,a). When the load moves
from the equilibrium position by the amount acts on it in the horizontal direction restoring elastic force
(Hooke's law).

It is assumed that the horizontal support along which the load slides
during its vibrations, it is absolutely smooth (no friction).

b) Vertical pendulum(Fig. 15, b). The equilibrium position in this case is characterized by the condition:

Where - the magnitude of the elastic force acting on the load
when the spring is statically stretched by under the influence of gravity of the load
.

Fig. 15. Spring pendulum: A– horizontal and b– vertical

If you stretch the spring and release the load, it will begin to oscillate vertically. If the displacement at some point in time is
, then the elastic force will now be written as
.

In both cases considered, the spring pendulum performs harmonic oscillations with a period

(27)

and cyclic frequency

. (28)

Using the example of a spring pendulum, we can conclude that harmonic oscillations are motion caused by a force that increases in proportion to the displacement . Thus, if the restoring force resembles Hooke's law
(she got the namequasi-elastic force ), then the system must perform harmonic oscillations. At the moment of passing the equilibrium position, no restoring force acts on the body; however, the body, by inertia, passes the equilibrium position and the restoring force changes direction to the opposite.

Math pendulum

Fig. 16. Math pendulum

Oscillations of such a pendulum at small angles of deflection
(not exceeding 5º) can be considered harmonic, and the cyclic frequency of a mathematical pendulum:

, (29)

and period:

. (30)

2.3. Body energy during harmonic oscillations

The energy imparted to the oscillatory system during the initial push will be periodically transformed: the potential energy of the deformed spring will transform into the kinetic energy of the moving load and back.

Let the spring pendulum perform harmonic oscillations with the initial phase
, i.e.
(Fig. 17).

Fig. 17. Law of conservation of mechanical energy

when a spring pendulum oscillates

At the maximum deviation of the load from the equilibrium position, the total mechanical energy of the pendulum (the energy of a deformed spring with a stiffness ) is equal to
. When passing the equilibrium position (
) the potential energy of the spring will become equal to zero, and the total mechanical energy of the oscillatory system will be determined as
.

Figure 18 shows graphs of the dependences of kinetic, potential and total energy in cases where harmonic vibrations are described by trigonometric functions of sine (dashed line) or cosine (solid line).

Fig. 18. Graphs of time dependence of kinetic

and potential energy during harmonic oscillations

From the graphs (Fig. 18) it follows that the frequency of change in kinetic and potential energy is twice as high as the natural frequency of harmonic oscillations.

When oscillations take place in school, they are illustrated with two simplest examples: a weight on a spring and a mathematical pendulum (that is, a point weight on an inextensible thread) in a gravitational field. In both cases, an important regularity is observed in the oscillations: their period does not depend on the amplitude - at least as long as this amplitude remains small - but is determined only by the mechanical properties of the system.

Now let's combine these two examples and consider the oscillations of a weight suspended on an extensible spring in a gravitational field (Fig. 1).

For simplicity, we neglect the third dimension and assume that this spring pendulum oscillates strictly in the plane of the figure. In this case, the weight (which is also considered a point weight) can move in a vertical plane in any direction, and not just up-down or left-right, as shown in Fig. 2. But if we again limit ourselves to only small deviations from the equilibrium position, then horizontal and vertical oscillations occur almost independently, with their own periods T x And T y.

It would seem that since these oscillations are determined by completely different forces and characteristics of the system, then their periods can be completely arbitrary, in no way related to each other. It turns out - no!

Task

Prove that in such a pendulum the period of horizontal oscillations is always greater than the period of vertical ones: T x > T y.

Clue

The problem may at first surprise you in that it seems like nothing is given, but something needs to be proven. But there's nothing wrong with that. When a problem is formulated in this way, it means that you can introduce for yourself some notations that you need, calculate with them what is required, and then come to a conclusion that is already does not depend from these values. Do this for this task. Take the formulas for the periods of oscillation, think about what quantities they include, and compare the two periods with each other, dividing one by the other.

Solution

Period of oscillation of a mass bob m on a stiffening spring k and length L 0 is

.

This formula does not change even if the weight is suspended in a gravitational field with free fall acceleration g. Of course, the equilibrium position of the weight will shift downward by a height Δ L = mg/k- it is with this elongation of the spring that the elastic force compensates for the force of gravity. But the period of vertical oscillations relative to this new equilibrium position with the stretched spring will remain the same.

The period of horizontal oscillations of a stretched pendulum is expressed in terms of the acceleration of gravity g and him full length L = L 0 +Δ L:

.

It is thanks to the additional stretching in the gravitational field that we find out that

That's the solution.

Afterword

Despite its apparent simplicity, a pendulum on a spring is a system quite rich in phenomena. This is one of the simplest examples of a nice phenomenon - the Fermi resonance. This is what it boils down to: Generally speaking, if the weight is somehow pulled back and released, it will oscillate both vertically and horizontally. These two types of vibrations will simply overlap and not interfere with each other. But if the periods of vertical and horizontal oscillations are related by the relation T x = 2T y, then horizontal and vertical vibrations, as if against their will, will gradually begin to transform into each other, as in the animation on the right. The energy of vibrations will be pumped, as it were, from vertical vibrations to horizontal ones and vice versa.

It looks like this: you pull the weight down and release it. At first it oscillates only up and down, then on its own it begins to sway sideways, for a moment the oscillation becomes almost completely horizontal, and then returns to vertical again. Surprisingly, a strictly vertical oscillation turns out to be unstable.

An explanation of this remarkable effect, as well as the magical ratio T x:T y= 2:1, that's it. Let us denote by x And y deviation of the weight from the equilibrium position (axis y pointing upward). With such a deviation, the potential energy increases by the amount

This is an accurate formula, it is suitable for any deviations, large or small. But if x And y small, significantly less L, then the expression is approximately equal to

plus other terms containing even higher degrees of deviation. Quantities U y And U x- these are ordinary potential energies from which vertical and horizontal vibrations are obtained. And here is the value highlighted in blue U xy is a special additive that generates interaction between these fluctuations. Thanks to this small interaction, vertical vibrations affect horizontal vibrations and vice versa. This becomes completely transparent if you carry out the calculations further and write the equation of vibrations horizontally and vertically:

where the notation is introduced

Without the blue additive, we would have the usual independent vertical and horizontal oscillations with frequencies ωy And ωx. This supplement plays a role coercive force, additionally rocking the vibrations. If the frequencies ωy And ωx are arbitrary, then this small force does not lead to any significant effect. But if the relation holds ωy = 2ωx, resonance occurs: the driving force for both types of oscillations contains a component with the same frequency as the oscillation itself. As a result, this force slowly but steadily swings one type of vibration and suppresses the other. This is how horizontal and vertical vibrations flow into each other.

Additional beauties arise if we honestly take into account the third dimension in this example. We will assume that the weight can compress and decompress the spring vertically and swing like a pendulum in two horizontal directions. Then, when the resonance condition is met, when viewed from above, the weight writes out a star-shaped trajectory, as, for example, in Fig. 3. This happens because the plane of oscillation does not remain stationary, but rotates - but not smoothly, but as if in jumps. While the oscillation goes from side to side, this plane more or less holds, and the rotation occurs during that short period when the oscillation is almost vertical. We invite readers to think for themselves what are the reasons for this behavior and what determines the angle of rotation of the plane. And those who want to plunge headlong into this rather deep problem can look through the article Stepwise Precession of the Resonant Swinging Spring, which not only provides a detailed analysis of the problem, but also talks about its history and the connection of this problem with other branches of physics, in particular with atomic physics.

A spring pendulum is an oscillatory system consisting of a material point of mass m and a spring. Consider a horizontal spring pendulum (Fig. 13.12, a). It consists of a massive body, drilled in the middle and placed on a horizontal rod, along which it can slide without friction (an ideal oscillating system). The rod is fixed between two vertical supports. A weightless spring is attached to the body at one end. Its other end is fixed to a support, which in the simplest case is at rest relative to the inertial reference frame in which the pendulum oscillates. At the beginning, the spring is not deformed, and the body is in the equilibrium position C. If, by stretching or compressing the spring, the body is taken out of the equilibrium position, then an elastic force will begin to act on it from the side of the deformed spring, always directed towards the equilibrium position. Let us compress the spring, moving the body to position A, and release \((\upsilon_0=0).\) Under the action of the elastic force, it will begin to move accelerated. In this case, in position A the maximum elastic force acts on the body, since here the absolute elongation x m of the spring is greatest. Therefore, in this position the acceleration is maximum. As the body moves toward the equilibrium position, the absolute elongation of the spring decreases, and consequently, the acceleration imparted by the elastic force decreases. But since the acceleration during a given movement is co-directed with the speed, the speed of the pendulum increases and in the equilibrium position it will be maximum. Having reached the equilibrium position C, the body will not stop (although in this position the spring is not deformed and the elastic force is zero), but having speed, it will move further by inertia, stretching the spring. The elastic force that arises is now directed against the movement of the body and slows it down. At point D, the speed of the body will be equal to zero, and the acceleration will be maximum, the body will stop for a moment, after which, under the influence of the elastic force, it will begin to move in the opposite direction, to the equilibrium position. Having passed it again by inertia, the body, compressing the spring and slowing down the movement, will reach point A (since there is no friction), i.e. will complete a complete swing. After this, the body movement will be repeated in the described sequence. So, the reasons for the free oscillations of a spring pendulum are the action of the elastic force that occurs when the spring is deformed and the inertia of the body.

By Hooke's law \(~F_x=-kx.\) By Newton's second law \(~F_x = ma_x.\) Therefore, \(~ma_x = -kx.\) Hence

\(a_x = -\frac(k)(m)x\) or \(a_x + -\frac(k)(m)x = 0 \) - dynamic equation of motion of a spring pendulum.

We see that the acceleration is directly proportional to the mixing and is directed oppositely to it. Comparing the resulting equation with the equation of harmonic oscillations \(~a_x + \omega^2 x = 0,\) we see that the spring pendulum performs harmonic oscillations with a cyclic frequency \(\omega = \sqrt \frac(k)(m)\) Since \(T = \frac(2 \pi)(\omega),\) then

\(T = 2 \pi \sqrt( \frac(m)(k) )\) is the period of oscillation of the spring pendulum.

Using the same formula, you can calculate the period of oscillation of a vertical spring pendulum (Fig. 13.12. b). Indeed, in the equilibrium position, due to the action of gravity, the spring is already stretched by a certain amount x 0, determined by the relation \(~mg=kx_0.\) When the pendulum is displaced from the equilibrium position O on X projection of the elastic force \(~F"_(ynpx) = -k(x_0 + x)\) and according to Newton’s second law \(~ma_x=-k(x_0+ x) + mg.\) Substituting here the value \(~kx_0 =mg,\) we obtain the equation of motion of the pendulum \(a_x + \frac(k)(m)x = 0,\) coinciding with the equation of motion of the horizontal pendulum.

Literature

Aksenovich L. A. Physics in secondary school: Theory. Assignments. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P. 377-378.