Movement of a spring pendulum. Free vibrations

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 represents massive body, drilled in the middle and placed on a horizontal rod along which it can slide without friction (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 inertial system reference point 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 body is affected by maximum strength elasticity, 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 is maximum, the body will stop for a moment, after which, under the influence of elastic force, it will begin to move in reverse side, 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 makes harmonic vibrations with cyclic frequency

Bodies under the influence of elastic force, potential energy which is proportional to the square of the displacement of the body from the equilibrium position:

When free mechanical vibrations kinetic and potential energies change periodically. At maximum deviation body from its equilibrium position, its speed, and therefore kinetic energy go to zero. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a horizontal spring, potential energy is the energy of elastic deformation of the spring.

When a body in its motion passes through the equilibrium position, its speed is maximum. At this moment it has maximum kinetic and minimum potential energy. An increase in kinetic energy occurs due to a decrease in potential energy. At further movement potential energy begins to increase due to a decrease in kinetic energy, etc.

Thus, during harmonic oscillations, a periodic transformation of kinetic energy into potential energy and vice versa occurs.

If there is no friction in the oscillatory system, then the total mechanical energy during free oscillations remains unchanged.

For spring weight:

Launch oscillatory motion body is carried out using the Start button. The Stop button allows you to stop the process at any time.

Graphically shows the relationship between potential and kinetic energies during oscillations at any time. Note that in the absence of attenuation total energy oscillatory system remains unchanged, the potential energy reaches a maximum with the maximum deviation of the body from the equilibrium position, and the kinetic energy takes maximum value when a body passes through a position of equilibrium.

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 high school: Theory. Assignments. Tests: Textbook. benefits 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.

1. The action on a body of an elastic force proportional to the displacement of the body x from the equilibrium position and always directed towards this position.

2. Inertia of an oscillating body, due to which it does not stop in the equilibrium position (when the elastic force becomes zero), but continues to move in the same direction.

The expression for the cyclic frequency is:

where w is the cyclic frequency, k is the spring stiffness, m is the mass.

This formula shows that the frequency of free vibrations does not depend on initial conditions and is completely determined own characteristics the oscillatory system itself - in in this case stiffness k and mass m.

This expression defines period of free oscillation of a spring pendulum.

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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 - magnitude 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.