What is the total energy of oscillation of a spring pendulum? Formula for the oscillation frequency of a spring pendulum
The study of pendulum oscillations is carried out using a setup, the diagram of which is shown in Fig. 5. The installation consists of a spring pendulum, a vibration recording system based on a piezoelectric sensor, a forced vibration excitation system, and an information processing system on a personal computer. The spring pendulum under study consists of a steel spring with a stiffness coefficient k and pendulum bodies
m
, in the center of which a permanent magnet is mounted. The movement of the pendulum occurs in a liquid and at low oscillation speeds the resulting friction force can be approximated with sufficient accuracy by a linear law, i.e. Fig.5 Block diagram of the experimental setup,
To increase the resistance force when moving in a liquid, the body of the pendulum is made in the form of a washer with holes.
To record vibrations, a piezoelectric sensor is used, to which a pendulum spring is suspended. During the movement of the pendulum, the elastic force is proportional to the displacement X Since the EMF arising in the piezoelectric sensor is in turn proportional to the pressure force, the signal received from the sensor will be proportional to the displacement of the pendulum body from the equilibrium position. Oscillations are excited using a magnetic field. The harmonic signal created by the PC is amplified and fed to an excitation coil located under the pendulum body. As a result of this coil, a magnetic field that is variable in time and non-uniform in space is formed. During the movement of the pendulum, the elastic force is proportional to the displacement.
The information processing system consists of an analog-to-digital converter and a personal computer. The analog signal from the piezoelectric sensor is represented in digital form using an analog-to-digital converter and fed to a personal computer.
Controlling the experimental setup using a computer
After turning on the computer and loading the program, the main menu appears on the monitor screen, general form which is shown in Fig. 5. Using the cursor keys , , , , you can select one of the menu items. After pressing the button
ENTER
the computer begins to execute the selected operating mode. The simplest hints on the selected operating mode are contained in the highlighted line at the bottom of the screen. Let's consider the possible operating modes of the program: Using the cursor keys , , , , you can select one of the menu items. Statics Using the cursor keys , , , , you can select one of the menu items.- this menu item is used to process the results of the first exercise (see Fig. 5) After pressing the button Using the cursor keys , , , , you can select one of the menu items. the computer requests the mass of the pendulum bob. After the next button press a new picture with a blinking cursor appears on the screen. Sequentially write down on the screen the mass of the load in grams and, after pressing the space bar, the amount of tension of the spring. Pressing go to a new line and again write down the mass of the load and the amount of tension of the spring. Data editing within the last line is allowed. To do this, press the key Using the cursor keys , , , , you can select one of the menu items. Backspace
remove the incorrect mass or spring stretch value and write the new value. To change data in other lines, you must successively press Esc Using the cursor keys , , , , you can select one of the menu items. And
, and then repeat the result set. After entering the data, press the function key
F2. 0
.
The values of the spring stiffness coefficient and the frequency of free oscillations of the pendulum, calculated using the least squares method, appear on the screen. After clicking on Using the cursor keys , , , , you can select one of the menu items. the program begins to remove the experimental dependence of the pendulum's deviation on time. In the case when the frequency of the driving force is zero, a picture of damped oscillations appears on the screen. Pressing The values of the oscillation frequency and damping constant are recorded in a separate window. If the frequency of the exciting force is not zero, then along with the graphs of the dependences of the deviation of the pendulum and the driving force on time, the values of the frequency of the driving force and its amplitude, as well as the measured frequency and amplitude of the pendulum oscillations, are recorded on the screen in separate windows.
Pressing a key you can exit to the main menu.
Save- if the result of the experiment is satisfactory, then it can be saved by pressing the corresponding menu key. Using the cursor keys , , , , you can select one of the menu items. New Series
- this menu item is used if there is a need to abandon the data of the current experiment. After pressing the key in this mode, the results of all previous experiments are erased from the machine’s memory, and a new series of measurements can be started.
After the experiment, they switch to the mode Measurements
. This menu item has several sub-items (Fig. 7)
Frequency response graph- this menu item is used after the end of the experiment to study forced oscillations. The amplitude-frequency characteristic of forced oscillations is plotted on the monitor screen.
FFC schedule - In this mode, after the end of the experiment to study forced oscillations, a phase-frequency characteristic is plotted on the monitor screen. Table
Exit 7-10 - end of the program (see, for example, Fig. 7) 20 Exercise 1. 150 Determination of the spring stiffness coefficient using the static method. Measurements are carried out by determining the elongation of a spring under the action of loads with known masses. It is recommended to spend at least measurements of spring elongation by gradually suspending weights and thereby changing the load from before d. Using the program operation menu item
Statistics
Free vibrations can occur under the influence of internal forces only after the entire system is removed from the equilibrium position.
In order for oscillations to occur according to the harmonic law, it is necessary that the force returning the body to the equilibrium position be proportional to the displacement of the body from the equilibrium position and directed in the direction opposite to the displacement.
F (t) = m a (t) = - m ω 2 x (t) .
The relationship says that ω is the frequency of a harmonic oscillation. This property characteristic of elastic force within the limits of applicability of Hooke’s law:
F y p r = - k x .
Definition 2
Forces of any nature that satisfy the condition are called quasi-elastic.
That is, a load with mass m attached to a spring of stiffness k with a fixed end, shown in Figure 2. 2. 1, constitute a system capable of performing harmonic free vibrations in the absence of friction.
Definition 3
A weight placed on a spring is called a linear harmonic oscillator.
Drawing 2 . 2 . 1 . Oscillations of a load on a spring. There is no friction.
Circular frequency
The circular frequency ω 0 is found by applying the formula of Newton’s second law:
m a = - k x = m ω 0 2 x .
So we get:
Definition 4
The frequency ω 0 is called natural frequency of the oscillatory system.
Determining the period harmonic vibrations the load on the spring T is found from the formula:
T = 2 π ω 0 = 2 π m k .
The horizontal arrangement of the spring-load system, the force of gravity is compensated by the support reaction force. When hanging a load on a spring, the direction of gravity goes along the line of movement of the load. The equilibrium position of the stretched spring is equal to:
x 0 = m g k , while oscillations occur around a new equilibrium state. The formulas for the natural frequency ω 0 and the oscillation period T in the above expressions are valid.
Definition 5
Given the existing mathematical connection between the acceleration of the body a and the coordinate x, the behavior of the oscillatory system is characterized by a strict description: acceleration is the second derivative of the body coordinate x with respect to time t:
The description of Newton's second law with a load on a spring will be written as:
m a - m x = - k x, or x ¨ + ω 0 2 x = 0, where free frequency ω 0 2 = k m.
If physical systems depend on the formula x ¨ + ω 0 2 x = 0, then they are able to perform free oscillatory harmonic movements with different amplitudes. This is possible because x = x m cos (ω t + φ 0) is used.
Definition 6An equation of the form x ¨ + ω 0 2 x = 0 is called equations of free vibrations. Their physical properties can only determine the natural frequency of oscillations ω 0 or the period T.
The amplitude x m and the initial phase φ 0 are found using a method that brought them out of the equilibrium state of the initial moment of time.
Example 1
In the presence of a displaced load from the equilibrium position to a distance ∆ l and a moment of time equal to t = 0, it is lowered without an initial speed. Then x m = ∆ l, φ 0 = 0. If the load was in an equilibrium position, then the initial speed ± υ 0 is transmitted during the push, hence x m = m k υ 0, φ 0 = ± π 2.
The amplitude x m with the initial phase φ 0 is determined by the presence of initial conditions.
Figure 2. 2. 2. Model of free oscillations of a load on a spring.
Mechanical oscillatory systems are distinguished by the presence of elastic deformation forces in each of them. Figure 2. 2. 2 shows the angular analogue of a harmonic oscillator performing torsional oscillations. The disk is positioned horizontally and hangs on an elastic thread attached to its center of mass. If it is rotated through an angle θ, then a moment of force of elastic torsional deformation M y p p arises:
M y p r = - x θ .
This expression does not correspond to Hooke's law for torsional deformation. The value x is similar to the spring stiffness k. Writing Newton's second law for rotational movement disk takes the form
I ε = M y p p = - x θ or I θ ¨ = - x θ, where the moment of inertia is denoted by I = IC, and ε is the angular acceleration.
Similarly with the formula of a spring pendulum:
ω 0 = x I , T = 2 π I x .
The use of a torsion pendulum is seen in mechanical watches. It is called a balancer, in which the moment of elastic forces is created using a spiral spring.
Figure 2. 2. 3. Torsion pendulum.
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Vibrations of a massive body caused by the action of elastic forceAnimation
Description
When an elastic force acts on a massive body, returning it to an equilibrium position, it oscillates around this position.
Such a body is called a spring pendulum. Oscillations occur under the influence of an external force. Oscillations that continue after the external force has ceased to act are called free. Oscillations caused by the action of an external force are called forced. In this case, the force itself is called forcing.
In the simplest case, a spring pendulum is a thing moving along a horizontal plane solid, attached by a spring to the wall (Fig. 1).
Spring pendulum
Rice. 1
The rectilinear motion of a body is described by the dependence of its coordinates on time:
x = x(t). (1)
If all the forces acting on the body in question are known, then this dependence can be established using Newton’s second law:
md 2 x /dt 2 = S F , (2)
where m is body mass.
The right side of equation (2) is the sum of the projections onto the x axis of all forces acting on the body.
In the case under consideration, the main role is played by the elastic force, which is conservative and can be represented in the form:
F (x) = - dU (x)/dx, (3)
where U = U (x) is the potential energy of the deformed spring.
Let x be the extension of the spring. It has been experimentally established that at small values of the relative elongation of the spring, i.e. provided that:
½ x ½<< l ,
where l is the length of the undeformed spring.
The following relationship is approximately true:
U (x) = k x 2 /2, (4)
where the coefficient k is called the spring stiffness.
From this formula follows the following expression for the elastic force:
F (x) = - kx. (5)
This relationship is called Hooke's law.
In addition to the elastic force, a friction force can act on a body moving along a plane, which is satisfactorily described by the empirical formula:
F tr = - r dx /dt , (6)
where r is the friction coefficient.
Taking into account formulas (5) and (6), equation (2) can be written as follows:
md 2 x /dt 2 + rdx /dt + kx = F (t), (7)
where F(t) is the external force.
If only the Hooke force (5) acts on the body, then the free vibrations of the body will be harmonic. Such a body is called a harmonic spring pendulum.
Newton's second law in this case leads to the equation:
d 2 x /dt 2 + w 0 2 x = 0, (8)
w 0 = sqrt(k/m) (9)
Oscillation frequency.
The general solution to equation (8) has the form:
x (t) = A cos (w 0 t + a), (10)
where the amplitude A and the initial phase a are determined by the initial conditions.
When the body in question is acted upon only by elastic force (5), its total mechanical energy does not change over time:
mv 2 / 2 + k x 2 /2 = const . (eleven)
This statement constitutes the content of the law of conservation of energy of a harmonic spring pendulum.
Suppose that in addition to the elastic force that returns it to the equilibrium position, a frictional force acts on a massive body. In this case, the free vibrations of the body excited at some point in time will decay over time and the body will tend to an equilibrium position.
In this, Newton's second law (7) can be written as follows:
m d 2 x /dt 2 + rdx /dt + kx = 0, (12)
where m is body mass.
The general solution to equation (12) has the form:
x(t) = a exp(- b t )cos (w t + a ), (13)
w = sqrt(w o 2 - b 2 ) (14)
Oscillation frequency
b = r / 2 m (15)
The oscillation damping coefficient, amplitude a and initial phase a are determined by the initial conditions. Function (13) describes the so-called damped oscillations.
The total mechanical energy of the spring pendulum, i.e. the sum of its kinetic and potential energies
E = m v 2 /2 + kx 2 / 2 (16)
changes over time according to the law:
dE/dt = P, (17)
where P = - rv 2 - the power of the friction force, i.e. energy converted into heat per unit time.
Timing characteristics
Initiation time (log to -3 to -1);
Lifetime (log tc from 1 to 15);
Degradation time (log td from -3 to 3);
Time of optimal development (log tk from -3 to -2).
), one end of which is rigidly fixed, and on the other there is a load of mass m.
When an elastic force acts on a massive body, returning it to an equilibrium position, it oscillates around this position. Such a body is called a spring pendulum. Oscillations occur under the influence of an external force. Oscillations that continue after the external force has ceased to act are called free. Oscillations caused by the action of an external force are called forced. In this case, the force itself is called forcing.
In the simplest case, a spring pendulum is a rigid body moving along a horizontal plane, attached by a spring to a wall.
Newton's second law for such a system, provided there are no external forces and friction forces, has the form:
If the system is influenced by external forces, then the vibration equation will be rewritten as follows:
, Where f(x)- this is the resultant of external forces related to a unit mass of the load.In the case of attenuation proportional to the oscillation speed with the coefficient c:
see also
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