A spring pendulum is a material point with a 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 
.
|
A |
|
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.
is an idealized system in the form of a material point suspended on a weightless inextensible thread of length
, which makes small oscillations under the influence of gravity (Fig. 16). 
Oscillations of such a pendulum at small angles of deflection
,
(29)
(not exceeding 5º) can be considered harmonic, and the cyclic frequency of a mathematical pendulum:
.
(30)
and period:
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.
Definition 1
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 is 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. 2 . 2 . 1 . Drawing
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.
The period of harmonic oscillations of the load on the spring T is determined 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. The recording of Newton's second law for the rotational motion of a 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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Goal of the work. Familiarize yourself with the main characteristics of undamped and damped free mechanical vibrations.
Task. Determine the period of natural oscillations of a spring pendulum; check the linearity of the dependence of the square of the period on the mass; determine the spring stiffness; determine the period of damped oscillations and the logarithmic damping decrement of a spring pendulum.
Devices and accessories. A tripod with a scale, a spring, a set of weights of various weights, a vessel with water, a stopwatch.
1. Free oscillations of a spring pendulum. General information
Oscillations are processes in which one or more physical quantities that describe these processes periodically change. Oscillations can be described by various periodic functions of time. The simplest oscillations are harmonic oscillations - such oscillations in which the oscillating quantity (for example, the displacement of a load on a spring) changes over time according to the law of cosine or sine. Oscillations that occur after the action of an external short-term force on the system are called free.
If the load is removed from the equilibrium position by deflecting by an amount x, then the elastic force increases: F control = – kx 2= – k(x 1 + x). Having reached the equilibrium position, the load will have a speed different from zero and will pass the equilibrium position by inertia. As the movement continues, the deviation from the equilibrium position will increase, which will lead to an increase in the elastic force, and the process will repeat in the opposite direction. Thus, the oscillatory motion of the system is due to two reasons: 1) the desire of the body to return to the equilibrium position and 2) inertia, which does not allow the body to instantly stop in the equilibrium position. In the absence of friction forces, the oscillations would continue indefinitely. The presence of friction forces leads to the fact that part of the oscillation energy turns into internal energy and the oscillations gradually die out. Such oscillations are called damped.
Undamped free oscillations
First, let us consider the oscillations of a spring pendulum, which is not affected by friction forces - undamped free oscillations. According to Newton’s second law, taking into account the signs of projections onto the X axis
From the equilibrium condition, the displacement caused by gravity: . Substituting into equation (1), we obtain: Differential" href="/text/category/differentcial/" rel="bookmark">differential equation
https://pandia.ru/text/77/494/images/image008_28.gif" width="152" height="25 src=">. (3)
This equation is called harmonic equation. The greatest deviation of the load from the equilibrium position A 0 called the amplitude of oscillations. The quantity in the cosine argument is called oscillation phase. The constant φ0 represents the phase value at the initial time ( t= 0) and is called initial phase of oscillations. Magnitude
is it circular or cyclic? natural frequency related to period of oscillation T ratio https://pandia.ru/text/77/494/images/image012_17.gif" width="125" height="55">. (5)
Damped oscillations
Let us consider free oscillations of a spring pendulum in the presence of friction force (damped oscillations). In the simplest and at the same time most common case, the friction force is proportional to the speed υ movements:
Ftr = – rυ, (6)
Where r– a constant called the resistance coefficient. The minus sign shows that the friction force and speed are in opposite directions. Equation of Newton's second law in projection onto the X axis in the presence of elastic force and friction force
ma = – kx – rυ. (7)
This differential equation taking into account υ = dx/ dt can be written down
https://pandia.ru/text/77/494/images/image014_12.gif" width="59" height="48 src="> – damping coefficient; – cyclic frequency of free undamped oscillations of a given oscillatory system, i.e. in the absence of energy losses (β = 0). Equation (8) is called differential equation of damped oscillations.
To get the displacement dependence x from time t, it is necessary to solve the differential equation (8)..gif" width="172" height="27">, (9)
Where A 0 and φ0 – initial amplitude and initial phase of oscillations; 
– cyclic frequency of damped oscillations at ω >> https://pandia.ru/text/77/494/images/image019_12.gif" width="96" height="27 src=">. (10)
On the graph of function (9), Fig. 2, the dotted lines show the change in amplitude (10) of damped oscillations.
Rice. 2. Displacement dependence X load from time to time t in the presence of friction force
To quantitatively characterize the degree of attenuation of oscillations, a value is introduced equal to the ratio of amplitudes that differ by a period, and is called damping decrement:
. (11)
The natural logarithm of this quantity is often used. This parameter is called logarithmic damping decrement:
The amplitude decreases in n times, then from equation (10) it follows that
From here we get the expression for the logarithmic decrement
If during the time t" amplitude decreases in e once ( e= 2.71 – the base of the natural logarithm), then the system will have time to complete the number of oscillations
Rice. 3. Installation diagram
The installation consists of a tripod 1 with measuring scale 2 . To a tripod with a spring 3 loads are suspended 4 of various masses. When studying damped oscillations in task 2, a ring is used to enhance the damping 5 , which is placed in a transparent container 6 with water.
In task 1 (performed without a vessel with water and a ring), to a first approximation, the damping of oscillations can be neglected and considered harmonic. As follows from formula (5) for harmonic oscillations, the dependence T 2 = f (m) – linear, from which the spring stiffness coefficient can be determined k according to the formula
where is the slope of the straight line T 2 from m.
Exercise 1. Determination of the dependence of the period of natural oscillations of a spring pendulum on the mass of the load.
1. Determine the period of oscillation of a spring pendulum at different values of the mass of the load m. To do this, use a stopwatch for each value m measure time three times t full n fluctuations ( n≥10) and according to the average time value https://pandia.ru/text/77/494/images/image030_6.gif" width="57 height=28" height="28">. Enter the results in Table 1.
2. Based on the measurement results, construct a graph of the square of the period T2 by weight m. From the slope of the graph, determine the spring stiffness k according to formula (16).
Table 1
Measurement results to determine the period of natural oscillations
3. Additional task. Estimate random, total and relative ε t time measurement errors for mass value m = 400 g.
Task 2. Determination of the logarithmic damping decrement of a spring pendulum.
1. Hang a mass on a spring m= 400 g with ring and place in a vessel with water so that the ring is completely submerged in water. Determine the period of damped oscillations for a given value m according to the method outlined in paragraph 1 of task 1. Repeat the measurements three times and enter the results on the left side of the table. 2.
2. Remove the pendulum from the equilibrium position and, noting its initial amplitude on a ruler, measure the time t" , during which the amplitude of oscillations decreases by 2 times. Take measurements three times. Enter the results on the right side of the table. 2.
table 2
Measurement results
to determine the logarithmic damping decrement
Measuring the period of oscillation | Measuring time reducing the amplitude by 2 times |
|||||
4. Test questions and assignments
1. What oscillations are called harmonic? Define their main characteristics.
2. What oscillations are called damped? Define their main characteristics.
3. Explain the physical meaning of the logarithmic damping decrement and damping coefficient.
4. Derive the time dependence of the speed and acceleration of a load on a spring performing harmonic oscillations. Provide graphs and analyze.
5. Derive the time dependence of kinetic, potential and total energy for a load oscillating on a spring. Provide graphs and analyze.
6. Obtain the differential equation of free vibrations and its solution.
7. Construct graphs of harmonic oscillations with initial phases π/2 and π/3.
8. Within what limits can the logarithmic damping decrement vary?
9. Give the differential equation of damped oscillations of a spring pendulum and its solution.
10. According to what law does the amplitude of damped oscillations change? Are damped oscillations periodic?
11. What motion is called aperiodic? Under what conditions is it observed?
12. What is the natural frequency of oscillations? How does it depend on the mass of the oscillating body for a spring pendulum?
13. Why is the frequency of damped oscillations less than the frequency of natural oscillations of the system?
14. A copper ball suspended from a spring performs vertical oscillations. How will the period of oscillation change if instead of a copper ball, an aluminum ball of the same radius is suspended from a spring?
15. At what value of the logarithmic damping decrement do the oscillations decay faster: at θ1 = 0.25 or θ2 = 0.5? Provide graphs of these damped oscillations.
Bibliography
1. AND. Physics course / . – 11th ed. – M.: Academy, 2006. – 560 p.
2. IN. General physics course: 3 volumes / . – St. Petersburg. : Lan, 2008. – T. 1. – 432 p.
3. WITH. Laboratory workshop in physics / .
– M.: Higher. school, 1980. – 359 p.
10.4. Law of conservation of energy during harmonic oscillations
10.4.1. Energy conservation at mechanical harmonic vibrations
Conservation of energy during oscillations of a mathematical pendulum
During harmonic vibrations, the total mechanical energy of the system is conserved (remains constant).
Total mechanical energy of a mathematical pendulum
E = W k + W p ,
where W k is kinetic energy, W k = = mv 2 /2; W p - potential energy, W p = mgh; m is the mass of the load; g - free fall acceleration module; v - load speed module; h is the height of the load above the equilibrium position (Fig. 10.15).
During harmonic oscillations, a mathematical pendulum goes through a number of successive states, so it is advisable to consider the energy of a mathematical pendulum in three positions (see Fig. 10.15):
Rice. 10.15
1) in equilibrium position
potential energy is zero; The total energy coincides with the maximum kinetic energy:
E = W k max ;
2) in emergency situation(2) the body is raised above the initial level to a maximum height h max, therefore the potential energy is also maximum:
W p max = m g h max ;
kinetic energy is zero; total energy coincides with maximum potential energy:
E = W p max ;
3) in intermediate position(3) the body has an instantaneous speed v and is raised above the initial level to a certain height h, therefore the total energy is the sum
E = m v 2 2 + m g h ,
where mv 2 /2 is kinetic energy; mgh - potential energy; m is the mass of the load; g - free fall acceleration module; v - load speed module; h is the height of the load above the equilibrium position.
During harmonic oscillations of a mathematical pendulum, the total mechanical energy is conserved:
E = const.
The values of the total energy of the mathematical pendulum in its three positions are reflected in the table. 10.1.
| № | Position | Wp | Wk | E = W p + W k |
|---|---|---|---|---|
| 1 | Equilibrium | 0 | m v max 2 / 2 | m v max 2 / 2 |
| 2 | Extreme | mgh max | 0 | mgh max |
| 3 | Intermediate (instant) | mgh | mv 2 /2 | mv 2 /2 + mgh |
The values of total mechanical energy presented in the last column of the table. 10.1, have equal values for any position of the pendulum, which is a mathematical expression:
m v max 2 2 = m g h max;
m v max 2 2 = m v 2 2 + m g h ;
m g h max = m v 2 2 + m g h ,
where m is the mass of the load; g - free fall acceleration module; v is the module of the instantaneous speed of the load in position 3; h - height of lifting of the load above the equilibrium position in position 3; v max - module of the maximum speed of the load in position 1; h max - maximum height of lifting the load above the equilibrium position in position 2.
Thread deflection angle mathematical pendulum from the vertical (Fig. 10.15) is determined by the expression
cos α = l − hl = 1 − hl ,
where l is the length of the thread; h is the height of the load above the equilibrium position.
Maximum angle deviation α max is determined by the maximum height of lifting the load above the equilibrium position h max:
cos α max = 1 − h max l .
Example 11. The period of small oscillations of a mathematical pendulum is 0.9 s. What is the maximum angle at which the thread will deviate from the vertical if, passing the equilibrium position, the ball moves at a speed of 1.5 m/s? There is no friction in the system.
Solution . The figure shows two positions of a mathematical pendulum:
- equilibrium position 1 (characterized by the maximum speed of the ball v max);
- extreme position 2 (characterized by the maximum lifting height of the ball h max above the equilibrium position).
The required angle is determined by the equality
cos α max = l − h max l = 1 − h max l ,
where l is the length of the pendulum thread.
We find the maximum height of the pendulum ball above the equilibrium position from the law of conservation of total mechanical energy.
The total energy of the pendulum in the equilibrium position and in the extreme position is determined by the following formulas:
- in a position of balance -
E 1 = m v max 2 2,
where m is the mass of the pendulum ball; v max - module of the ball velocity in the equilibrium position (maximum speed), v max = 1.5 m/s;
- in extreme position -
E 2 = mgh max,
where g is the gravitational acceleration module; h max is the maximum height of the ball lifting above the equilibrium position.
Law of conservation of total mechanical energy:
m v max 2 2 = m g h max .
Let us express from here the maximum height of the ball's rise above the equilibrium position:
h max = v max 2 2 g .
We determine the length of the thread from the formula for the oscillation period of a mathematical pendulum
T = 2 π l g ,
those. thread length
l = T 2 g 4 π 2 .
Let's substitute h max and l into the expression for the cosine of the desired angle:
cos α max = 1 − 2 π 2 v max 2 g 2 T 2
and perform the calculation taking into account the approximate equality π 2 = 10:
cos α max = 1 − 2 ⋅ 10 ⋅ (1.5) 2 10 2 ⋅ (0.9) 2 = 0.5 .
It follows that the maximum deflection angle is 60°.
Strictly speaking, at an angle of 60° the oscillations of the ball are not small and it is unlawful to use the standard formula for the period of oscillation of a mathematical pendulum.
Conservation of energy during oscillations of a spring pendulum
Total mechanical energy of a spring pendulum consists of kinetic energy and potential energy:
E = W k + W p ,
where W k is kinetic energy, W k = mv 2 /2; W p - potential energy, W p = k (Δx ) 2 /2; m is the mass of the load; v - load speed module; k is the stiffness (elasticity) coefficient of the spring; Δx - deformation (tension or compression) of the spring (Fig. 10.16).
In the International System of Units, the energy of a mechanical oscillatory system is measured in joules (1 J).
During harmonic oscillations, the spring pendulum goes through a number of successive states, so it is advisable to consider the energy of the spring pendulum in three positions (see Fig. 10.16):
1) in equilibrium position(1) the speed of the body has a maximum value v max, therefore the kinetic energy is also maximum:
W k max = m v max 2 2 ;
the potential energy of the spring is zero, since the spring is not deformed; The total energy coincides with the maximum kinetic energy:
E = W k max ;
2) in emergency situation(2) the spring has a maximum deformation (Δx max), so the potential energy also has a maximum value:
W p max = k (Δ x max) 2 2 ;
the kinetic energy of the body is zero; total energy coincides with maximum potential energy:
E = W p max ;
3) in intermediate position(3) the body has an instantaneous speed v, the spring has some deformation at this moment (Δx), so the total energy is the sum
E = m v 2 2 + k (Δ x) 2 2 ,
where mv 2 /2 is kinetic energy; k (Δx) 2 /2 - potential energy; m is the mass of the load; v - load speed module; k is the stiffness (elasticity) coefficient of the spring; Δx - deformation (tension or compression) of the spring.
When the load of a spring pendulum is displaced from its equilibrium position, it is acted upon by restoring force, the projection of which onto the direction of movement of the pendulum is determined by the formula
F x = −kx ,
where x is the displacement of the spring pendulum load from the equilibrium position, x = ∆x, ∆x is the deformation of the spring; k is the stiffness (elasticity) coefficient of the pendulum spring.
During harmonic oscillations of a spring pendulum, the total mechanical energy is conserved:
E = const.
The values of the total energy of the spring pendulum in its three positions are reflected in the table. 10.2.
| № | Position | Wp | Wk | E = W p + W k |
|---|---|---|---|---|
| 1 | Equilibrium | 0 | m v max 2 / 2 | m v max 2 / 2 |
| 2 | Extreme | k (Δx max) 2 /2 | 0 | k (Δx max) 2 /2 |
| 3 | Intermediate (instant) | k (Δx ) 2 /2 | mv 2 /2 | mv 2 /2 + k (Δx ) 2 /2 |
The values of total mechanical energy presented in the last column of the table have equal values for any position of the pendulum, which is a mathematical expression law of conservation of total mechanical energy:
m v max 2 2 = k (Δ x max) 2 2 ;
m v max 2 2 = m v 2 2 + k (Δ x) 2 2 ;
k (Δ x max) 2 2 = m v 2 2 + k (Δ x) 2 2 ,
where m is the mass of the load; v is the module of the instantaneous speed of the load in position 3; Δx - deformation (tension or compression) of the spring in position 3; v max - module of the maximum speed of the load in position 1; Δx max - maximum deformation (tension or compression) of the spring in position 2.
Example 12. A spring pendulum performs harmonic oscillations. How many times is its kinetic energy greater than its potential energy at the moment when the displacement of the body from the equilibrium position is a quarter of the amplitude?
Solution . Let's compare two positions of the spring pendulum:
- extreme position 1 (characterized by the maximum displacement of the pendulum load from the equilibrium position x max);
- intermediate position 2 (characterized by intermediate values of displacement from the equilibrium position x and velocity v →).
The total energy of the pendulum in the extreme and intermediate positions is determined by the following formulas:
- in extreme position -
E 1 = k (Δ x max) 2 2,
where k is the stiffness (elasticity) coefficient of the spring; ∆x max - amplitude of oscillations (maximum displacement from the equilibrium position), ∆x max = A;
- in an intermediate position -
E 2 = k (Δ x) 2 2 + m v 2 2 ,
where m is the mass of the pendulum load; ∆x - displacement of the load from the equilibrium position, ∆x = A /4.
The law of conservation of total mechanical energy for a spring pendulum has the following form:
k (Δ x max) 2 2 = k (Δ x) 2 2 + m v 2 2 .
Let us divide both sides of the written equality by k (∆x) 2 /2:
(Δ x max Δ x) 2 = 1 + m v 2 2 ⋅ 2 k Δ x 2 = 1 + W k W p ,
where W k is the kinetic energy of the pendulum in an intermediate position, W k = mv 2 /2; W p - potential energy of the pendulum in an intermediate position, W p = k (∆x ) 2 /2.
Let us express the required energy ratio from the equation:
W k W p = (Δ x max Δ x) 2 − 1
and calculate its value:
W k W p = (A A / 4) 2 − 1 = 16 − 1 = 15 .
At the indicated moment of time, the ratio of the kinetic and potential energies of the pendulum is 15.
