Mathematical problems find their application in many sciences. These include not only physics, chemistry, engineering and economics, but also medicine, ecology and other disciplines. One of the important concepts that should be mastered in order to find solutions to important dilemmas is the derivative of a function. The physical meaning of it is not at all as difficult to explain as it may seem to the uninitiated in the essence of the issue. It is enough to find suitable examples of this in real life and normal everyday situations. In fact, any motorist copes with a similar task every day when he looks at the speedometer, determining the speed of his car at a particular instant of a fixed time. After all, it is in this parameter that the essence of the physical meaning of the derivative lies.

How to find speed

Any fifth grader can easily determine the speed of a person on the road, knowing the distance traveled and travel time. To do this, the first of the given values ​​\u200b\u200bis divided by the second. But not every young mathematician knows that in this moment finds the increment ratio of a function and an argument. Indeed, if we imagine the movement in the form of a graph, laying the path along the y-axis, and the time along the abscissa, it will be just that.

However, the speed of a pedestrian or any other object that we determine on a large section of the path, considering the movement to be uniform, may well change. There are many forms of motion in physics. It can be performed not only with a constant acceleration, but slow down and increase in an arbitrary way. It should be noted that in this case the line describing the movement will no longer be a straight line. Graphically, it can take on the most complex configurations. But for any of the points on the graph, we can always draw a tangent represented by a linear function.

To refine the parameter of displacement change depending on time, it is necessary to reduce the measured segments. When they become infinitely small, the calculated speed will be instantaneous. This experience helps us to define the derivative. Its physical meaning also logically follows from such reasoning.

In terms of geometry

It is known that what more speed body, the steeper the graph of the dependence of displacement on time, and hence the angle of inclination of the tangent to the graph at some particular point. An indicator of such changes can be the tangent of the angle between the x-axis and the tangent line. It is he who determines the value of the derivative and is calculated by the ratio of the lengths of the opposite to the adjacent leg in right triangle, formed by a perpendicular dropped from some point to the x-axis.

This is geometric sense first derivative. The physical one is revealed in the fact that the value of the opposite leg in our case is the distance traveled, and the adjacent one is the time. Their ratio is speed. And again we come to the conclusion that the instantaneous speed, determined when both gaps tend to infinitely small, is the essence, pointing to its physical meaning. The second derivative in this example will be the acceleration of the body, which in turn demonstrates the degree of change in speed.

Examples of finding derivatives in physics

The derivative is an indicator of the rate of change of any function, even when we are not talking about movement in the literal sense of the word. To demonstrate this clearly, let's take a few concrete examples. Suppose the current strength, depending on time, changes according to the following law: I= 0.4t2. It is required to find the value of the rate at which this parameter changes at the end of the 8th second of the process. Note that the desired value itself, as can be judged from the equation, is constantly increasing.

For the solution, it is required to find the first derivative, the physical meaning of which was considered earlier. Here dI/ dt = 0,8 t. Next, we find it at t=8 , we obtain that the rate at which the change in current strength occurs is equal to 6,4 A/ c. Here it is considered that the current strength is measured in amperes, and the time, respectively, in seconds.

Everything is changeable

Visible the world, consisting of matter, is constantly undergoing changes, being in motion flowing in it various processes. To describe them, you can use the most different parameters. If they are united by dependence, then they are mathematically written as a function that clearly shows their changes. And where there is movement (in whatever form it may be expressed), there also exists a derivative, the physical meaning of which we are considering at the present moment.

In this regard, the following example. Suppose the body temperature changes according to the law T=0,2 t 2 . You should find the rate of its heating at the end of the 10th second. The problem is solved in a manner similar to that described in the previous case. That is, we find the derivative and substitute into it the value for t= 10 , we get T= 0,4 t= 4. This means that the final answer is 4 degrees per second, that is, the heating process and the change in temperature, measured in degrees, occur at exactly this rate.

Solution of practical problems

Of course, in real life, everything is much more complicated than in theoretical problems. In practice, the value of quantities is usually determined during the experiment. In this case, instruments are used that give readings during measurements with a certain error. Therefore, in calculations, one has to deal with approximate values ​​of the parameters and resort to rounding inconvenient numbers, as well as other simplifications. Having taken this into account, we will again proceed to problems on the physical meaning of the derivative, given that they are only a kind of mathematical model of the most complex processes occurring in nature.

Eruption

Imagine that a volcano erupts. How dangerous can he be? To answer this question, many factors need to be considered. We will try to take into account one of them.

From the mouth of the "fiery monster" stones are thrown vertically upwards, having an initial speed from the moment they go outside. It is necessary to calculate how high they can reach.

To find the desired value, we compose an equation for the dependence of the height H, measured in meters, on other quantities. These include initial speed and time. The acceleration value is considered known and approximately equal to 10 m/s 2 .

Partial derivative

Let us now consider the physical meaning of the derivative of a function from a slightly different angle, because the equation itself can contain not one, but several variables. For example, in the previous problem, the dependence of the height of the stones ejected from the vent of the volcano was determined not only by the change in time characteristics, but also by the value initial speed. The latter was considered a constant, fixed value. But in other tasks with completely different conditions, everything could be different. If there are several quantities on which a complex function depends, the calculations are made according to the formulas below.

The physical meaning of the frequent derivative should be determined as in the usual case. This is the rate at which the function changes at some particular point as the parameter of the variable increases. It is calculated in such a way that all other components are taken as constants, only one is considered as a variable. Then everything happens according to the usual rules.

Understanding the physical meaning of the derivative, it is not difficult to give examples of solving intricate and complex problems, the answer to which can be found with such knowledge. If we have a function that describes the fuel consumption depending on the speed of the car, we can calculate at what parameters of the latter the gasoline consumption will be the least.

In medicine, you can predict how it will react human body to the medicine prescribed by the doctor. Taking the drug affects a variety of physiological parameters. These include changes blood pressure, pulse, body temperature and much more. All of them depend on the dose taken. medicinal product. These calculations help to predict the course of treatment, both in favorable manifestations and in undesirable accidents that can fatally affect changes in the patient's body.

Undoubtedly, it is important to understand the physical meaning of the derivative in technical matters, in particular in electrical engineering, electronics, design and construction.

Braking distances

Let's consider the next task. Moving at a constant speed, the car, approaching the bridge, had to slow down 10 seconds before the entrance, as the driver noticed road sign, prohibiting movement at a speed of more than 36 km / h. Did the driver break the rules if the braking distance can be described by the formula S = 26t - t 2 ?

Having calculated the first derivative, we find the formula for the speed, we get v = 28 - 2t. Next, we substitute the value t=10 into the specified expression.

Since this value was expressed in seconds, the speed turns out to be 8 m / s, which means 28.8 km / h. This makes it possible to understand that the driver began to slow down on time and did not violate the traffic rules, and hence the limit indicated on the speed sign.

This proves the importance of the physical meaning of the derivative. An example of solving this problem demonstrates the breadth of the use of this concept in various spheres of life. Including in everyday situations.

Derivative in economics

Before the 19th century, economists mostly dealt with averages, whether it was labor productivity or the price of output. But from some point on, limiting values ​​became more necessary for making effective forecasts in this area. These include marginal utility, income or cost. Understanding this gave impetus to the creation of a completely new tool in economic research which has existed and developed for more than a hundred years.

To make such calculations, where such concepts as minimum and maximum predominate, it is simply necessary to understand the geometric and physical meaning of the derivative. Among the creators theoretical basis These disciplines can be called such prominent English and Austrian economists as W. S. Jevons, K. Menger and others. Of course, limit values ​​in economic calculations are not always convenient to use. And, for example, quarterly reports do not necessarily fit into existing scheme, but still the application of such a theory in many cases is useful and effective.

Lesson Objectives:

Educational:

• To create conditions for meaningful assimilation by students of the physical meaning of the derivative.
• To promote the formation of skills and abilities of the practical use of the derivative for solving various physical problems.

Developing:

• To promote the development of mathematical horizons, cognitive interest among students through the disclosure of the practical necessity and theoretical significance of the topic.
• Provide conditions for improving the mental skills of students: compare, analyze, generalize.

Educational:

• Promote interest in mathematics.

Lesson type: A lesson in mastering new knowledge.

Forms of work: frontal, individual, group.

Equipment: Computer, interactive whiteboard, presentation, textbook.

Lesson structure:

1. Organizing time setting the goal of the lesson
2. Learning new material
3. Primary fixation of new material
4. Independent work
5. Summary of the lesson. Reflection.

During the classes

I. Organizational moment, setting the goal of the lesson (2 min.)

II. Learning new material (10 min.)

Teacher: In previous lessons, we got acquainted with the rules for calculating derivatives, learned how to find derivatives of a linear, power, trigonometric functions. We learned what the geometric meaning of the derivative is. Today in the lesson we will learn where this concept is applied in physics.

For this, we recall the definition of the derivative (Slide 2)

Now let's turn to the course of physics (Slide 3)

Students discuss and remember physical concepts and formulas.

Let the body move according to the law S(t)=f(t) Consider the path traveled by the body during the time from t 0 to t 0 + Δ t, where Δt is the increment of the argument. At the moment of time t 0 the body passed the path S(t 0), at the moment t 0 +Δt - the path S(t 0 +Δt). Therefore, during the time Δt, the body has traveled the path S(t 0 +Δt) –S(t 0), i.e. we got a function increment. The average speed of the body for this period of time υ==

The shorter the time interval t, the more accurately we can find out with what speed the body is moving at the moment t. Letting t → 0, we get the instantaneous speed - numerical value speed at the moment t of this movement.

υ= , at Δt→0 speed is the derivative of the distance with respect to time.

slide 4

Recall the definition of acceleration.

Applying the above material, we can conclude that at t a(t)= υ’(t) acceleration is the derivative of speed.

Further, formulas for current strength, angular velocity, EMF, etc. appear on the interactive whiteboard. Students complete the instantaneous values ​​of these physical quantities through the concept of a derivative. (With absence interactive whiteboard use presentation)

Slides 5-8

The conclusion is made by the students.

Conclusion:(Slide 9) The derivative is the rate of change of the function. (Functions of path, coordinates, speed, magnetic flux, etc.)

υ (x) \u003d f '(x)

Teacher: We see that the relationship between quantitative characteristics a wide variety of processes investigated by physics, technical sciences, chemistry, is analogous to the relationship between path and speed. You can give a lot of problems, for the solution of which it is also necessary to find the rate of change of a certain function, for example: finding the concentration of a solution at a certain moment, finding the flow rate of a liquid, the angular velocity of rotation of a body, the linear density at a point, etc. We will now solve some of these problems.

III. Consolidation of acquired knowledge (work in groups) (15 min.)

With subsequent analysis at the blackboard

Before solving problems, clarify the units of measurement of physical quantities.

Speed ​​- [m/s]
Acceleration - [m / s 2]
Strength - [N]
Energy - [J]

Task 1 group

The point moves according to the law s(t)=2t³-3t (s is the distance in meters, t is the time in seconds). Calculate the speed of the point, its acceleration at time 2s

Task 2 group

The flywheel rotates around the axis according to the law φ(t)= t 4 -5t. Find its angular velocity ω at time 2s (φ is the angle of rotation in radians, ω is the angular velocity rad/s)

Task 3 group

A body with a mass of 2 kg moves in a straight line according to the law x (t) \u003d 2-3t + 2t²

Find the speed of the body and its kinetic energy 3 s after the start of the movement. What force is acting on the body at this moment in time? (t is measured in seconds, x is in meters)

Task 4

Dot Commits oscillatory movements according to the law x(t)=2sin3t. Prove that the acceleration is proportional to the x-coordinate.

IV. Independent solution of problems No. 272, 274, 275, 277

[A.N.Kolmogorov, A.M.Abramov et al. "Algebra and the beginning of analysis grades 10-11"] 12 min

Given:	Decision:
x(t)=- ______________ t=? υ(t)=?	υ(t)=x’(t); υ(t)= (-)’= 3t²+6t= +6t; a(t)=υ'(t) a(t)=( +6t)’= 2t+6=-t+6; a(t)=0; -t+6=0; t=6; υ(6)=+6 6=-18+36=18m/s Answer: t=6c; υ(6)= 18m/s

The derivative of the function f (x) at the point x0 is the limit (if it exists) of the ratio of the increment of the function at the point x0 to the increment of the argument Δx, if the increment of the argument tends to zero and is denoted by f ‘(x0). The action of finding the derivative of a function is called differentiation.
The derivative of a function has the following physical meaning: the derivative of a function in given point- the rate of change of the function at a given point.

The geometric meaning of the derivative. The derivative at the point x0 is equal to the slope of the tangent to the graph of the function y=f(x) at this point.

The physical meaning of the derivative. If a point moves along the x-axis and its coordinate changes according to the x(t) law, then the instantaneous speed of the point:

The concept of a differential, its properties. Differentiation rules. Examples.

Definition. The differential of a function at some point x is the main, linear part of the increment of the function. The differential of the function y = f(x) is equal to the product of its derivative and the increment of the independent variable x (argument).

It is written like this:

or

Or


Differential Properties
The differential has properties similar to those of the derivative:





To basic rules of differentiation include:
1) taking the constant factor out of the sign of the derivative
2) derivative of the sum, derivative of the difference
3) derivative of the product of functions
4) derivative of a quotient of two functions (derivative of a fraction)

Examples.
Let's prove the formula: By the definition of the derivative, we have:

An arbitrary factor can be taken out of the sign of the passage to the limit (this is known from the properties of the limit), therefore

For example: Find the derivative of a function
Decision: We use the rule of taking the multiplier out of the sign of the derivative :

Quite often, it is necessary to first simplify the form of a differentiable function in order to use the table of derivatives and the rules for finding derivatives. The following examples clearly confirm this.

Differentiation formulas. Application of the differential in approximate calculations. Examples.

The use of the differential in approximate calculations allows the use of the differential for approximate calculations of function values.
Examples.
Using the differential, calculate approximately
To calculate given value apply the formula from the theory
Let us introduce a function and represent the given value in the form
then Calculate

Substituting everything into the formula, we finally get
Answer:

16. L'Hopital's rule for disclosure of uncertainties of the form 0/0 Or ∞/∞. Examples.
The limit of the ratio of two infinitesimal or two infinitely large quantities is equal to the limit of the ratio of their derivatives.

1)

17. Increasing and decreasing function. extremum of the function. Algorithm for studying a function for monotonicity and extremum. Examples.

Function increases on an interval if for any two points of this interval, related relationship, the inequality is true. I.e, greater value argument corresponds to a larger value of the function, and its graph goes “from bottom to top”. The demo function grows over the interval

Likewise, the function decreases on an interval if for any two points of the given interval, such that , the inequality is true. That is, a larger value of the argument corresponds to a smaller value of the function, and its graph goes “from top to bottom”. Ours decreases on intervals decreases on intervals .

Extremes The point is called the maximum point of the function y=f(x) if the inequality is true for all x from its neighborhood. The value of the function at the maximum point is called function maximum and denote .
The point is called the minimum point of the function y=f(x) if the inequality is true for all x from its neighborhood. The value of the function at the minimum point is called function minimum and denote .
The neighborhood of a point is understood as the interval , where is a sufficiently small positive number.
The minimum and maximum points are called extremum points, and the function values ​​corresponding to the extremum points are called function extrema.

To explore a function for monotony use the following diagram:
- Find the scope of the function;
- Find the derivative of the function and the domain of the derivative;
- Find the zeros of the derivative, i.e. the value of the argument at which the derivative is equal to zero;
- Mark on the number line general part the domain of the function and the domain of its derivative, and on it - the zeros of the derivative;
- Determine the signs of the derivative on each of the obtained intervals;
- By the signs of the derivative, determine at which intervals the function increases and at which it decreases;
- Record the appropriate gaps separated by semicolons.

Algorithm for studying a continuous function y = f(x) for monotonicity and extrema:
1) Find the derivative f ′(x).
2) Find stationary (f ′(x) = 0) and critical (f ′(x) does not exist) points of the function y = f(x).
3) Mark the stationary and critical points on the real line and determine the signs of the derivative on the resulting intervals.
4) Draw conclusions about the monotonicity of the function and its extremum points.

18. Convexity of a function. Inflection points. Algorithm for examining a function for convexity (Concavity) Examples.

convex down on the X interval, if its graph is located not lower than the tangent to it at any point of the X interval.

The differentiable function is called convex up on the X interval, if its graph is located no higher than the tangent to it at any point of the X interval.

The point formula is called graph inflection point function y \u003d f (x), if at a given point there is a tangent to the graph of the function (it can be parallel to the Oy axis) and there is such a neighborhood of the point formula, within which the graph of the function has different directions of convexity to the left and to the right of the point M.

Finding intervals for convexity:

If the function y=f(x) has a finite second derivative on the interval X and if the inequality (), then the graph of the function has a convexity directed down (up) on X.
This theorem allows you to find the intervals of concavity and convexity of a function, you only need to solve the inequalities and, respectively, on the domain of definition of the original function.

Example: Find out the intervals at which the graph of the functionFind out the intervals at which the graph of the function has a convexity directed upwards and a convexity directed downwards. has a convexity directed upwards and a convexity directed downwards.
Decision: The domain of this function is the entire set of real numbers.
Let's find the second derivative.

The domain of definition of the second derivative coincides with the domain of definition of the original function, therefore, in order to find out the intervals of concavity and convexity, it is enough to solve and respectively. Therefore, the function is downward convex on the interval formula and upward convex on the interval formula.

19) Asymptotes of a function. Examples.

Direct called vertical asymptote graph of the function if at least one of the limit values ​​or is equal to or .

Comment. The line cannot be a vertical asymptote if the function is continuous at . Therefore, vertical asymptotes should be sought at the discontinuity points of the function.

Direct called horizontal asymptote graph of the function if at least one of the limit values ​​or is equal to .

Comment. A function graph can only have a right horizontal asymptote or only a left one.

Direct called oblique asymptote graph of the function if

EXAMPLE:

Exercise. Find asymptotes of the graph of a function

Decision. Function scope:

a) vertical asymptotes: a straight line is a vertical asymptote, since

b) horizontal asymptotes: we find the limit of the function at infinity:

that is, there are no horizontal asymptotes.

c) oblique asymptotes:

Thus, the oblique asymptote is: .

Answer. The vertical asymptote is a straight line.

The oblique asymptote is a straight line.

20) General scheme function studies and plotting. Example.

a.
Find the ODZ and breakpoints of the function.

b. Find the points of intersection of the graph of the function with the coordinate axes.

2. Conduct a study of the function using the first derivative, that is, find the extremum points of the function and the intervals of increase and decrease.

3. Investigate the function using the second-order derivative, that is, find the inflection points of the function graph and the intervals of its convexity and concavity.

4. Find the asymptotes of the graph of the function: a) vertical, b) oblique.

5. On the basis of the study, build a graph of the function.

Note that before plotting, it is useful to establish whether a given function is even or odd.

Recall that a function is called even if the value of the function does not change when the sign of the argument changes: f(-x) = f(x) and a function is called odd if f(-x) = -f(x).

In this case, it is enough to study the function and plot its graph for positive values argument belonging to the ODZ. At negative values argument, the graph is completed on the basis that for an even function it is symmetrical about the axis Oy, and for odd with respect to the origin.

Examples. Explore functions and build their graphs.

Function scope D(y)= (–∞; +∞). There are no break points.

Axis intersection Ox: x = 0,y= 0.

The function is odd, therefore, it can be studied only on the interval , and its argument is in units of [x], then the derivative (speed) is measured in units of .

Task 6

x(t) = 6t 2 − 48t+ 17, where x t t= 9s.

Finding the derivative
x"(t) = (6t 2 − 48t + 17)" = 12t − 48.
Thus, we have obtained the dependence of speed on time. To find the speed at a given point in time, you need to substitute its value in the resulting formula:
x"(t) = 12t − 48.
x"(9) = 12 9 − 48 = 60.

Answer: 60

Comment: Let's make sure that we did not make a mistake with the dimensions of the quantities. Here, the unit of distance (function) [x] = meter, the unit of time (function argument) [t] = second, hence the unit of derivative = [m/s], i.e. the derivative gives the speed just in those units that are mentioned in the question of the problem.

Task 7

The material point moves in a straight line according to the law x(t) = −t 4 + 6t 3 + 5t+ 23, where x- distance from the reference point in meters, t- time in seconds, measured from the start of the movement. Find its speed (in meters per second) at the time t= 3s.

Finding the derivative
x"(t) = (−t 4 + 6t 3 + 5t + 23)" = −4t 3 + 18t 2 + 5.
We substitute the given moment of time in the resulting formula
x"(3) = −4 3 3 + 18 3 2 + 5 = −108 + 162 + 5 = 59.

Answer: 59

Task 8

The material point moves in a straight line according to the law x(t) = t 2 − 13t+ 23, where x- distance from the reference point in meters, t- time in seconds, measured from the start of the movement. At what point in time (in seconds) was her speed equal to 3 m/s?

Finding the derivative
x"(t) = (t 2 − 13t + 23)" = 2t − 13.
We equate the speed given by the obtained formula to the value of 3 m/s.
2t − 13 = 3.
Solving this equation, we determine at what time the equality is true.
2t − 13 = 3.
2t = 3 + 13.
t = 16/2 = 8.

Answer: 8

Task 9

The material point moves in a straight line according to the law x(t) = (1/3)t 3 − 3t 2 − 5t+ 3, where x- distance from the reference point in meters, t- time in seconds, measured from the start of the movement. At what point in time (in seconds) was her speed equal to 2 m/s?

Finding the derivative
x"(t) = ((1/3)t 3 − 3t 2 − 5t + 3)" = t 2 − 6t − 5.
We also make an equation:
t 2 − 6t − 5 = 2;
t 2 − 6t − 7 = 0.
This is a quadratic equation that can be solved using the discriminant or Vieta's theorem. Here, in my opinion, the second way is easier:
t 1 + t 2 = 6; t one · t 2 = −7.
It's easy to guess that t 1 = −1; t 2 = 7.
We put only the positive root in the answer, because time cannot be negative.

Consider an arbitrary straight line passing through the point of the graph of the function - the point A (x 0, f (x 0)) and intersecting the graph at some point B(x; f(x )). Such a straight line (AB) is called a secant. From ∆ABC: ​​AC = ∆ x; BC \u003d ∆y; tgβ =∆y /∆x .

Since AC || Ox , then Р ALO = Р BAC = β (as corresponding with parallel). ButÐ ALO is the angle of inclination of the secant AB to the positive direction of the Ox axis. Means, tgβ = k - slope direct AB.

Now we will decrease ∆x, i.e. ∆x→ 0. In this case, point B will approach point A according to the graph, and the secant AB will rotate. The limiting position of the secant AB at ∆х→ 0 will be a straight line ( a ), called the tangent to the graph of the function y = f(x) at point A.

If we pass to the limit as ∆х → 0 in the equality tg β =∆ y /∆ x , then we get

or tg a \u003d f "(x 0), since
a - angle of inclination of the tangent to the positive direction of the Ox axis

, by definition of a derivative. But tg a = k is the slope of the tangent, so k = tg a \u003d f "(x 0).

So, the geometric meaning of the derivative is as follows:

The derivative of the function at the point x 0 is equal to the slope tangent to the graph of the function drawn at the point with the abscissa x 0 .

The physical meaning of the derivative.

Consider the movement of a point along a straight line. Let the coordinate of the point at any moment of time be given x(t ). It is known (from the course of physics) that average speed for a period of time [ t0; t0 + ∆t ] is equal to the ratio of the distance traveled during this period of time to the time, i.e.

Vav = ∆x /∆t . Let us pass to the limit in the last equality as ∆ t → 0.

lim V cf (t) = n (t 0 ) - instantaneous speed at time t 0 , ∆t → 0.

and lim \u003d ∆ x / ∆ t \u003d x "(t 0 ) (by definition of a derivative).

So, n(t) = x "(t).

The physical meaning of the derivative is as follows: the derivative of the function y = f( x) at the pointx 0 is the rate of change of the function f(x) at the pointx 0

The derivative is used in physics to find the speed from a known function of coordinates from time, acceleration from a known function of speed from time.

u (t) \u003d x "(t) - speed,

a(f) = n "(t ) - acceleration, or

a (t) \u003d x "(t).

If the law of motion of a material point along a circle is known, then it is possible to find the angular velocity and angular acceleration during rotational motion:

φ = φ (t ) - change of angle from time,

ω = φ "(t ) - angular velocity,

ε = φ "(t ) - angular acceleration, orε \u003d φ "(t).

If the distribution law for the mass of an inhomogeneous rod is known, then the linear density of the inhomogeneous rod can be found:

m \u003d m (x) - mass,

x н , l - rod length,

p = m "(x) - linear density.

With the help of the derivative, problems from the theory of elasticity and harmonic vibrations are solved. Yes, according to Hooke's law

F = - kx , x - variable coordinate, k - coefficient of elasticity of the spring. Puttingω 2 = k / m , we get differential equation spring pendulum x "( t ) + ω 2 x(t ) = 0,

where ω = √k /√m oscillation frequency ( l/c ), k - spring stiffness ( H/m).

An equation of the form y" +ω 2 y = 0 is called the equation of harmonic oscillations (mechanical, electrical, electromagnetic). The solution of such equations is the function

y \u003d Asin (ωt + φ 0 ) or y \u003d Acos (ωt + φ 0 ), where

A is the amplitude of oscillations,ω - cyclic frequency,

φ 0 - initial phase.