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>>Mathematics: What does the notation y = f(x) mean in mathematics?

What does the notation y = f(x) mean in mathematics?

When studying any real process, we usually pay attention to two quantities involved in the process (more complex processes not two quantities are involved, but three, four, etc., but we are not considering such processes yet): one of them changes as if by itself, regardless of anything (we denoted such a variable by the letter x), and the other quantity takes values ​​that depend on the selected values ​​of the variable x (we denoted such a dependent variable by the letter y). Mathematical model of a real process is precisely the recording in mathematical language of the dependence of y on x, i.e. connections between variables x and y. Let us remind you once again that by now we have studied the following mathematical models: y = b, y = kx, y = kx + m, y = x 2.

Do these mathematical models have anything in common? Eat! Their structure is the same: y = f(x).

This entry should be understood as follows: there is an expression f(x) with the variable x, with the help of which the values ​​of the variable y are found.

Mathematicians prefer the notation y = f(x) for a reason. Let, for example, f(x) = x 2, i.e. we're talking about O function y = x 2. Suppose we need to select several argument values ​​and corresponding function values. So far we have written like this:

if x = 1, then y = I 2 = 1;
if x = - 3, then y = (- 3) 2 = 9, etc.

If we use the notation f(x) = x 2, then the notation becomes more economical:

f(1) = 1 2 =1;
f(-3) = (-3) 2 = 9.

So, we got acquainted with one more fragment mathematical language: the phrase “the value of the function y = x 2 at the point x = 2 is 4” is written shorter:

“if y = f(x), where f(x) = x 2, then f(2) = 4.”

And here is a sample reverse translation:

If y = f(x), where f(x) = x 2, then f(- 3) = 9. In other words, the value of the function y = x 2 at the point x = - 3 is 9.

Example 1. Given a function y = f(x), where f(x) = x 3. Calculate:

a) f(1); b) f(- 4); c) f(o); d) f(2a);
e) f(a-1); e) f(3x); g) f(-x).

Solution. In all cases, the action plan is the same: you need to substitute in the expression f(x) for x the value of the argument that is indicated in parentheses, and perform the appropriate calculations and transformations. We have:

Comment. Of course, instead of the letter f, you can use any other letter (mostly from Latin alphabet): g(x), h(x), s(x), etc.

Example 2. Two functions are given: y = f(x), where f(x) = x 2, and y = g (x), where g (x) = x 3. Prove that:

a) f(-x) = f(x); b) g(-x)= -g(x).

Solution. a) Since f(x) = x 2, then f(- x) = (- x) 2 = x 2. So, f(x) = x 2, f(- x) = x 2, which means f(- x) = f (x)

b) Since g(x) = x 3, then g(- x) = -x 3, i.e. g(-x) = -g(x).

Usage mathematical model of the form y = f(x) turns out to be convenient in many cases, in particular, when the real process is described various formulas at different intervals of change in the independent variable.

Using the graph constructed in Figure 68, we will describe some properties of the function y - f(x) - such a description of properties is usually called reading the graph.

Reading a graph is a kind of transition from geometric model(from a graphical model) to a verbal model (to a description of the properties of a function). A
constructing a graph is a transition from an analytical model (it is presented in the condition of example 4) to a geometric model.

So, let's start reading the graph of the function y = f(x) (see Fig. 68).

1. The independent variable x runs through all values ​​from - 4 to 4. In other words, for each value of x from the interval [- 4, 4], the value of the function f(x) can be calculated. They say this: [-4, 4] is the domain of definition of the function.

Why did we say when solving Example 4 that f(5) cannot be found? Yes, because the value x = 5 does not belong to the domain of definition of the function.

2. y max = -2 (the function reaches this value at x = -4); At Nanb. = 2 (the function reaches this value at any point in the half-interval (0, 4].

3. y = 0 if 1 = -2 and if x = 0; at these points the graph of the function y = f(x) intersects the x-axis.

4. y > 0 if x є (-2, 0) or if x є (0, 4]; on these intervals, the graph of the function y = f(x) is located above the x axis.

5. y< 0, если же [- 4, - 2); на этом промежутке график функции у = f(x) расположен ниже оси х.

6. The function increases on the interval [-4, -1], decreases on the interval [-1, 0] and is constant (neither increases nor decreases) on the half-interval (0.4].

As we learn new properties of functions, the process of reading a graph will become more rich, meaningful and interesting.

Let's discuss one of these new properties. The graph of the function discussed in example 4 consists of three branches (three “pieces”). The first and second branches (the straight line segment y = x + 2 and part of the parabola) are “joined” successfully: the segment ends at the point (-1; 1), and the parabola section begins at the same point. But the second and third branches are less successfully “joined”: the third branch (“piece” of the horizontal line) begins not at the point (0; 0), but at the point (0; 4). Mathematicians say this: “the function y = f(x) undergoes a discontinuity at x = 0 (or at the point x = 0).” If a function does not have discontinuity points, then it is called continuous. So, all the functions that we met in the previous paragraphs (y = b, y = kx, y = kx + m, y = x2) are continuous.

Example 5. The function is given. You need to build and read its graph.

Solution. As you can see, the function here is sufficiently defined complex expression. But mathematics is a single and integral science, its sections are closely related to each other. Let's use what we learned in Chapter 5 and reduce algebraic fraction

valid only under the restriction Therefore, we can reformulate the problem as follows: instead of the function y = x 2
we will consider the function y = x 2, where Let's build on coordinate plane xOy parabola y = x 2 .
The straight line x = 2 intersects it at the point (2; 4). But according to the condition, this means that we must exclude point (2; 4) of the parabola from consideration, for which purpose we mark this point with a light circle in the drawing.

Thus, the graph of the function is constructed - this is a parabola y = x 2 with a “pricked out” point (2; 4) (Fig. 69).

Let's move on to describing the properties of the function y = f (x), i.e., reading its graph:

1. The independent variable x takes any value except x = 2. This means that the domain of definition of the function consists of two open rays(- 0 o, 2) and

2. y max = 0 (achieved at x = 0), y max _ does not exist.

3. The function is not continuous; it undergoes a discontinuity at x = 2 (at the point x = 2).

4. y = 0 if x = 0.

5. y > 0 if x є (-oo, 0), if x є (0, 2) and if x є (B,+oo).
6. The function decreases on the ray (- co, 0], increases on the half-interval .

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A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

Instructions

Example Problem 3: A 1 kg block slides off the top inclined plane in 5 seconds, the path is 10 meters. Determine the friction force if the angle of inclination of the plane is 45°. Consider also the case when the block was subjected to an additional force of 2 N applied along the angle of inclination in the direction of movement.

Find the acceleration of the body similarly to examples 1 and 2: a = 2*10/5^2 = 0.8 m/s2. Calculate the friction force in the first case: Ftr = 1*9.8*sin(45о)-1*0.8 = 7.53 N. Determine the friction force in the second case: Ftr = 1*9.8*sin(45о) +2-1*0.8= 9.53 N.

Case 6. The body moves along inclined surface evenly. This means that according to Newton's second law, the system is in equilibrium. If the sliding is spontaneous, the movement of the body obeys the equation: mg*sinα = Ftr.

If an additional force (F) is applied to the body, preventing uniformly accelerated movement, the expression for motion has the form: mg*sinα–Ftr-F = 0. From here, find the friction force: Ftr = mg*sinα-F.

Sources:

• slip formula

At relative motion two bodies, friction arises between them. It can also occur when moving in gaseous or liquid medium. Friction can both hinder and promote normal movement. As a result of this phenomenon, a force acts on the interacting bodies.

Instructions

Most general case is considering force, when one of the bodies is fixed and at rest, and the other slides along its surface. From the side of the body along which the moving body slides, the support reaction force directed perpendicular to the sliding plane acts on the latter. This force is the letter N. A body can also be at rest relative to a fixed body. Then strength friction, acting on it Ftrfriction. It depends on the materials of the rubbing surfaces, the degree of their polishing and a number of other factors.

In the case of body motion relative to the surface of a fixed body, the force friction slip becomes equal to the product of the coefficient friction on force support reactions: Ftr = ?N.

If the surface is horizontal, then the force of the support reaction in modulus is equal to the force of gravity on the body, that is, N = mg, where m is the mass of the sliding body, g is the free acceleration, equal to approximately 9.8 m/(s^2) on Earth . Hence, Ftr = ?mg.

Let now a constant force F>Ftr = ?N act on the body, parallel to the surface of the contacting bodies. When a body slides, the resulting component of the force in the horizontal direction will be equal to F-Ftr. Then, according to Newton’s second law, the acceleration of the body will be related to the resulting force according to the formula: a = (F-Ftr)/m. Hence, Ftr = F-ma. The acceleration of a body can be found from kinematic considerations.

A frequently considered special case of force friction when a body slips from a fixed plane. Let it? - the angle of inclination of the plane and let the body slide evenly, that is, without . Then the equations of motion of the body will look like this: N = mg*cos?, mg*sin? = Ftr = ?N. Then from the first equation of motion force friction can be expressed as Ftr = ?mg*cos?. If the body moves along an inclined plane with a, then the second equation will have the form: mg*sin?-Ftr = ma. Then Ftr = mg*sin?-ma.

Video on the topic

If the force directed parallel to the surface on which the body stands exceeds the static friction force, then movement will begin. It will continue as long as the driving force exceeds the sliding friction force, which depends on the friction coefficient. You can calculate this coefficient yourself.

You will need

• Dynamometer, scales, protractor or protractor

Instructions

Find the mass of the body in kilograms and place it on a flat surface. Attach a dynamometer to it and start moving your body. Do this in such a way that the dynamometer readings stabilize, maintaining a constant speed. In this case, the traction force measured by the dynamometer will be equal, on the one hand, to the traction force, which is shown by the dynamometer, and on the other hand, the force multiplied by the sliding.

The measurements taken will allow us to find this coefficient from the equation. To do this, divide the traction force by body weight and the number 9.81 (gravitational acceleration) μ=F/(m g). The resulting coefficient will be the same for all surfaces of the same type as those on which the measurement was made. For example, if a body was moving on a wooden board, then this result will be valid for all wooden bodies moving by sliding on the tree, taking into account the quality of its processing (if the surfaces are rough, the value of the sliding friction coefficient will change).

You can measure the sliding friction coefficient in another way. To do this, place the body on a plane that can change its angle relative to the horizon. It could be an ordinary board. Then start carefully at one edge. At the moment when the body begins to move, sliding down a plane like a sled down a hill, find the angle of its inclination relative to the horizon. It is important that the body does not move with acceleration. In this case, the measured angle will be extremely small, at which the body will begin to move under . The sliding friction coefficient will be equal to the tangent of this angle μ=tg(α).

Video on the topic

Strength reactions supports refers to elastic forces, and is always directed perpendicular to the surface. It resists any force that causes the body to move perpendicular to the support. In order to calculate it, you need to identify and find out the numerical value of all the forces that act on the body standing on the support.

You will need

• - scales;
• - speedometer or radar;
• - goniometer.

Instructions

Determine body weight using scales or any other method. If the body is on a horizontal surface (and it does not matter whether it is moving or at rest), then the force of support is equal to the force of gravity on the body. In order to calculate it, multiply the body mass by the acceleration of gravity, which is equal to 9.81 m/s² N=m g.

When a body moves along an inclined plane directed at an angle to the horizontal, the ground reaction force is at an angle to the force of gravity. At the same time, it compensates only for that component of gravity that acts perpendicular to the inclined plane. To calculate the support reaction force, use a protractor to measure the angle at which the plane is located to the horizontal. Calculate force support reaction, multiplying the body mass by the acceleration of gravity and the cosine of the angle at which the plane is located to the horizon N=m g Cos(α).

If a body moves along a surface that is a part of a circle with a radius R, for example, a bridge, then the support reaction force takes into account the force in the direction from the center of the circle, with an acceleration equal to the centripetal one, acting on the body. To calculate the reaction force of the support at the top point, subtract the square of the velocity from the acceleration of gravity to the radius.

Multiply the resulting number by the mass of the moving body N=m (g-v²/R). Speed ​​should be measured in meters per second and radius in meters. At a certain speed, the value of the acceleration directed from the center of the circle can become equal, and even the acceleration of gravity, at which point the adhesion of the body to the surface will disappear, therefore, for example, motorists need to clearly control the speed on such sections of the road.

If it is directed downward and the body’s trajectory is concave, then calculate the support reaction force by adding to the acceleration of free fall the ratio of the square of the speed and the radius of curvature of the trajectory, and multiply the resulting result by the mass of the body N=m (g+v²/R).

Sources:

• strength support

Movement in real conditions cannot continue indefinitely. The reason for this is friction. It occurs when a body comes into contact with other bodies and is always directed opposite to the direction of movement. This means that the strength friction always performs negative work, which must be taken into account when making calculations.

You will need

• - tape measure or rangefinder;
• - table for determining the friction coefficient;
• - concept of kinetic energy;
• - scales;
• - calculator.

Instructions

If a body moves uniformly and in a straight line, find the force that causes it to move. She compensates for the strength friction, therefore numerically equal to it, but to the side. Using a tape measure or rangefinder, measure the distance S by which the force F moved the body. Then work strength friction will be equal to the product strength to a distance with a minus sign A=-F∙S.

Example. The car moves along the road evenly and in a straight line. Which work strength friction at a distance of 200 m, if the engine thrust is 800 N? With a uniform rectilinear force, the traction force of the engine is equal in magnitude to the force friction. Then its work will be equal to A=-F∙S =-800∙200=-160000 J or -160 kJ.

In this example, we will consider how the reliability of the resulting regression equation is assessed. The same test is used to test the hypothesis that the regression coefficients are simultaneously equal to zero, a=0, b=0. In other words, the essence of the calculations is to answer the question: can it be used for further analysis and forecasts?

To determine whether the variances in two samples are similar or different, use this t-test.

So, the purpose of the analysis is to obtain some estimate with which it could be stated that at a certain level of α the resulting regression equation is statistically reliable. For this coefficient of determination R 2 is used.
Testing the significance of a regression model is carried out using Fisher's F test, the calculated value of which is found as the ratio of the variance of the original series of observations of the indicator being studied and the unbiased estimate of the variance of the residual sequence for this model.
If the calculated value with k 1 =(m) and k 2 =(n-m-1) degrees of freedom is greater than the tabulated value at a given significance level, then the model is considered significant.

where m is the number of factors in the model.
The statistical significance of paired linear regression is assessed using the following algorithm:
1. A null hypothesis is put forward that the equation as a whole is statistically insignificant: H 0: R 2 =0 at the significance level α.
2. Next, determine the actual value of the F-criterion:


where m=1 for pairwise regression.
3. The tabulated value is determined from the Fisher distribution tables for a given significance level, taking into account that the number of degrees of freedom for the total sum of squares (larger variance) is 1 and the number of degrees of freedom for the residual sum of squares (smaller variance) in linear regression is n-2 (or through the Excel function FRIST(probability,1,n-2)).
F table is the maximum possible value of the criterion under the influence of random factors with given degrees of freedom and significance level α. The significance level α is the probability of rejecting the correct hypothesis, provided that it is true. Typically α is taken to be 0.05 or 0.01.
4. If the actual value of the F-test is less than the table value, then they say that there is no reason to reject the null hypothesis.
Otherwise, the null hypothesis is rejected and with probability (1-α) the alternative hypothesis about the statistical significance of the equation as a whole is accepted.
Table value of the criterion with degrees of freedom k 1 =1 and k 2 =48, F table = 4

Conclusions: Since the actual value F > F table, the coefficient of determination is statistically significant ( the found regression equation estimate is statistically reliable) .

Analysis of variance

Regression equation quality indicators

Example. Based on a total of 25 trading enterprises, the relationship between the following characteristics is studied: X - price of product A, thousand rubles; Y is the profit of a trading enterprise, million rubles. When assessing the regression model, the following intermediate results were obtained: ∑(y i -y x) 2 = 46000; ∑(y i -y avg) 2 = 138000. What correlation indicator can be determined from these data? Calculate the value of this indicator based on this result and using Fisher's F test draw conclusions about the quality of the regression model.
Solution. From these data we can determine the empirical correlation ratio: , where ∑(y avg -y x) 2 = ∑(y i -y avg) 2 - ∑(y i -y x) 2 = 138000 - 46000 = 92,000.
η 2 = 92,000/138000 = 0.67, η = 0.816 (0.7< η < 0.9 - связь между X и Y высокая).

Fisher's F test: n = 25, m = 1.
R 2 = 1 - 46000/138000 = 0.67, F = 0.67/(1-0.67)x(25 - 1 - 1) = 46. F table (1; 23) = 4.27
Since the actual value F > Ftable, the found estimate of the regression equation is statistically reliable.

Question: What statistics are used to test the significance of a regression model?
Answer: For the significance of the entire model as a whole, F-statistics (Fisher's test) are used.

Definition. Let the function \(y = f(x) \) be defined in a certain interval containing the point \(x_0\) within itself. Let's give the argument an increment \(\Delta x \) such that it does not leave this interval. Let's find the corresponding increment of the function \(\Delta y \) (when moving from the point \(x_0 \) to the point \(x_0 + \Delta x \)) and compose the relation \(\frac(\Delta y)(\Delta x) \). If there is a limit to this ratio at \(\Delta x \rightarrow 0\), then the specified limit is called derivative of a function\(y=f(x) \) at the point \(x_0 \) and denote \(f"(x_0) \).

$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x_0) $$

The symbol y is often used to denote the derivative. Note that y" = f(x) is a new function, but naturally related to the function y = f(x), defined at all points x at which the above limit exists . This function is called like this: derivative of the function y = f(x).

Geometric meaning of derivative is as follows. If it is possible to draw a tangent to the graph of the function y = f(x) at the point with abscissa x=a, which is not parallel to the y-axis, then f(a) expresses the slope of the tangent:
\(k = f"(a)\)

Since \(k = tg(a) \), then the equality \(f"(a) = tan(a) \) is true.

Now let’s interpret the definition of derivative from the point of view of approximate equalities. Let the function \(y = f(x)\) have a derivative at a specific point \(x\):
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x) $$
This means that near the point x the approximate equality \(\frac(\Delta y)(\Delta x) \approx f"(x)\), i.e. \(\Delta y \approx f"(x) \cdot\Delta x\). The meaningful meaning of the resulting approximate equality is as follows: the increment of the function is “almost proportional” to the increment of the argument, and the coefficient of proportionality is the value of the derivative at a given point x. For example, for the function \(y = x^2\) the approximate equality \(\Delta y \approx 2x \cdot \Delta x \) is valid. If we carefully analyze the definition of a derivative, we will find that it contains an algorithm for finding it.

Let's formulate it.

How to find the derivative of the function y = f(x)?

1. Fix the value of \(x\), find \(f(x)\)
2. Give the argument \(x\) an increment \(\Delta x\), go to a new point \(x+ \Delta x \), find \(f(x+ \Delta x) \)
3. Find the increment of the function: \(\Delta y = f(x + \Delta x) - f(x) \)
4. Create the relation \(\frac(\Delta y)(\Delta x) \)
5. Calculate $$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) $$
This limit is the derivative of the function at point x.

If a function y = f(x) has a derivative at a point x, then it is called differentiable at a point x. The procedure for finding the derivative of the function y = f(x) is called differentiation functions y = f(x).

Let us discuss the following question: how are continuity and differentiability of a function at a point related to each other?

Let the function y = f(x) be differentiable at the point x. Then a tangent can be drawn to the graph of the function at point M(x; f(x)), and, recall, the angular coefficient of the tangent is equal to f "(x). Such a graph cannot “break” at point M, i.e. the function must be continuous at point x.

These were “hands-on” arguments. Let us give a more rigorous reasoning. If the function y = f(x) is differentiable at the point x, then the approximate equality \(\Delta y \approx f"(x) \cdot \Delta x\) holds. If in this equality \(\Delta x \) tends to zero, then \(\Delta y \) will tend to zero, and this is the condition for the continuity of the function at a point.

So, if a function is differentiable at a point x, then it is continuous at that point.

The reverse statement is not true. For example: function y = |x| is continuous everywhere, in particular at the point x = 0, but the tangent to the graph of the function at the “junction point” (0; 0) does not exist. If at some point a tangent cannot be drawn to the graph of a function, then the derivative does not exist at that point.

Another example. The function \(y=\sqrt(x)\) is continuous on the entire number line, including at the point x = 0. And the tangent to the graph of the function exists at any point, including at the point x = 0. But at this point the tangent coincides with the y-axis, i.e., it is perpendicular to the abscissa axis, its equation has the form x = 0. Such a straight line does not have an angle coefficient, which means that \(f"(0)\) does not exist.

So, we got acquainted with a new property of a function - differentiability. How can one conclude from the graph of a function that it is differentiable?

The answer is actually given above. If at some point it is possible to draw a tangent to the graph of a function that is not perpendicular to the abscissa axis, then at this point the function is differentiable. If at some point the tangent to the graph of a function does not exist or it is perpendicular to the abscissa axis, then at this point the function is not differentiable.

Rules of differentiation

The operation of finding the derivative is called differentiation. When performing this operation, you often have to work with quotients, sums, products of functions, as well as “functions of functions,” that is, complex functions. Based on the definition of derivative, we can derive differentiation rules that make this work easier. If C is a constant number and f=f(x), g=g(x) are some differentiable functions, then the following are true differentiation rules:

Table of derivatives of some functions

The word “power” is so comprehensive that giving it a clear concept is an almost impossible task. The variety from muscle strength to mind strength does not cover the entire spectrum of concepts included in it. Force, considered as a physical quantity, has a clearly defined meaning and definition. The force formula specifies a mathematical model: the dependence of force on basic parameters.

The history of the study of forces includes the determination of dependence on parameters and experimental proof of the dependence.

Power in Physics

Force is a measure of the interaction of bodies. The mutual action of bodies on each other fully describes the processes associated with changes in speed or deformation of bodies.

As a physical quantity, force has a unit of measurement (in the SI system - Newton) and a device for measuring it - a dynamometer. The principle of operation of the force meter is based on comparing the force acting on the body with the elastic force of the dynamometer spring.

A force of 1 newton is taken to be the force under the influence of which a body weighing 1 kg changes its speed by 1 m in 1 second.

Strength as defined:

• direction of action;
• application point;
• module, absolute value.

When describing interaction, be sure to indicate these parameters.

Types of natural interactions: gravitational, electromagnetic, strong, weak. Gravitational universal gravitation with its variety - gravity) exist due to the influence of gravitational fields surrounding any body that has mass. The study of gravitational fields has not yet been completed. It is not yet possible to find the source of the field.

A larger number of forces arise due to the electromagnetic interaction of the atoms that make up the substance.

Pressure force

When a body interacts with the Earth, it exerts pressure on the surface. The force of which has the form: P = mg, is determined by body mass (m). The acceleration due to gravity (g) has different values ​​at different latitudes of the Earth.

The vertical pressure force is equal in magnitude and opposite in direction to the elastic force arising in the support. The formula of force changes depending on the movement of the body.

Change in body weight

The action of a body on support due to interaction with the Earth is often called body weight. Interestingly, the amount of body weight depends on the acceleration of movement in the vertical direction. In the case where the direction of acceleration is opposite to the acceleration of gravity, an increase in weight is observed. If the acceleration of the body coincides with the direction of free fall, then the weight of the body decreases. For example, being in an ascending elevator, at the beginning of the ascent a person feels an increase in weight for some time. There is no need to say that its mass changes. At the same time, we separate the concepts of “body weight” and its “mass”.

Elastic force

When the shape of a body changes (its deformation), a force appears that tends to return the body to its original shape. This force was given the name "elasticity force". It arises as a result of the electrical interaction of the particles that make up the body.

Let's consider the simplest deformation: tension and compression. Tension is accompanied by an increase in the linear dimensions of bodies, compression - by their decrease. The quantity characterizing these processes is called body elongation. Let's denote it "x". The elastic force formula is directly related to elongation. Each body undergoing deformation has its own geometric and physical parameters. The dependence of the elastic resistance to deformation on the properties of the body and the material from which it is made is determined by the elasticity coefficient, let's call it rigidity (k).

The mathematical model of elastic interaction is described by Hooke's law.

The force arising during deformation of the body is directed against the direction of displacement of individual parts of the body and is directly proportional to its elongation:

• F y = -kx (in vector notation).

The “-” sign indicates the opposite direction of deformation and force.

In scalar form there is no negative sign. The elastic force, the formula of which has the following form F y = kx, is used only for elastic deformations.

Interaction of magnetic field with current

The effect of a magnetic field on a direct current is described. In this case, the force with which the magnetic field acts on a conductor with current placed in it is called the Ampere force.

The interaction of the magnetic field with causes force manifestation. Ampere's force, the formula of which is F = IBlsinα, depends on (B), the length of the active part of the conductor (l), (I) in the conductor and the angle between the direction of the current and the magnetic induction.

Thanks to the last dependence, it can be argued that the vector of action of the magnetic field can change when the conductor is rotated or the direction of the current changes. The left hand rule allows you to determine the direction of action. If the left hand is positioned in such a way that the magnetic induction vector enters the palm, the four fingers are directed along the current in the conductor, then the thumb bent 90 ° will show the direction of the magnetic field.

Mankind has found applications for this effect, for example, in electric motors. Rotation of the rotor is caused by a magnetic field created by a powerful electromagnet. The force formula allows you to judge the possibility of changing engine power. As the current or field strength increases, the torque increases, which leads to an increase in motor power.

Particle trajectories

The interaction of a magnetic field with a charge is widely used in mass spectrographs in the study of elementary particles.

The action of the field in this case causes the appearance of a force called the Lorentz force. When a charged particle moving at a certain speed enters a magnetic field, the formula of which is F = vBqsinα, causes the particle to move in a circle.

In this mathematical model, v is the modulus of the velocity of a particle whose electric charge is q, B is the magnetic induction of the field, α is the angle between the directions of velocity and magnetic induction.

The particle moves in a circle (or arc of a circle), since the force and speed are directed at an angle of 90 ° to each other. Changing the direction of linear velocity causes acceleration to appear.

The rule of the left hand, discussed above, also occurs when studying the Lorentz force: if the left hand is positioned in such a way that the magnetic induction vector enters the palm, four fingers extended in a line are directed along the speed of a positively charged particle, then bent by 90 ° the thumb will indicate the direction of the force.

Plasma problems

The interaction of a magnetic field and matter is used in cyclotrons. Problems associated with the laboratory study of plasma do not allow it to be kept in closed vessels. A highly ionized gas can only exist at high temperatures. Plasma can be kept in one place in space using magnetic fields, twisting the gas in the form of a ring. Controlled ones can also be studied by twisting high-temperature plasma into a cord using magnetic fields.

An example of the effect of a magnetic field under natural conditions on ionized gas is the Aurora Borealis. This majestic spectacle is observed above the Arctic Circle at an altitude of 100 km above the surface of the earth. The mysterious colorful glow of the gas could only be explained in the 20th century. The earth's magnetic field near the poles cannot prevent the solar wind from entering the atmosphere. The most active radiation, directed along magnetic induction lines, causes ionization of the atmosphere.

Phenomena associated with charge movement

Historically, the main quantity characterizing the flow of current in a conductor is called current strength. It is interesting that this concept has nothing to do with force in physics. The current strength, the formula of which includes the charge flowing per unit time through the cross section of the conductor, has the form:

• I = q/t, where t is the flow time of charge q.

In fact, current is the amount of charge. Its unit of measurement is Ampere (A), as opposed to N.

Definition of work of force

The force exerted on a substance is accompanied by the performance of work. The work of a force is a physical quantity numerically equal to the product of the force and the displacement passed under its action and the cosine of the angle between the directions of force and displacement.

The required work of force, the formula of which is A = FScosα, includes the magnitude of the force.

The action of a body is accompanied by a change in the speed of the body or deformation, which indicates simultaneous changes in energy. The work done by a force directly depends on the magnitude.