Find the coordinates of the center of the mass of a homogeneous line. How to calculate the center of gravity of a flat circumscribed figure for the help of an underwire integral? The order of the vikonanny of a typical rozrahunka

We will guide the butt of the target to the center of the mass of the body by the method of podіlu yoga on the border of the body, the center of the mass of those in the house.

butt 1. Designate the coordinates of the center of the mass of a homogeneous plate (Fig. 9). Calculate the tasks in millimeters baby 9.

Solution: We show the coordinate axes i . We break the plate into pieces, made with three straight cuts. For the dermal rectus, diagonals are drawn, the points of the crossbar of which indicate the position of the center of the dermal rectus mass. In the adopted coordinate system, it is not easy to calculate the values of the coordinates and points. And to herself:

(-1; 1), (1; 5), (5; 9). The areas of the skin body are moderately improved:

; ; .

The area of all the plates is good:

For assignment of coordinates to the center of the mass of the given plate, it is necessary to virazi (21). We represent the value of all known quantities this man is equal, taken

Vіdpovіdno up to otrimanih the value of the coordinates to the center of the mass of the plate, you can specify a point on the little one. As you can see, the center of the mass (geometric point) of the plate is located behind the boundaries.

Addition method. Tsej sposіb є chastkovy vpadkom way podіlu. Vіn can zastosovuvatisya to tіl, yakі mаyut virіzi (empty). Moreover, without a vir_zanoї part of the position of the center of the mass of the body is visible. Let's look at, for example, zastosuvannya such a method.

butt 2. Designate the position of the center of the mass of the vaga of a round plate with a radius R, de є virіz with a radius r (Fig. 10). Come on.

Solution: Like Bachimo, from Fig.10 the center of the mass of the plate lies on the axis of symmetry of the plate, that is, on a straight line, the shards are straight, all the symmetry. In this way, to assign the position to the center of the mass of the plate, it is necessary to assign only one coordinate, but the other coordinates will be drawn on the axis of symmetry and equal to zero. Let's show the coordinate axes . It is accepted that the plate is folded into two bodies - from a new stake (without viriz) that body, like nibi vikonane with viriz. In the adopted coordinate system, the coordinates for the designation of bodies are: .Areas of bodies are: ; . The total area of the whole body is more equal to the difference between the areas of the first and the other body, and

Calculate the values m, and it is necessary to match formulas (4), (5) and (7). As a result, we take formulas for coordinates to the center of the mass of the thin plate :

Butt 4 (calculation of coordinates to the center of the mass of a uniform dress)

Find the coordinates of the center of the mass of a homogeneous figure, surrounded by lines and .

Having inspired the figure, we note that it is geometrically out and symmetrical like a straight line. right-handed. Then, behind the given physical powers, we lay down the center of the mass that the wines are located on the axis of symmetry, so that

To calculate, add up the static moment and win the formulas (4) and (5):

;

Suggestion: C .

Additives of the third integrals

Programs for additional integrations are similar to supplements of sub-integrals, but only for trivi- mers.

If you want to win one of the powers of the triple integral (about the same value of the function, which is also the same value of one), then go formula for the calculation of the obligation to be spacious body :

Let's write down the formula for obyagu through third integral and calculable loss integral in cylindrical coordinates:

Vidpovid: (alone oblige).

The formula for calculating the mass of a trivi- mer object that borrows volume V, may look:

(13)

Here is the volume of schіlnіst rozpodіlu masi.

Butt 6

Know the mass of cool radius R how space is proportional to the cube in the center and on the single wall k.

V: elementary volume ta .

It is worth noting that when calculating the three-time integral, there were no more integrals, the chips of the internal integrals appeared to be fallows in the case of the change of the outer integrals.

Vidpovid: (single masi).

Mechanical characteristics for binding V(Static moments, moments of inertia, coordinates to the center of mass) are calculated according to formulas, like

folded by analogy with formulas for two-world bodies.

Elementary static moments and moments of inertia along the coordinate axes:

elementary moments of inertia along the coordinate planes and points on the cob of coordinates:

Dali, to calculate the mechanical characteristics of the whole obyagu V,It is necessary to sum up the elementary additions of the characteristics for all parts of the breakdown (the characteristics of the greatest power of additivity are counted), and then go to the boundary in the sum, which was beyond the mind, that all the elementary parts of the breakdown will change (contract into points). Quantities are described as an integration of an elementary addendum of mechanical characteristics, which are calculated, for obligatory V.

As a result, come formulas for calculating the static moments М and the moments of inertia I trivi- mer tіl :

Truly, they formulated the formulas like victorious as they are ready, and lead them into virishuvaniy tasks.

Apply 7 (calculation of the mechanical characteristics of three-dimensional bodies)

Find the moment of inertia of a uniform cylinder, the height of which h and base radius R, how to axis, which zbіgaєtsya with the diameter of the base.

We know d for a partial point of the cylinder:

move to a point with coordinates to the axis – the length of the perpendicular drawn from the center of the point to the axis . Let's make the plane perpendicular to the axis so that the point lies on this plane. Then be straight, which crosses everything and lies on this plane, it will be perpendicular . Zokrema, straight line, which connects a point and a point, will be perpendicular to the axis, and if you stand between these points, you will be a shukana d. Calculate yoga for the given formula between two points.

3 Additions of underlying integrals

3.1 Theoretical introduction

Let's take a look at the programs underwire integral to the top of low geometric tasks and tasks of mechanics.

3.1.1 Flat plate area calculation

Let's look at a thin material plate D, expanded in the flat Ohu. area S tsієї plates can be found for the help of the undercurrent integral for the formula:

3.1.2 Static moments. Flat plate mass center

static moment M x shodo axis Ox material points P(x;y) that lie near the flat Oxy and maє masu m, It is called dobutok massi points on the її ordinate, tobto. M x = my. Similarly, the static moment M y shodo axis Ouch: M y = mx. Static moments flat plates with a surface slot γ = γ (x, y) are calculated using the formulas:

As seen from mechanics, coordinates x c ,y c the centers of mass of a flat material system are defined by equalities:

de m- Masa system, and M xі M y- Static moments of the system. Flat plate weight m determined by formula (1), the static moments of a flat plate can be calculated using formulas (3) and (4). Todi, zgіdno z formulas (5), it is taken viraz for coordinates to the center of the mass of the flat plate:

Typical rozrahunok avenge two tasks. The skin doctor is given a flat plate D, surrounded by lines, shown for the mind of the task. G(x,y) - surface clearance of the plate D. To know the number of plates: 1. S- Square; 2. m- Masu; 3. M y , M x- Static moments for axes Oyі Oh obviously; 4. , - Coordinates of the center of mass.

3.3 The order of vykonannya typical rozrahunku

When performing a skin task, it is necessary to: 1. Remove the chair from a given area. Choose a coordinate system, for which the sub-integrals will be calculated. 2. Record the area of the visual system of irregularities in the chosen coordinate system. 3. Calculate the area S ta masu m plates following formulas (1) and (2). 4. Calculate the static moments M y , M x formulas (3) and (4). 5. Calculate the coordinates of the center mass using formulas (6). Apply the center of the mass on the armchair. We blame the visual (yakish) control of taking away the results. Numerical numbers may be taken away from the trio of numbers.

3.4 Apply a typical robe

Task 1. plate D surrounded by lines: y = 4 – x 2 ; X = 0; y = 0 (x ≥ 0; y≥ 0) Surface thickness γ 0 = 3. Solution. The area specified in the problem is surrounded by a parabola y = 4 – x 2 , coordinate axes i lie at the first quarter (Fig. 1). The task is modified in the Cartesian coordinate system. This area can be described by a system of irregularities:

Rice. one

area S plates are more stable (1): Since the plate is uniform, m = γ 0 S= 3 = 16. Behind formulas (3), (4) we know the static moments of the plate: The coordinates of the center mas are given by formula (6): Suggestion: S ≈ 5,33; m = 16; M x = 25,6; M y = 12; = 0,75; = 1,6.

Task 2. plate D surrounded by lines: X 2 + at 2 = 4; X = 0, at = X (X ≥ 0, at≥ 0). Surface thickness γ (x,y) = at. Solution. The plate is surrounded by a stake and straight lines that pass through the cob of coordinates (Fig. 2). Therefore, for the completion of the task, it is necessary to manually transcribe the polar coordinate system. polar kut φ change from π/4 to π/2. Promin, passing from the pole through the plate, “enter” before it at ρ = 0 and “enter” the stake, equal to: X 2 + at 2 = 4 <=>p = 2.

Rice. 2

Again, a given area can be written with a system of irregularities: The area of the plate is known from the formula (1): The mass of the plate is known by formula (2), substituting γ (x,y) = y = ρ sin φ :
For the calculation of the static moments of the plate, we can use formulas (3) and (4):
The coordinates of the center of mass are taken from the formulas (6): Suggestion: S ≈ 1,57; m ≈ 1,886; M x = 2,57; M y = 1; = 0,53; = 1,36.

3.5 Designing the sound

The stars may have a representation of all the vikonan rozrahunka, neatly vikonan armchairs. Numerical numbers may be taken away from the trio of numbers.

– calculation for the center of gravity is flat fringed figure . Rich reader intuitively understand what is the center of gravity, I recommend repeating the material of one of the lessons analytical geometry de i solved zavdannya about the center of gravity of trikutnik and in an accessible form deciphering the physical term.

At independent and control tasks, for perfection, as a rule, the simplest vipadok is propagated - the flat is surrounded homogeneous a figure, in order to post a postiynoї physical strength - glass, derev'yana, pewter'yana chavunnі іgry, hard childishness is thin. Dali for umovchannyam mova pіde tіlki about such figures =)

The first rule is the simplest butt: even though the figure is flat symmetry center, then vin є the center of gravity tsієї figure. For example, the center of a round uniform plate. It is logical and life-conscious - the mass of such a figure is “fairly distributed on all sides” like the center. Believe - I don't want to.

However, in reality, you are unlikely to give licorice eliptic chocolate bar Tom will have to deal with a serious kitchen tool:

The coordinates of the center of gravity of a uniform flat circumscribed figure are covered by the advancing formulas:

, or:

, de - area of the region (figures); but let's briefly:

, de

Integral is mentally called “ixovim” integral, and integral is “igrom” integral.

Acceptance-finishing : for flat furrowed heterogeneous figures, the width of which is given by the function, folding formulas:
, de - Masa figures;in times of uniform strength, the stink will be forgiven to the introduction of more formulas.

On formulas, vlasne, all novelty and ends, reshta - all your vminnya virishuvati subvincial integrals, to the point of speech, at once one hopes for a miraculous ability to work out and perfect one’s technique. And the thoroughness, as it seems, there is no difference =)

Throwing a big portion of parabolas:

butt 1

Find the coordinates of the center of the vaga of a uniform flat figure, surrounded by lines.

Solution: lines here are elementary: set the entire abscissa, and equal - a parabola, so that it will be easy for you to get help geometric transformation of graphics:

– parabola, Pushed 2 units to the left and 1 unit down.

I am stitching the whole armchair with a ready point to the center of the vagary of the figure:

Rule a friend: what does the figure have all symmetry, then the center of gravity of this figure is bound to lie on its axis.

Our figure is symmetrical to shodo straight so we actually already know the “ix” coordinate of the point “em”.

It is also important to respect that the center of gravity of the displacements is closer to the abscissa axis along the vertical, the oscillators there are a massive figure.

So, perhaps, not everyone has yet figured out what the center of the vaga is like: be kind, raise your finger uphill and put a dot on the new shaded “foot”. Theoretically, the figure is not guilty of falling.

The coordinates of the center of gravity of the figure are known by the formulas de .

The order of bypassing the area (figures) is obvious here:

Respect! It is determined by the most viable order of bypass once- I vikoristovuemo yoga for all integrated!

1) On the back, I calculate the area of the figures. Through the obvious simplicity of the integral, the solution can be arranged compactly, smut, not to get lost in the calculations:

We marvel at the armchair and pretend to be on the square square. Viyshlo bіlya do it.

2) The x-coordinate of the center of gravity has already been found by the “graphic method”, so you can refer to the symmetry and go to the next point. However, it’s still not a rajah to work like that - it’s great to think that it’s a good idea to reject the formula “win the formula”.

Respect that here it is possible to deal with wine-coloured calculations - sometimes it’s not obligatory to bring fractions to a double standard and torment a calculator.

In this manner:
, What and need to take.

3) We know the ordinate of the center of gravity. Let's calculate the Greek integral:

And the axis here without a calculator would have been hard. I will comment about every change that as a result of the multiplicity of richly divided members there are 9 members, moreover, the deacons are similar to them. Similar dodanki I vaccinated orally (how to sound like a robiti at such vipadkas) and immediately writing down the sum of the bag.

As a result:
which is more and more similar to the truth.

At the final stage, it is marked on the armchair speck. For the mind, it was not necessary to make an armchair, but in greater numbers, I want to depict a figure, even if I don’t want to. Natomist is an insane plus - visual and effective re-verification of the result.

Vidpovid:

Come two butts of an independent solution.

butt 2

Find the coordinates of the center of the vaga of a uniform flat figure, surrounded by lines

Before the speech, as you can see, like a parabola is ruffled and the dots are turned, in which everything is overturned, then here you can really do without an armchair.

І folding:

butt 3

Find the center of the vaga of a uniform flat figure, surrounded by lines

In times of difficulty from the post-budget schedule, vivchit (repeat) parabolic lesson and/or butt No. 11 statti Suspended integrals for teapots.

Srazkovі zrazki solution like a lesson.

In addition, a dozen or so similar applications can be found in the archives on the side Ready solutions for your mathematics.

Well, I can't help but please the lovers advanced mathematics, How often to ask me to sort out and important tasks:

butt 4

Find the center of the vaga of a uniform flat figure, surrounded by lines. The figure of that її center of gravity is depicted on the armchair.

Solution: umova tsієї zadachi vzhe categorically vmagaє vykonannya armchair Aje vimoga not nastіlki і formally! - Tsyu figure zdatna to reveal in the mind a person from the middle level of training:

Straight roz_kaє kolo on 2 parts, and additional guard (Div. linear irregularities) I point out to those that I can go myself about a small shading shmatochok.

The figure is symmetrical and visually straight (depicted by a dotted line), the center of gravity is to blame for lying on this line. І it is obvious that yogo coordinates are equal behind the module. A guideline that practically turns on the pardon!

Now filthy novelty =) On the horizon, a low-receipt integral from the root looms, which we reportedly took from Applied No. 4 to the lesson Efficient methods for solving integrations. And who knows what else is painted there. It would be given, through the presence cola Obviously, not everything is so simple. Rivnyannya straight line transforms at a glance and the integration may not be true (if the fanatics want trigonometric integrals evaluate). At zvyazku z tsim usually zupinitsya on Cartesian coordinates.

The order of bypassing the figure:

1) Calculate the area of the figure:

First integral of rational take p_dvedennyam p_d sign of the differential:

And in another integral, we will carry out a standard replacement:

Let's count the new inter-integration:

2) We know.

Here at the 2nd integral there is a new twist method. Vіdpratsyuyte ta vіzmіt on ozbroєnnya qі optimal (on my mind) accept the development of typical integrals.

After the difficult and trivial, calculate again the beastly look at the armchair (remember that the points We still don't know! ) and otrimuemo to a certain degree of moral satisfaction in view of the value found.

3) Vykhodyachi z carried out earlier analysis, lost reconciliation, scho.

Note:

Representable point on the chair. Vіdpovіdno to formulary mind, we will write down її as residual proof:

Similar task for an independent vision:

butt 5

Find the center of the vaga of a uniform flat figure, surrounded by lines. Vikonati armchair.

We are interested in the fact that in it the figure is given to do small remembrances, and if there is some time for pardon, then the high level of fire will be “not spent” in the region. What, bezperechno, good from the point of view of the control solution.

Srazkovy zrazok designed like a lesson.

Іnodi buvaє dotsіlnim transition to polar coordinates at lower integrals. See figure. Shukav-shukav at home far butt If you don't know, I'll show you the solution for the 1st demo task of the assigned lesson:

Guess what in that butt we went to polar coordinates, explained the procedure for bypassing the area and virahuvali її area

Let's know the center of gravity of the figures. The scheme is the same: . The value is visible directly from the armchair, and the “ix” coordinate can be shifted a little closer to the y-axis, the shards there are scattered by a massive part of the beer.

In integrals, we can use the standard formulas for the transition:

Ymovirno, better for everything, they didn’t have mercy.