Maxwell's pendulum force thread tension. The law of conservation of mechanical energy for the Maxwell pendulum
1. Purpose of work: assigned to the moment of inertia of Maxwell's pendulum. The designation of the force was the tension of the threads under the hour of the turn and at the moment of "rivka" (lower point of the trajectory).
2. Theoretical ambush robots.
Maxwell's pendulum with a single disk mounted on a cylindrical shaft (Fig. 1); center mass of the disk and the shaft to lie on the axis of the wrap. Threads are wound on the shaft with a radius of r, which are fastened to the brackets. When unwinding the threads, Maxwell's pendulum creates a flat swing. Such a movement is called flat, when all the points of the body move near the parallel planes. The flat swing of the pendulum is possible as a sum of two swings - progressive swing to the center of the mass uzdovzh axis OY, zі shvidkіstyu V that overt rhu with kutovoy swidkistyu w shodo axis O’ Z, to pass through the center mas of the pendulum.
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å M iZ = mac
Here index W means the center of mass of the system.
The main equalization of the dynamics of the wraparound swing for the Maxwell pendulum on the middle axis O" Z, to pass through the center of the mass
å M iZ = J Z E Z
Here J Z- moment of inertia of the pendulum O" Z.
EZ- projection of the apex acceleration on the whole O"Z; the left part is equal - the algebraic sum of the moments of the external forces along the axis O"Z.
If the thread does not slip, then the tightness to the center of the mass of the pendulum is the top tightness w pov'yazanі kіnematichnіm spіvvіdshennyam
Vc = w r
a) Assigned to the moment of inertia of Maxwell's pendulum.
Vikoristovuyuchi the law of conservation of mechanical energy, you can experimentally designate the moment of inertia of the pendulum. For whom the hour is being fought t lowering the pendulum m from above h.
We accept the potential energy of Maxwell's pendulum Wp.s. = 0 in position, if the pendulum is in bottom point. Kinetic energy in which camp
W before . n . = mV 2 /2 + J w 2 /2 (1)
Here V- Shvidk_st to the center of the mass of the pendulum; w- kutova swidkіst;
J- the moment of inertia of the pendulum about the axis to pass through the center of mass: m = mV + md + ml- Mass of the pendulum; mV, md,ml- mass the shaft, the disk and the ring, which enters the pendulum warehouse. The upper position of the pendulum has a potential energy
W P . V . = mgh ,
and kinetic energy is equal to zero. From the law of conservation of mechanical energy for Maxwell's pendulum (by dissipative forces, rubbing it with forces, the support is broken again and again) slid.
mgh = mV 2 /2 + J w 2 /2 (2)
Oscilki the center of the mass of the pendulum is collapsing in a straight line and evenly accelerated, then
h = a t 2 /2; V = a t (3)
Z (3) is required V = 2 h / g (4)
Substituting the spin of the pendulum (4) in (2) and vicorist spin of the pendulum between the center of the mass and the top swidk of the pendulum about the axis of symmetry, we take away the formula for the calculation of the experimental moment of inertia of the Maxwell pendulum
J e = mr 2 (g t 2 /2h-1) (5)
Here r is the radius of the shaft
Subtracting the result is equal to the value of the moment of inertia, which depends on the theoretical measurement. The theoretical moment of inertia of Maxwell's pendulum can be calculated using the Formula
J T = J B + J D + J K (6)
Here J B, J D, J K- moments of inertia of the storage parts of the pendulum: the shaft, the disk, and the ring. Vikoristovuyuchi zagalnu formula for assigning the moment of inertia
J= ∫ r 2 dm (7)
we know the moments of inertia of the elements of Maxwell's pendulum.
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Moment of inertia of the disk
Here R1- the radius of the disk, the inner diameter of the ring (Fig. 1). Moment of inertia of the ring
J K \u003d m K * (R 1 2 + R 2 2) / 2 (10)
Here R2- Zovnishhnіy diameter of the ring
b) Determining the strength of the tension of the threads for the hour of the swing of the Maxwell pendulum T D i at the moment of the "rivka" - T R.
The swing of Maxwell's pendulum is described by the equal system
-ma = 2T - mg (11); J E = 2Tr (12); h = a t 2 /2 (13)
Z (11) and (12) show that with the Russian pendulum of Maxwell, the tension force of the thread is more
T D = mg / 2 (mr 2 / J + 1) (14)
de the moment of inertia of the pendulum J is assigned to sp_v_dnosheniyam (5).
Laboratory robot №1*
Maxwell pendulum
Meta roboti: Calculate the moment of inertia of the Maxwell pendulum with the dynamic building data and equalize it with the theoretical values
Attach and materials: Maxwell's pendulum, electronic stopwatch, changing rings.
Laboratory attachment
Maxwell's pendulum with a small disk (handwheel) is firmly planted on the whole. Under the influence of force, the weight of the vines is lowered on two threads, the back of the leg is wound on the entire handwheel (Fig. 1). The thread under the turn of the disk is unwound down to the top, the handwheel, which has untwisted, continues the wrapping move at the same time straight and winds the threads on the whole, after which the wine rises uphill, improving with its wrapping. Move to the upper point, the disk will again go down and so on. The flywheel oscillates down and uphill, to which such an attachment is called a pendulum.
Laboratory installation
At the laboratory installation, Maxwell's pendulum is reinforced on brackets, which allow you to adjust the length of the suspension and its parallelism. Photoelectric sensors are attached to the upper and lower brackets, functionally tied to an electronic stopwatch, which controls the swing of the pendulum. Changes are superimposed on the mahovichi, which change the moment of inertia of the pendulum. On the top bracket
an electromagnet that fixes the position of the handwheel with the ring when the "START" key is pressed.
Theoretical description of the robot and the visualization of the working formula
The pendulum at the process of coliving zdіysnyuє forward and obertal ruhi, as described in vіdpovіdnim rivnyannami. To fold the arm, let's look at the forces and moments of forces that are on the handwheel (Fig. I). Come on 
- force of gravity, 
- Tension force of one thread. 
- The radius of the axis of the pendulum. 
10 mm - diameter of the axis of the pendulum, 
- Mass of the pendulum. 
- Moment of inertia of the handwheel. To the same extent as the progressive movement, according to another Newton's law, it can be written as follows:
.
(1)
Rivnian (1) cost a sub-strength value 
, because two threads are wound on the whole handwheel, the tension force is caused by the skin 
.
Under the force of the tension, the disk is zdijsnyuє obertalny ruh. The moment of these forces is rising:
.
(2)
Shoulder of strength 
є radius 
axis of the pendulum, the diameter of the thread is negligible.
The ratio of the overwrap handwheel can be written as follows:
,
(3)
de 
- Kutove priskornnya disk wrapper.
Kutove priskorennya 
ta priskrennya to the center of mass 
pov'yazanі spіvvіdnoshennyam:
.
(4)
priskorennya 
, the center of the mass can be known by knowing the length of the track and the hour of the handwheel from the upper to the lower point (with the correction of zero cob roughness):
.
(5)
.
(6)
Substituting (6) to (4), we take:
.
(7)
Z urahuvannyam (6) that (7) equal (1) that (3) will look:
.
(8)
.
(9)
Virishingly equal (8) and (9), we can work out the formula for assigning the moment of inertia of the Maxwell pendulum by an experimental way:
.
(10)
Formula (10) has mass 
є the main mass of the pendulum, which includes the mass of the axis of the pendulum, the disk and the ring. 
-?
-?
-?
-?
-?
The order of vikonannya roboti
1. Remove the installation in the merezha.
2. Put enough vibra- tions on the handwheel, pressing it all the way.
3. On the whole of the pendulum, wind the thread of the pidviska, giving respect to those. so that the won was wound evenly, coil to coil.
4. Fix the pendulum at the upper bracket by pressing the START button of the stopwatch.
5. Press the stopwatch drop button.
6. Press the "START" key, with which the electronic stopwatch will rise to the pendulum's swing to the bottom bracket. Vimiryuvannya repeat 5 times and enter in the second column of the table.
7. Behind the scale on the vertical column, mark the doujin 
pendulum.
8. Vimiryuvannya hour (point 6) repeat for different planted kіlets and bring it to the table.
9. Designate the weight of the pendulum. The meaning of the weight of the okremih elements on them.
10. For the formula (10) calculate the moment of inertia 
- pendulum for all
series vimiriv.
11. Calculate the obvious and absolute errors of the moment
inertia for taking away independently formulas. Differential formula may look
12. Calculate the theoretical values of the moment in the inertia of the pendulum using the formulas (11) and equalize with the calculated ones using the formulas (10):
,
(11)
de 
- Moment of inertia of the axis of the pendulum.
- Mass of the axis of the pendulum, 
= 10 mm - axis diameter
- Moment of inertia of the disk.
- Mass of the disk, 
86 mm - the actual disc diameter
- Moment of inertia of the mounted ring.
- weight of the ring, 
105 mm - the outer diameter of the ring.
13. Residual results of the determination of the moment in the inertia of the pendulum of tribute for such a look:
,
.
14. For otrimanimi results zrobiti vysnovki.
Table of results
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1. Give the moment of inertia of a material point of that solid body.
2. How is the main equalization of the dynamics of the overt rhu recorded?
3. What kind of physical attachment is called Maxwell's pendulum? Name the main yoga elements and explain the principle of yoga work.
4. Enter a working formula for assigning the moment of inertia of Maxwell's pendulum.
5. Explain formula (11) for the theoretical values of the moments of inertia of the pendulum.
6. Enter the formula for the visual and absolute deviations of the specified moments of inertia.
Meta Robot.
On the butt of Maxwell's pendulum, one can learn from the calculation and experimentally the moment of inertia of a cylindrical solid body along the axis of symmetry.
Settlement.
Maxwell's pendulum.
Themes of the show.
IN laboratory work on the butt of Maxwell's pendulum, the laws of translational and wraparound momentum were examined, a working formula for the calculation of the moment of inertia of Maxwell's pendulum was taken away, a description of the experimental setup was made to control the moment of inertia of the pendulum.
The laboratory work is recognized for students, who are concentrating on a high-level physical workshop at the laboratory of mechanics.
Short theory.
M
Maxwell's pendulum is a massive disk, all suspended on two threads wound on it (Fig. 1).
As soon as the pendulum is released, then it causes a reversible forward movement near the vertical plane with a one-hour wrapping of the disk around the axis.
The forces that make the pendulum are shown in fig. 2.
To describe the swing of Maxwell's pendulum, manually select the system in front of it, tie it to the center of the mass of the pendulum and let one whole, straight down.
The center of the mass of the system is called the apparent point, the radius vector, which is defined by the size
de T - the mass of the system, - the mass of material points, how to establish the whole system, - їх vector radii. Value shvidkіst ruhu tsієї vyavnoї point. Impulse of the system with urahuvannyam (I) is recorded as
Tobto є tvir masi system on swidkіst її mas center, which is similar to the momentum of the material point. In this rank, behind the movement of the center of the mass, you can stitch as if behind the movement of the material point. Vykhodyachi s tsgogo, ruh to the center of mass of Maxwell's pendulum can be described equal:
de m - Mass of the pendulum, - linear acceleration to the center of the mass - the resulting tension force of both threads.
The overt swing of the pendulum is described by the main dynamics of the overt swing that can be seen:
de ℐ - moment of inertia, - the resultant hour of forces, which blow on the pendulum at any singing point, which lie and the axis of wrapping, - kutove hastened. Under the kuta vector, the vector is recognizable, modulo equal to the kutu rotation and straightening of the axis of the wrapping so that the rotation of the posterigavsya follows the year's arrow on the cob.
The moment of inertia of the body is how the actual axis of wrapping is called the value
, (4) (4)
de - masi material points that make the body whole, - remove from these points to the axis of wrapping. Later, the moment of inertia characterizes the distribution of the mass of the body along the axis of the wrap. From (4) it can be seen that the hour of inertia is an additive quantity, so the moment of inertia of the body is equal to the sum of the moments of inertia of the yogo elements. Yakscho speech in it is divided without interruption, then the calculation of the moment of inertia is reduced to the calculation of the integral
; (5) (5)
de r - stand in the elementary mass dm.
to the axis wrap. The integration is carried out according to all weights of the body. Maxwell's pendulum can be seen from the combination of empty cylinders and the succulent cylinder - the axis of the pendulum. Rozrahuєmo, scho moments of inertia of such bodies. Whether it be z tsikh tіl can be broken into thin cylindrical balls, the particles of which are located on the same line of sight. Rosіb'ёmo cylinder radius R on the concentric balls of the shirt dr . Let the radius of any ball r, todі masa particles, laying at tsimu shаrі, dorіvnyuє
de dV - volume of the ball, h- Cylinder height, - Shchilnist speech cylinder. Mustache particles of the ball are on the surface r vіd axis, otzhe, moment of inertia of the ball
The moment of inertia of the entire cylinder can be found by integrating over all balls:
Cylinder pins , then the moment of inertia of the contractile cylinder is more
Moment of inertia of an empty cylinder, what is the maximum inner radius , and the current one can also be calculated using the formula (6), changing in the integral between the integration
Note that the mass of an empty cylinder
, let's write down the moment of inertia of the empty cylinder in the following order:
(8) - ( 8)
Prote, the analytical calculation of integrals (5) is less possible for the simplest types of bodies of regular geometric shape. For solids of irregular shape, such integrals are known numerically or indirectly by methods of determining the moment of inertia.
To know the moment of inertia of Maxwell's pendulum, it is possible to speed up the movement
For the variance of differential equations (2) and (3) we pass from the vector form to the scalar one. The design is equal (2) to move all the way to the center of the mass of the pendulum. I’ll look ahead to see:
Let's look at the projections of vectors and on all coordinates, which zbіgaєtsya z vіssyu wrapping and straightening.
Skladova moment of force at the point vzdovzh osі, scho to pass through the qiu point, is called the moment of force schodo
The vector can be written in this way;
de - single vector, straightening vzdovzh , A 5. Todі kutove prikorennya
shards straight vector ^ when the pendulum is lowered, it changes from time to time.
In this rank, equal (Z) is projected for the whole wrapping of the coming rank:
(10) (10)
de - the radius of the axis of the disk, a thread is wound around the yak - the apex of the disk. The skids of the center of the mass are lowered by the flooring, the thread is spinning on the skids, it is moving x tied with a kut, turning spіvvіdshennyam
Differentiation tse spіvvіdnoshnja dvіchі, otrimaemo
Spіlne solution rivnyan (9) - (11) gives the same virazi for linear acceleration to the center of the mass system and the resulting tension force:
(13)
From (12), (13) it can be seen that the speed of the disk is that the force of the tension of the thread is constant and the speed is straight down. Also, as when the pendulum is lowered, the coordinate of the th center of the mass is lifted from the point of the th anchoring point, then the coordinate changes according to the law
Substituting (14) in (12), it is important for the moment of inertia of Maxwell's pendulum to move forward
, de (15)
New include quantities that can be easily measured experimentally: - the outer diameter of the axis of the pendulum at once with a thread wound on a new thread, t - pendulum lowering hour, x - vіdstan, passed by the center of the mass of the pendulum, m. - the mass of the pendulum, which is composed of the mass of the axis of the pendulum, the mass of the disk and the mass of the ring, dressed on the disk. The outer diameter of the axis of the pendulum at once from a thread wound on a new thread
follow the formula
de D - pendulum axis diameter, - thread diameter.
Mechanical design of the accessory.
The glaring view of Maxwell's pendulum is shown in fig. 3. Base I is equipped with adjustable feet 2, which allow the adjustment of the fittings. Column 3 is fixed in the stand, until the fastening of the non-violent upper bracket 4 and the lower lower bracket 5. On the upper bracket there is an electromagnet 6, a photoelectric sensor 7 and a compir 8 for fixing and adjusting the thread of the swing of the pendulum. The lower bracket at once attached to the new photoelectric sensor 9 can move the column and fix it in the chosen position.
The pendulum 10 is the disk fixed on the axis, on which the rings 11 are pressed, thus changing the moment of inertia of the system.
The pendulum with the upper ring is pressed down at the upper position by an electromagnet. The length of the thread of the pendulum is marked by a millimeter scale on the column of the fixture. Photoelectric sensors connected with milliseconds. View of the front panel of the stopwatch 12 representations in fig. 4.
On the front panel of the milisecond watch there are step handles
"MEREZHA" - vimikach merezha. The onslaught of these keys turns on the voltage of life. When this occurs, zeros appear on the digital indicators and the light bulbs of the photoelectric sensors turn on.
"REDUCTION" - setting the stopwatch to zero. The onslaught of the keys clicks off the electronic circuits of the millisecond watch, zeros appear on the digital indicators.
"PIT" - electromagnet control. When pressing the key, the electromagnet vibrates, the circuit of the millisecond watch generates a pulse for an hour.
Vikonanny roboti.
The lower bracket of the attachment is slipped and fixed at the extreme lower position.
On the disk of the pendulum, put on one ring, pressing it all the way.
Screw in the comor nut to adjust the length of the suspension thread. Pick up the length of the thread in such a way that the edge of the steel ring after the lowering of the pendulum is two millimeters lower than the optical axis of the lower photoelectric sensor. At once, work on adjusting the installation of the pendulum, savage respect for those, so that all of it was a parallel basis for the fixture. Squeeze the comedian.
Press the "MEREZHA" key.
Wind the thread of the pendulum on the entire pendulum, with bestial respect for those who won it wound evenly, turn to turn.
Fix the pendulum behind the help of the electromagnet, with bestial respect for the clock, so that the thread in this position is not twisted too much.
Turn the pendulum at the straight line of the possible wrapping on the tip of about 5 °.
Press the button "REDUCTION".
Repeat vimir ten times to mark the middle hour of the fall of the pendulum.
Behind the scale on the vertical column, I will mark the length of the thread of the pendulum.
Vymiryavshi thread diameter and axis of the pendulum D at different cuts, to know the average values of these values and after them, assign the formula (16) the diameter of the axle with a thread wound on it. For vimiryuvannya Dі you can vikoristovuvati micrometer.
Appreciate the weight of the pendulum at once from over the ring. The value of the weight of the other elements applied by them.
Behind the formula (15) assign the moment of inertia of Maxwell's pendulum. Calculate the moment of inertia of the pendulum theoretically, using formulas (7), (8), and subtract the result with the value allocated for formula (15).
Repeat vimiryuvannya for two kіlets, which are lost.
Dovirchy interval △ ℐ you can expand on the formula
de ΔD, , △ t, △ x - dovirchі intervals for direct vitiruvan values D, , t і x, vrakhovuyut like vipadkovі, i systematic pokibki. Methods for the analysis of these values instructed by the helper of L.P.
Safety technology.
When working with an attachment, it is necessary to comply with the safety rules, which include attachments, which have a voltage of up to 250 volts. The operation of the accessory is only allowed for obvious grounding.
Control food.
Formulate a theorem about the center of mass of a system of material points.
Give the moment of inertia of one material point of the system of material points.
Write down the swing of Maxwell's pendulum.
How do the speed, speed and strength of the tension of the threads change under the hour of the swing of the pendulum?
How does the mechanical energy of Maxwell's pendulum change under the hour?
Federal State Autonomous Lighting Mortgage
higher professional education
Far-Away Federal University
School of Natural Sciences
MAXWELL PENDULUM
Navchalno-methodical help
to laboratory work No. 1.10
Metoyu robotiє the development of the laws of the dynamics of the overturned momentum of a solid body, the familiarization with the Maxwell pendulum and the method of vibrating at a new moment of inertia of the wheel of the Maxwell pendulum how to pass through the center of the mass mass, as well as the experimental significance of accelerating the progressive movement to the center of the mass of the wheel of the Maxwell pendulum.
1. Basic concepts of the overt ruhu of a hard body .
Under a solid body, the mechanics understand the model absolutely solid body - Tila, the deformations of which in the minds of this leader can be resented. Such a body is possible as a system of hard-fixed material points. Whether it be a folding movement of a solid body, it can be divided into two main types of movement - translational and wrapping.
Progressive Rukh is called a ruh of a solid body, in some way it is straight, drawn through two points of the body, parallel to itself for the whole hour (Fig. 1). With such a Russian, all points of a solid body collapse absolutely the same way, so that the very swidkity, quickening, trajectory of the rush, change the same movement and go through the same path. Therefore, the forward movement of a solid body can be like the movement of a material point. Such a point can be buti, zokrema, the center of mass (center of inertia) of the body. Under the center of mass the body is conscious of the point of the resultant mass forces that are on the body. Masovі forces - tse forces, proportional to the masses of the elements of the body, where do the forces develop, for the wisdom of the forces that all the elements of the body develop, parallel to one.
Shards in translational Russia, all elementary masses m i of a solid body crumble with the same speeds and speeds, then another Newton's law is valid for the skin of them:
de - the sum of all internal forces, which will operate on the elementary mass Δm i (there will be i-1 of such forces, the shards cannot be part of the activity on themselves), and the sum of all the external forces, which will operate on the elementary mass Δm i from the side of other bodies . Having summed up the equality (1) over the whole body and vrakhovuchi, that the sum of all internal forces zgіdno with Newton's third law is equal to zero, we take away the law of dynamics of the progressive motion of a solid body:
de - scho is the result of all ovnishnіh forces, scho blow on the body as a whole, - impulse (kіlkіst ruhu) of the body. Otrimane river (3) progressive ruhu a solid body rises from equal dynamics of the material point.
overt Rukh is called the arm of a solid body, with which all the points of the body describe a stake, the centers of which lie on the same straight line, which is called the whole wrapping of the body. In obertal Russia, all points of the body collapse with one and the same vertex swidkistyu and vertex accelerations and the same vertex moves. However, as a proof, in the case of an overt Russian of a solid body, the axis of the mass is not so much fixed, but the strength is not sufficient to characterize the external inflow. So, for the sake of proof, it’s obvious that it’s quicker to lie down in obertal Russia like a body of body, and її rozpodіlu shdo osі wrapping; to deposit not only in strength, but in the form of a point її zastosuvannya and direct diї. To this end, new characteristics were introduced for the description of the overwrap of a solid body, such as moment of force, moment of impulse and moment of inertia of the body . If so, follow the mother on the uvazi, there are two different understandings of these values: schodo osі and vіdno be-like points (poles, cob), taken on the tsіy axis.
Moment of force some unbreakable point Pro The vector value is called, which corresponds to the vector creation of the radius vector drawn from the point O to the point of reporting the resulting force to the vector of the force vector:
The vector of the moment of force is always perpendicular to the plane, in a certain expansion of the vector i , and it is directly related to the direction of the plane according to the rule of the vector creation or to the rule of the gimlet. Appropriate to the rule of the gimlet: if you wrap the handle of the gimlet behind the direct force, then the translational movement of the gimlet will bend with the direction of the force moment vector (Fig. 2). Vectors, directly tying them up from a straight wrap (kut swidkіst, kutove quickening, moment of force, moment of impulse thin), name pseudovectors
or axial
V on the vіdmіnu vіd vіd zvuchaynykh vektorіv (swidkіst, radius-vector, accelerated thinly), how to name polar
.
Value the vector of the moment of force (the numerical value of the moment of force) depends on the formula of the vector creation (4), that is. , de a -
4
cut between straight lines vector_v that . Rozmir p= r·Sinα is called the shoulder of strength (Fig. 2). shoulder strength p - the shortest distance from the point O to the line of force.
Moment of force , called projection on the entire vector of the moment of force, found at any point where to lie on this axis. It was understood that the axis of the moment of force is a scalar quantity.
In system SI, the moment of force is reduced by Nm.
To introduce the understanding of the moment of the momentum of the body, we introduce the beginning of the understanding for the material point, what to put on a solid body, what to wrap.
Moment of impulse material point Δ m i schodo non-destructive point O is called the vector addition of the radius-vector drawn from the point O to the point Δm i on the momentum vector of the material point:
de - the momentum of the material point.
The angular momentum of a rigid body (or a mechanical system) like a non-violent point is called a vector , equal geometrical sum of moments in the momentum of a given point About all material points of a given body, tobto. .
The moment of the momentum of the solid body called the projection on the qiu of the entire vector to the momentum of the body's momentum at any point, chosen on the axis. Obviously, at times the moment of the impulse is a scalar quantity. In the system СІ the moment of the impulse is reduced to
To the world of inertness tіl for progressive movement є їх mass. The inertness of the body in the case of obertal Russia is not only deposited in the mass of the body, but also in the form of її rozpodіlu in the expanse of the body wrapping. The world inertness of the body in case of wrapping Russia is the moment of inertia of the body I, but the axis of wrapping or the point. The moment of inertia, like mass, is a scalar value.
The moment of inertia of the body is good for the axis of wrapping the physical value is called equal to the sum of the masses of the material points, on which you can break the whole body, on the squares of the skin from them to the axis of the wrap:
de - Moment of inertia of a material point.
The moment of inertia of the body is like a point that lies on the axis, a scalar quantity is called, which is equal to the sum of the creations of the mass of the skin material point of a given body per square її distance to the point O. Rozrakhunkov's formula for the moment of inertia is similar to formula (6).
In system СІ the moment of inertia is reduced kgm 2 .
2. The basic law of the dynamics of the wraparound motion of a solid body .
We know the connection between the moment of force and the moment of momentum of a solid body, which wraps around in a somewhat unbreakable axis of GO. Therefore, thoughts rose on the elementary parts (masi), as they are taken into account by material points.
The skin from the material points, which enter into the firm body, will collapse along the stake in the plane perpendicular to the axis of the wrap, and the centers of all of them will lie on this axis. I realized that all points of the body at a given hour may have the same top speed and the same top speed. Let's look at the i-material point, the mass is Δm i , and the radius of the stake, according to which it collapses, r i . On it, there are like outer forces from the side of other bodies, so internal - from the side of other material points that lie on this body. We spread the resulting force, which is directed to the material point of the mass Δm i on two mutually perpendicular warehouses of force, moreover, so that the force vector runs straight from the dot to the trajectory of the particle, and the force is perpendicular to the dot (Fig. 3). As a whole, it is obvious that the wrapping of the material point is less than dotary warehouse strength, the magnitude of which is possible for the sum of internal and external forces. Which viewpoint for the point Δm i has a different Newton's law for the scalar view 
(7)
For the sake of the fact that in the case of an overt Russian of a solid body, it is about the axis, linear roughness of the material points along circular trajectories is different for the magnitude of that straight, and the apex of the roughness w for all these points is the same (and for the magnitude of that straight), replaceable in equal (7) linear smoothness on the apex (vi = wr i):
. (8)
We introduce up to equal (8) the moment of force, which is equal to the part. For which we multiply the left and right part of the alignment (8) by the radius r i , which, according to the ratio to the resulting force, is the shoulder:
. (9)
, (10)
de skin member at the right part of the curve (10) - the moment of the double force along the axis of the wrap. How to introduce a quick wrapping of the material point of the mass Δm i
tsії ΔI i shodo tsієї w axis (=ΔI i), then
I’ll look at the material points of the axis in the future:
Analogous equalities can be written down to all other material points, as if they enter into a firm body. We know the sum of these equals with the adjustment of the fact that the value of the apex acceleration for all material points of a given body, which wraps, if it is the same, is taken away:
The total moment of the internal forces is equal to zero, to that the skin internal force, according to Newton's third law, can be equal to the magnitude, but I will directly direct my own force, applied to the other material point of the body, with such a shoulder. Sumarny moment \u003d M - is the twisting moment of all the forces that are blowing on the body, that are turning around. The sum of the moments of inertia = determines the moment of inertia of a given body, as well as the axis of wrapping. After substitution of the values of the equalities (12), the remaining is taken:
Rivnyannia (13) is called the main linearity of the dynamics of the overt movement of a solid body like an axis. Oskіlki =, and the moment of inertia of the body is shodo tsієї osі wrapping є with a constant value і, also, it is possible to introduce the sign of the differential, then equal (13) can be written at the sight:
Value
is called the momentum of the body's momentum along the axis. Z urahuvannyam (15) equal (14) can be written at the sight:
Rivnyannya (13-16) be of a scalar character and zastosovuyutsya only for the description of the overt ruhu tіl schodo osі. When describing the wraparound rotation of the body, as well as the points (either the poles or the cob), which lie on the given axis, the designation of the alignment is clearly recorded in the vector view:
(13 *); (14 *); (15 *); (16 *).
When the alignment of the translational and the overt body movement is equal, it is clear that in the case of overt Russ, the replacement of the force is the moment of force, the replacement of the mass of the body is the moment of inertia of the body, the replacement of the impulse (or the amount of the rotation) is the moment of the impulse (or the moment of the intensity of the rotation). Z equals (16) and (16 *) is clearly equal to the moment at any axis and at the point:
dL=Mdt(17); (17*).
Vіdpovіdno to vіnnyannja momentіv schodo osі (17) - change the moment of impulse
sa body like a non-violent axis is equal to the moment of the impulse of the outer strength, which is on the body of a well-balanced axis. For a point (17 *) equal momentum is formulated: change of the momentum vector for the momentum for the point equal to the momentum for the moment of the force vector, which is on the body, while the points are.
Z equals (17) and (17*) the law of conservation of the momentum of the solid body's momentum, as well as the axis, and the number of points. Z equal (17) viplivaє, as the total moment of all the existing forces M should be equal to zero
(M=0, also dL=0) the momentum of the th body in the case of the axis of the th wrapping is filled with a constant value (L=Const).
Wherever the point is: just as the total vector of the moment of all the forces of the world, unless the wrapping point is immutable, then the vector of the momentum of the body, if the point is permanent.
The next step is to indicate that the system is looking at how the wrapping of the body is viewed, є non-inertial , then the moment of force M is included as the moment of forces of interaction, and the moment of forces of inertia
or points.
3 . Description of the installation. Vision of a working formula.
Fig.4. Laboratory installed.
Base 1, equipped with three adjustable supports, for the help of which the vertical position of supports 2 and 9 is installed. 
For the help of a millimeter line 3 and two transverse sights 4, there is a difference between the passages of the center of the pendulum 5 during the first fall. At the upper part of the tripods there are 2 reamings of the vuzol 6 regulation of the threads of the pendulum 5. On the lower loose bracket there are 7 installations of the "light bar" 8 - an electronic vimiruvach hour. At the station there are 9 re-stashings of "launcher attachment" 10.
The main element of the installation is the pendulum 5, which is folded from the disk, through the center of which the entire diameter D passes.
The installation is based on the law of conservation of mechanical energy: the absolute mechanical energy of the E system, on the basis of which there is less conservative force, it is permanently determined to equal:
de-kinetic energy of the pendulum wrap-around swing, I-moment of inertia of the pendulum, w-crown swirlness of the wrap-around swing disk.
Twisting threads on the entire pendulum , we lift it to the height of h and create it a store of potential energy. As soon as the pendulum is released, then the vine begins to sink under the force of gravity, swelling at once an overturnal rush. At the lower point, if the pendulum sinks to the full length of the threads, the progressive movement will snuggle down. With this disk, as it unwinds, with a shear, it continues the wraparound movement, and in addition, for inertia, I rewind the threads on the shear. As a result of which the disk of the shear begins to rise up the hill. After reaching the highest point, the cycle of kolivalny rush will be renewed. The shearing disk will swing uphill and down, such an extension is called Maxwell's pendulum.
For a working formula, we can look at the forces that make Maxwell's pendulum (Fig. 5).
By such forces є: the force of tension m, is applied to the center of the mass of the system, that force is the tension of the threads. Let's write down for tsієї system equal to the progressive swing of the pendulum. It is consistent with another Newton's law for progressive movement to the center of the mass of the pendulum, equal to the movement may look: 
m= m+2
The tension force of one thread. Designed for alignment on the entire OS, which runs directly from the center of the mass of the pendulum:
m= mg - 2T (19)
The pendulum of the progressive swing takes the fate of the overturned Russia for the swing of the air at a new moment of force T. Then for such a swing of the pendulum we write down the basic law of the dynamics of the wraparound swing for an absolutely solid body:
de I - the moment of inertia of the pendulum wheel around the axis of the wrap, - the speed of the pendulum, M - the resulting moment of external forces around the axis of the wrap around the wheel of the pendulum.
Even though there is no slicking between, after simple transformations, we take away the formula for raising the moment of inertia I at a glance:
Shards of the value I, m і r, which enter at the level (24), do not change during the process, the pendulum's swing can be adjusted to constant acceleration. For such a move, h, passed in an hour t, with russ with cob zero swedishness, it’s more expensive. Zvіdki. Substituting the known acceleration equalization (24) and replacing the value of the radius of the pendulum axis r by її diameter D, we still take the basic working formula for the calculation of the moment of inertia of the pendulum:
For the working formula (25):
m is the weight of the pendulum, which is the sum of the mass of the disk m d, that axis m pro;
D - old diameter of the axis of the pendulum at once from the thread wound on it
(D = D 0 + d o , de D o – pendulum axle diameter, d o – suspension thread diameter);
t - the hour of the pendulum passing through h at the time of the fall;
g - quickening of the free fall.
The order of vikonannya roboti.
Adjusting the length of the threads with the adjusting screws 6, install the horizontal position of the shear (axle), on which the wheel of the Maxwell pendulum is fixed.
Install the light bar'er 8 so that for an hour the swing of Maxwell's pendulum sheared (all of the pendulum) freely passing through the light bar'er.
Vymіryuvalnoy linіykoy 3 vznachte vіdstan h, how to move the center of Maxwell's mass wheel.
a thread of thread d o .
Behind the table data:
c) for the data of the tables h i і t i induce a graph of the fallow land, passed by the point to the center of the Maxwell mass wheel for the vertical movement down, at the hour.
Table D = (D o + d o) = ... ... m
| No. pp | h i , m | t i , h | I i , kg m 2 | ΔI i , kg m 2 | (∆Ii) 2 | A i , ms -2 | (Δ A i ,) | (Δ A i ,) 2 |
| 1. | ||||||||
| 2. | ||||||||
| ……… | ||||||||
| ……. | ||||||||
| 7. |
Nizhny Novgorod State Technical University
Viksunskiy Philia
Laboratory robot №1-4
from global physics
Maxwell pendulum
Vikonala:
Gerasimova E.M.
PTK-09
Revisited:
Maslov V.P.
1. Purpose of work .
Assigned to the moment of inertia of Maxwell's pendulum.
2.Short views on theory
Diya priladu is based on one of the main laws of mechanics - the law of conservation of mechanical energy: there is no more mechanical energy of the system, like there is less conservative force, more constant. Maxwell's pendulum is a solid body, planted on the whole. The axis is suspended on two threads that are wound on it (Fig. 1). Nekhtuyuchi forces rubbing, the system vvazhatimyutsya conservative. potential energy. When the pendulum is swinging, the wines begin to swing under the force of gravity: forward to the bottom and wrap around its own axis. At what potential energy is transformed into kinetic energy. Having sank at the extreme lower station, the pendulum wraps itself around it directly by inertia, the threads are wound around the whole and the pendulum rises. This is how the pendulum swings.
Malyunok 1
Let's write equal to the swing of the pendulum. With the translational Russian of the pendulum, after another Newton's law, with the improvement of the orderly pendulum of forces, one can write
,
de m - mass of the pendulum, g - acceleration of the force of gravity, a - acceleration of the forward movement to the center of the mass of the pendulum,
T-force of preload of a single thread ,
Designing the price, it’s important
ma=mg-2T. (1)
For a pendulum wrap-around swing, we write the basic law of wrap-around swing dynamics for an absolutely rigid body:
, de J is the moment of inertia of the pendulum along the axis of the wrap, is the speed of the pendulum, M is the resulting moment of external forces along the axis of the wrap.
Oskіlki moment of force tyazhennya shdo osі wrapping up to zero,
, (2)
de r-radius axis. so yak 
ta z (1) 2T \u003d m (g-a), we can write:
,
and then a change
.
Accelerated and maybe buti was taken away for the fading hour of the rush and the passable pendulum of the hz equal to the evenly accelerated rush without cob shvidkost:
. Todi 
І akscho suggest the diameter of axis D, taking into account the main Rosrakhun's formula
. (3)
3. Description of the experimental setup
W 
the diagram of the laboratory stand is shown in fig. 1. The main element of the stand is disk 1, through the center of which a wire must pass 2. Two symmetrically stitched threads are wound on the center of the disk Z. At the outer position (shown by a dotted line in Fig. 1), the disk is trimmed by electromagnets 4. When the electromagnet is turned on, the disk begins to collapse down one-hour twisting of threads.
The folding of the ruh of the disk can be like an overlay of two independent ruhіv - forward and wrap. When the center of inertia of the disk passes beyond the movement of the progressive movement, it moves behind the vertical scale 5. At the time of the progressive movement, it is carried out according to the millisecond 6, which is signaled from the photo sensor 7 at the moment when the edge of the disk, which is going down, changes the light of the photo sensor.
If necessary, change the leading zsuva path, which should be passed by the disk in progressive rus, adjust the length of the threads behind the additional screw 8. If the platform 9 with the photo sensor is also moved, changing the screw 10, so the disk does not fall, the light turns at the same platform of the photosensor.
The value of the forward speed of the disk can be changed by adding the disk to change the ring 11 .
m =(0.050 
0.003) kg
m d =(0,050
0.003) kg
m k1 =(0,158
0.003) kg
m k2 =(0,370
0.003) kg
m k2 =(0,670
0.003) kg
4. Weekly data
Table No. 1
de m \u003d d - mass of the shaft and disk,
m to - masa kіlets,
r-radius of the shaft,
R 1 - inner radius of the wheel,
R 2 - outer radius of the car,
h-height of the foot shaft.
5. Rozrahunki:
Experimentally, the moment of inertia of Maxwell's pendulum is determined by the following formula:
de m 1 \u003d m + m d + m to I \u003d 0.05 + 0.05 + 0.158 \u003d 0.258 kg
m 2 \u003d m + m d + m to II \u003d 0.05 + 0.05 + 0.370 \u003d 0.470 kg
m 3 \u003d m + m d + m to III \u003d 0.05 + 0.05 + 0.670 \u003d 0.770 kg
Table #2
|
№ dosvіdu |
m before ,kg |
J, kg m 2 |
The calculated value is practical,
Graph analysis (div. graph on millimeters):
Oskіlki zovnіshnі radii kіlets rіznі, then th
for skin Masi will be different, and then, matimemo three graphs. For skin graphics, we can use one point
, A 
we know the formula 
- refining lines of the graphic axis of ordinates,
on the graph, the lines of the graph change all the ordinates of the values:
- Change of life,
Estimated value theoretically:
4.Determine thread tension N і N max :
In order to balance the tension force of the threads with the force of gravity, then we believe that the tension force of the thread is approximately equal to the tension force of the pendulum, and the tension force of the thread is max 2-2.5 times greater than the force of gravity of the pendulum.
Appointment of kidnappings:
masa valu + small ring + disk:
mass shaft + middle ring + disk:
masa shaft + large ring + disk:
shaft radius:
loss of radius disk + ring:
small ring + disk:
middle ring + disk:
large ring + disk:
disc radius change:
hijack moment of inertia:
Visnovok: At the same time, robots got to know Maxwell's pendulum, learned how to determine the moment of inertia of Maxwell's pendulum. Vinikli razbіzhnostі mіzh practical and theoretical calculations are explained by the forces of support.
Enthusiasm...