What is Maxwell's pendulum. Assigned to the moment of inertia of Maxwell's pendulum. Determining the tension of the threads under the hour of the turn and at the moment of "Rivka" (lower point of the trajectory)

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 - axle 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

№,

, h

, kg

, m

Porivn. value

, h

, kg

, m

Control nutrition

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.

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. (one)

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 in order to provide the diameter of the axis D, we take away the main Rosrakhun formula

. (3)

3. Description of the experimental setup

Z 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 equalize 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 gravity 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.

Navchalno-methodical help

before laboratory work № 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 the movement of a solid body is called a movement, when it is straight, it is 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:

, (1)

de - the sum of all internal forces that can be exerted on an elementary mass Δm i the sum of all the equal forces that are on the elementary mass Δm i from the side of other bodies. Having summed up the equal (1) on the whole body, that vrakhovuychi, that is the sum of all internal forces zgidno with Newton's third law is equal to zero, we take away the law of dynamics of the progressive motion of a solid body:

Abo , (3)

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 infusion. 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:

(4)

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 in naming 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 -

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.

The momentum of the material point Δmischodo 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:

, (5)

de - Impulse of a material point.

The moment of momentum of a solid 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 points. 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:

, (6)

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 the sum of the creative masses of the skin material point of a given body per square її distance to the point Pro. Rosrakhun's formula 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 lie like a strong force from the side of other bodies, so and internal - from the side of other material points that lie on that body. We spread the resulting force, which is directed to the material point of the mass Δm i on two mutually perpendicular storage forces i, 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). It is quite obvious that the wrapping of a given material point is swayed by a superfluous warehouse power, the magnitude of which can be imagined by looking at the sum of the internal. the same forces. In which direction for the point Δm i another Newton's law in scalar

(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 wrap around the material point of the mass Δm i

ції ΔI i \u003d ΔI i), then the equalization of the overt

I’ll look at the material points of the axis in the future:

∆I i = (11)

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:

Sumarny moment of internal forces equal to zero, because the skin's internal force, according to Newton's third law, may be equal in magnitude, but I also direct my own force, applied to another material point of the body, with such a shoulder. A crazy moment \u003d M - is the twisting moment of all the forces that are blowing on the body, that are wrapping up. Sum of moments of inertia =I determine the moment of inertia of a given body as to 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 \u003d, and the moment of inertia of the body is shodo tsієї osі wrapping є with a constant value і, also, you can add the sign of the differential, then equality (13) can be written at the sight:

. (14)

Value

is called the momentum of the body's momentum along the axis. Z urahuvannyam (15) equal (14) can be written at the sight:

(16)

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:

E = + , (18)

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 slipping between the weave and the threads, and the thread can be unstretched, then the linear speeding up is tied to the kutov kinematic spіvvіdnoshennia.

yum:
, de v-linear speed to the center of the mass of the pendulum, r-radius of the axis of the pendulum. Todі kutov prikorennya can be written down like

(21)

Since the force of gravity m passes through the center of the mass of the system i, then, if the moment of force is equal to zero, then the moment of force M, which is on the pendulum, will be wised by the total tension force, which is equal to 2T. In this way, and from the improvement of the equal (21), equal (20) can be written in the view:

(22)

From equal (19) we know the resultant force 2T and we can represent її on equal (22):

. (23)

Having divided the right and the left part of the equation (23) by the value of the acceleration after simple transformations, we take the formula for the calculation of the moment of inertia I at the sight:

. (24)

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, it’s possible h, passed in an hour t, in Russia with zero cob speed, it’s more . Stars . 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:

. (25)

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.

Jobs

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 the “rivka” (the 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 the sum of two swings - progressive swing to the center of the mass 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.

Here index Z 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

Here J Z- moment of inertia of the pendulum O‘ Z.

EZ- Projection of 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

a) Assigned to the moment of inertia of Maxwell's pendulum.

Vikoristovuyuchi the law of conservation mechanical energy You can experimentally determine 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 at the position, if the pendulum is at the lower point. Kinetic energy in which camp

Here V- Shvidk_st to the center of the mass of the pendulum; w- kutova swidkіst;

J the moment of inertia of the pendulum how to axis, how to pass through the center mass: m = min + md + ml- Mass of the pendulum; min, md,ml- Masi shaft, disk that kіltsya, scho enter to the warehouse of the pendulum. The upper position of the pendulum has a potential energy

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 weakly weak)

Oscilki the center of the mass of the pendulum is collapsing in a straight line and evenly accelerated, then

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

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

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

we know the moments of inertia of the elements of Maxwell's pendulum.

MAXWELL PENDULUM

Meta roboti: learn from the laws of the flat motion of the body, find the moment of inertia of the disk of the Maxwell pendulum

Ownership: Maxwell's pendulum, stopwatch.

A flat motion of a solid body is such a motion, for which the trajectory of all points of the body lies near parallel planes.

We take away the equal kinetic energy of the flat ruh. A small part of the body, like laying down material points, is collapsing progressively and may kinetic energy. Imagine the speed of the part as the sum of the speed of the center of the mass V 0 that speed U i shodo axis Pro, to pass through the center of the mass perpendicularly to the plane of the ruh (Fig. 1). The total kinetic energy of all particles is more expensive.

Vimagaemo, shob middle member, then the sum of the impulses of the particles is about the axis O, reaching zero. So it will be, as if it were a visual ruh, it will be wraparound, ω. (In order to provide a good balance to the middle member, then we take away the formula for the rosette to the center of the mass).

As a result, the kinetic energy of the flat movement can be represented as the sum of the energy of the progressive movement of the body to the center of the mass and the overt movement of the axis to pass through the center of the mass

. (1)

Here m- masa tila, – moment of inertia of the body O, pass through the center

Let's look at another way of manifesting a flat ruffle, like wrapping around the so-called mitt axis. Let's create a diagram of the translational and wraparound Russian for the points of the body, which lie on the perpendicular to the vector V 0, (Fig. 2).

There is such a point in space Z, resulting swidkіst kakoї dorіvnyuє zero. Passing through it is called the mitteva of the whole wrapping, so that the body is more like a wrapping. Vіdstan mіzh center mas і mittєvoy vіssyu mozhna vyznachiti іz spіvvіdnoshennia mizh kutovoyu linіynoyu shvidkіstyu center mas.

Equalization of the kinetic energy of the wrapping ruhu around the mitteva axis may look

Here J with - moment of inertia of the body along the mitt axis . Having set equal (1) and (2), with , we take

. (3)

This viraz is called the Steiner theorem: the moment of inertia of the body is about the axis Z increase the sum of the moment of inertia about the axis Pro, scho to pass through the center of the mass and parallel to the given and the creation of the mass of the body on a square between the axes.

Let's look at the regularities of the flat swing with the butt of the Maxwell pendulum (Fig. 3). The pendulum is a disk, maybe with a stiffened ring, on the axis of any fastenings there is a round shear of small radius r. Two threads are wound on the ends of the shear, on which a pendulum of movement. Like a pendulum let go, you fall, turning around at once. The trajectories of all points lie near parallel planes, that is why the plane is flat. The center of mass spreading is on the axis of symmetry, and the mitt of the whole wrapping is folded with a shaping shear and pass through the dots of the dotik of the thread on the front r in the center of wt. At the lower point of the swing, the pendulum, continuing the inertia, wraps around, winds the threads on the haircut and begins to rise. In an ideal mood, for the presence of a support, I climbed up to the cob camp.

The system tіl pendulum - the Earth is closed, and the internal forces of gravity and the tension of the threads are conservative. If in the first vicinity it is possible to overcome the support forces, then it is possible to stop the energy conservation law: the potential energy of the pendulum at the upper outward position is transformed into the lower point to the kinetic energy of the flat movement (1):

. (4)

We can imagine the speed of the wrapping, and the speed of progressive movement behind the formula of the kinematics of evenly accelerated movement. After the transformation, we take the Rosrakhun formula for the moment of inertia along the axis of symmetry

. (5)

The hour of the fall is beaten by a stopwatch. When pressed on the "Start" button, an electromagnet vibrates, which stifles the pendulum and starts the clock. When retiring with a pendulum, the rahunok is attached to change the photocell. The height of the fall is behind the scale on the stand behind the station for the replacement of the photocell (Fig. 3)

The moment of inertia according to the axis of symmetry for the pendulum can be calculated theoretically as the sum of the moments of inertia of the shear, disk and ring:

1. Install the photocell at the lower position so that the pendulum, when lowered, bends the promin of the photocell. The length of the thread is adjustable with a screw from a lock nut on the bracket of the rack. Measure the height of the fall as the coordinate of the exchange for the scale on the station.

Turn on the unit up to 220 V, press the "Merezha" button.

2. Wrapping the shear, wind the thread around the shear, raising the disk to the electromagnet. The disk will be magnetized. Press the "Start" button. The magnet will let the pendulum go and the wines will drop down more often, the clock will start to rise for an hour with a stopwatch. Write down to table. 1 height of the fall, that hour of the fall.

Energy conservation law. Maxwell pendulum

1st Craiova Scientific and Practical Conference of Primary and Senior Researchers of Grades 9-11 “Applied and Fundamental Nutrition of Mathematics” Applied Nutrition of Mathematics The law of conservation of energy. Maxwell's pendulum Sokolov Dar'ya Vitaliivna, grade 10, MBOU "Lyceum 1", Perm, Savina Marina Vitaliivna, teacher of physics. Perm

2 Intro In the world we otochu stіlki tsіkavih speeches, as they have become for us the main and we do not mention their uniqueness. We are not to be teased by the movement of an electric kettle, a remote control for a TV set, a pilosos, even if we are chirping with these speeches today, and it doesn’t matter to us what the robot is grounded on. Sometimes it is necessary to add an hour to the production of something new. We will see a toy under the name of Yo-yo. For her help, there are many who win different effective tricks. First appointed Yo-Yo toy from two of the same for the size of that bag of disks, fastened with a string tied to it with a skein. The designation of the most recent variant of the toy, which you can learn and do. We began to wonder why the robot is running. It appeared that the Yo-Yo of this type works according to the principle of Maxwell's pendulum, it spins like a skein and turns back and so, until it snarls. James Clerk Maxwell

3 James Clerk Maxwell British physicist, mathematician and mechanic. Scot for travel. Maxwell laid the foundations of modern classical electrodynamics (Maxwell's equals), introduced the physicist to understand the struma of the displacement of the electromagnetic field, taking away a number of inheritances from his theory (the prophecy of electromagnetic waves, electromagnetic force, the nature of the force). One of the founders of the kinetic theory of gases (having placed the molecules of the gas behind the pins). One of the first advances in physics is statistical manifestations, showing the statistical nature of another cob of thermodynamics ("Maxwell's demon"), taking away a number of important results from molecular physics and thermodynamics (Maxwell's thermodynamic speed, Maxwell's rule for the phase transition of gas and others).

4 Maxwell's pendulum Maxwell's pendulum is a round, hard body, planted on the whole. The axis is suspended on two threads that are wound on it. Diya priladu is based on one of the main laws of mechanics - the law of conservation of mechanical energy: the mechanical energy of the system is the same, for which there is less conservative force, it is constant. Under the influence of the force of gravity, the pendulum oscillates at the vertical straight line, and at once the torsion oscillates along its own axis. Nekhtuyuchi forces rubbing, the system vvazhatimyutsya conservative. Twisting the threads, we raise the pendulum to height h, reminding you of a reserve of 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.

5 The law of conservation of energy Philosophical rethinking of the law was laid down by ancient philosophers. It is clearer, if not more succinct, the formula given by Rene Descartes in The Cobs of Philosophy (1644). An analogous view of the 18th century M. U. Lomonosov. At the sheet before Euler, I formulate my own “incredible natural law” (5 limes, 1748), repeating this in the dissertation “Mirkuvannya about hardness and homeland of bodies” (1760). One of the first experiments that confirmed the law of conservation of energy was the experiment of Joseph Louis Gay-Lussac, carried out in 1807. Trying to make sure that the heat capacity of the gas is deposited in an obligatory manner, having expanded the gas in an empty state and showing that the temperature does not change. Prote did not bother to explain this fact to youmu. At the beginning of the 19th century, a number of experiments showed that an electric jet could produce chemical, thermal, magnetic and electrodynamic forces. Such a different mind prompted M. Faraday to think that he believes in the fact that different forms, in which the forces of matter are manifested, can be combined, so that they can transform one on one. This thought, for its day, conveys the law of conservation of energy. The first robots from the installation of a kіlkisny connection between the victorious robot and the warmth, as seen, were carried out by the Sadi Carnot. In 1824, a small pamphlet was published “Think about the destructive power of fire and about machines that build building power”. A classic proof of the law was given by James Joule at the bottom of classical proofs. The results of such papers were presented at the physics and mathematics section of the British Association in Yoga Robots in 1843 “About the thermal effect of magnetoelectrics and the mechanical value of heat”. The first to notice that formulating the law of energy conservation was the German doctor Robert Meyer. The formulation of the exact terms of the law of conservation of energy was first given by Hermann Helmholtz. The law of conservation of energy is the basic law of nature, which means that the energy of a closed system is saved per hour. In other words, energy cannot be generated from anything and cannot appear anywhere, it can only pass from one form to another. Since the law of energy conservation is not brought to specific values and phenomena, but rather reflects a sharp, stagnant creak and establishes a law, then it is more correct to call it not a law, but the principle of energy conservation. Chastkovy vpadok The law of conservation of mechanical energy The mechanical energy of a conservative mechanical system is saved from the hour. Seemingly simpler, for the presence of dissipative forces (for example, rubbing forces), mechanical energy is not blamed for anything and cannot appear anywhere.

6 The eternal dvigun Іsnuє impersonal myths about the eternal dviguns, but, regardless of the number of tests, no one dared to induce the eternal dvigun to work the core of the robot without a dії zzovnі. The axis of the deaky model of perpetual engines: Lance sack on a tricot prism "Bird Hottabicha" Lance float

7 Archimedes screw and water wheel Magnet and zholobi Vcheni began to guess that the eternal engine could not be awakened. In the 19th century the science of thermodynamics was inspired. One of the foundations of thermodynamics, having become the law of conservation of energy, is a kind of recognition of a wealth of experimental facts. Thermodynamics can be used to describe the work and a number of mechanisms, for example, internal combustion engines or refrigeration units. As you can see, as for such minds, the mechanism is worked out, it is possible to unravel, cross the wines. In 1918, roci Emma Noether brought an important theorem for theoretical physics, zgіdno z zgіdno z kakoi in sistі, scho vodіє symmetrіy, z'yavlyayutsya quantities that are taken. Saving energy in the same time. How to understand the “uniformity of the hour”? Let us have some kind of attachment. If I learn yoga this year, tomorrow, or after a lot of rock, and it will work out anyway, then for such a system, the hour is the same, and in this practice, the law of conservation of energy. Unfortunately, there is not enough school knowledge to bring Noether's theorem. Ale, the proof is mathematically suvory, and the connection between the same time and energy savings is unambiguous. Trying to induce an eternal dvigun that works skilki for a long time, just trying to fool nature. Such a stupidity itself, like a test to cover 1000 kilometers for 10 hours on a car with a speed of 100 km / year (remember the formula s = vt?).

8 Why go out, the energy is saved? Chi did not put physicists between the recognition and the law of conservation of energy? Obviously, no! In the mood, as in the system there is no uniformity of the hour, energy is not saved. The butt of such a system is Vsesvit. We see that the All World is expanding. Today is not the same as in the past, and in the future it will change. In this rank, the All-World has no uniformity of time, and for it the law of conservation of energy does not stop. Moreover, the energy of the entire World is not saved. Chi give such an application to the daily savings of energy hope for the future eternal dvigun? Sorry, don't give. On the terrestrial scale, the expansion of the Universe is absolutely incomprehensible, and the Earth's law of conservation of energy is victorious with majestic accuracy. This is how physics explains the impossibility of inducing eternal movements. Under the hour of vikonanny tsієї robots squandered on the video on the Internet. It is called "Eternal Engine". The new one shows a clumsy construction made of cardboard, as if it were spinning without pinning. We have explained that it is one of the most ancient designs of the perpetual dvigun. Vaughn represents a gear wheel, in the pits of some kind of attached vantazh, which are visible on the hinges. The geometry of the teeth is such that the pitches in the left part of the wheel are closer to the axis, lower to the right. According to the author's idea, tse, obviously important to the law, it would not be enough to bring the wheel in a permanent wrapping. In case of wrapping vantages, they used the right-handed person and saved the rushyne zusilla.

9 However, if such a wheel is made ready, it will become unbreakable. The reason for this fact lies in the fact that if you want the right-handed vantazh to be more important, they are more important for the money. As a result, the moments of forces of the right hand and left hand are equal. We made such a cardboard construction itself and messed it up, it really doesn’t work.

10 Practical part

11 Later, now we know what Maxwell's pendulum is and what this robot is based on. Mi vyrishili vygotovity different pendulums, shchob z'yasuvati, vіd chogo to lay їhnya robot. In order to find out, how to lay down the robot of the pendulum in the thread, we prepared two identical pendulums with threads, different for the tovshchina: At the pendulum with a thick thread T \u003d 2.65s Visnovok: lay the robot of the pendulum along the thread. Also, the threads were divided by length: l = 46cm, T = 2.5s l = 92cm, T = 4.6s Visnovok: a period of proportional long thread.

12 In order to recognize the position of the pendulum robot in the shearing, we prepared two identical pendulums with shearings, different for tovshchina: as the thinnest shear of the pendulum, the period fit more.

13 So the shearers themselves were divided according to their length: l=11cm, T=2.5s l=6cm, T=2.5s Visnovok: The pendulum’s robot should lie down in the length of the shearing. In order to find out how to lay the pendulum robot in the disk, we prepared two identical pendulums, with different disks in width:

14 The width of the pendulum is 1 mm, T = 4.5 s For the pendulum, the width of the disc is 12 mm, T = 5 s In 12 times the width is increased, the period is increased insignificantly. Wisnowok: The width of the disc does not fit into the pendulum much. Also, the disks were divided for the mass:

15 m is large, T = 5.2s m is small, T = 5s Wisnowok: The mass of the disk is too small for the robot of the pendulum. Also, the discs are small with a different radius:

16 R=6, T=5s R=4, T=3.5s We changed R by 13 and the period also changed to about 13. Wisnovok: Period proportional to radius. In order to develop the mechanical energy of the pendulum, it is necessary to know its potential and the kinetic energy that is accumulated. The potential energy of the pendulum is taken into account according to the formula: Ep \u003d mg De m (mass of the pendulum) \u003d 0.054 kg g (acceleration of the free fall) \u003d 9.81 m / s2 h (height on the yak the pendulum is lowered) \u003d 0.21 m Ep \u003d 0.055 9.81 0 ,21 = 0.113 J The kinetic energy of the pendulum is determined by the formula: Ek = mv22 + Jω22 = mv22 + Jv22r2 = mv22 (1 + jmr2) r(radius of pendulum shear) = 0.0003m; v (lowering speed to the center of the pendulum mass) \u003d 2ht \u003d 2 0.212.6 \u003d 0.16 m / s; t(hour of lowering the pendulum) = 2.6 s

17 De a = 2ht2 - acceleration of the forward movement to the center of the pendulum mass J = 0.055 0.0003 0.0003 9.81 2.6 2.62 0.21-1 = 0 Now we can increase the kinetic energy of the pendulum: Ek = 0.05 ,16 0.055 0.003 0.003 = 0.11J Now it is easy to measure the mechanical energy of our pendulum: Em = En + Ek Em = 0.113 +0.11 = 0.223J We have found out that all warehouses are pouring into the robot of the pendulum. We responded to all questions, as if they blamed us on this topic.

Maxwell's pendulum. Appointed to the moment of inertia of bodies. that reverb to the law of conservation of energy

transcript

1 Laboratory robot 9 Maxwell's pendulum. Assigned to the moment of inertia tіl STATEMENT OF THE PROBLEM Maxwell's pendulum is a disk fixed on a horizontal axis and lifted in a bifilar way. Rings are put on the disk in order to be able to change the mass, and, also, the moment of inertia of the pendulum. Rice. Fig. 1. Scheme of the laboratory setup When the electromagnet is turned off, Maxwell's pendulum, wrapping around the horizontal axis, falls vertically down from the acceleration. Whom wins the law of conservation of energy, tobto. the potential energy of the lifted pendulum to pass from the kinetic energy of the translational and overt rotation. 1 h

2 mv mgh (1) m m 0 m mk mass of the Maxwell pendulum; m 0 weight of the axis of the pendulum; m disk mass; m k Otrimane viraz it is possible to beat the assigned moment of inertia of the pendulum. In this rank, for the help of Maxwell's pendulum, two experimental tasks can be accomplished: 1. Create a re-verification of the law of conservation of energy in mechanics; Calculate the moment of inertia of the pendulum. ACCESSORIES AND ACCESSORIES Maxwell's pendulum, stopwatch, vibrating line on a vertical column, electromagnet, caliper. SHORT THEORY Assigned to the moment of inertia of the pendulum 3 equal (1) significant moment of inertia of the pendulum. For the value of v i is dependent on the height of the pendulum. In respect of the translational swing of the pendulum to the bottom, we will evenly accelerate with the cob swidkistyu v 0. The level of kinematics: at h ; h v, t v a; v r t h () rt r radius of the disk axis. h

3 Then, representing the subtraction of the value of v i virase (1), it is possible: mgh 4m h 4 h (3) t r t The subtraction of virase is reversible according to the moment of inertia: gt mr 1 or h md gt exp 1 (4) h D D 0 DH ; D 0 diameter of the disk axis; D H thread diameter. Viraz (4) is a working formula for the experimental assignment of the moment of inertia of the pendulum. The theoretical value of the moment of inertia of the Maxwell pendulum є the sum of the moment of inertia: 1. The moment of inertia of the axis of the pendulum 1 0 m0d0, (5) m 0 і D 0 mass and the outer diameter of the pendulum axis.. The moment of inertia of the disk 1 m D0 D, (6) m D is the mass and current diameter of the disk. 3 h

4 3. Moment of inertia of the ring k 1 mk D Dk, (7) m k and D k mass and outer diameter of the ring. Let's write down the sum: theor 0 k theor 1 m0d 0 1 m 1 D D m D D 0 k k () Viraz () is a working formula for assigning the theoretical value to the moment of inertia of Maxwell's pendulum. Conversion to the law of conservation of energy The law of conservation of energy: most of the mechanical energy of a closed system of bodies, between which there is no more conservative strength, is stagnant. W W K W П const The potential energy of the raised pendulum is more: W П mgh, (9) m m 0 m mk mass of the pendulum. The kinetic energy of the pendulum is composed of the kinetic energy of the forward movement and the kinetic energy of the wrapping movement: 4

5 W K mv (10) After changing the value of v and equal () we take h t 4 m D0 W K (11) m m 0 m mk of the mass of the pendulum. If you don’t lose that support of the middle, then the values of w and W K are exactly the same. Rozrahunok is absolutely abducted by the sizes of the sizes of the subgrades of the virasi (4), the formula for the rosrahunki stroke at the vimyryuvanni moment: d0 ht (1) d ht is the abolotment of the vimyryvannya: experimental setup, it is necessary to match the experimental expert and the theoretical value of the moment of inertia of the pendulum. The consequences of the moment of inertia can be expressed as follows:

6 theor expert 100% (14) theor The loss at the specified energy is calculated according to the formula: WP WK W 100% (15) W ХІД ROBOTI P 1. Measure the diameter of the disk, ring, pendulum axle, thread with a caliper. The lower bracket will fit the lower position. 3. Adjust the length of the thread in such a way that the edge of the steel ring fixed on the disk, after lowering the pendulum, is one mm below the optical axis of the lower photocell. 4. Adjust the entire pendulum so that it is parallel to the base of the fixture. 5. Press the key "START" and "SHIPPING". 6. On the whole of the pendulum, wind the thread of the suspension and fix the pendulum behind the auxiliary electromagnet. Reverse the lower edge of the ring with the zero of the scale on the column. Yakshcho nі, vіdreguluvati. 7. Press the START key. Record the value of the fall of the pendulum and repeat the freeze for the hour 5 times with the same ring on the disk. Calculate the mean hour of the fall. 6 h

7. Behind the scale on the vertical column, the fixture indicates the height of the pendulum's fall, indicating the upper and lower position of the pendulum along the lower edge of the ring. 9. Variable formulas (4, 9, 11) to increase the expansion of the moment of inertia and energy of the pendulum exp, theor, W P, W K. deviations for the moment of inertia and the value of energy W for the additional formulas (1, 13, 14, 15), vicorist average value 11. exp, theor, W K, W P. Table h, m t, s m k, kg exp, kg m theor, kg m W P, J W K, J Mean value 7 s

8 POWER CONTROLS 1. What is called the moment of inertia of the body? The moment of inertia of the world of inertia of the body in obertovy rus. Explain the sense of which virus. 3. Why is the moment of inertia of the disk important? 4. Write down the formula for assigning the moment of inertia of the ring? 5. Why is the moment of inertia of a thin-walled cylinder important? 6. Enter the formula for the experimental value of the moment of inertia of Maxwell's pendulum. 7. Formulate the law of conservation of mechanical energy. Give the appointed potential energy. 9. Give an understanding of kinetic energy. 10. What is the law of conservation of energy for Maxwell's pendulum? h

fiz / maxwell pendulum 4-5

Ministry of Education and Science Russian Federation Derzhavne lighting mortgage viscogo

UFIMSKY STATE NAFTOVY TECHNICAL UNIVERSITY

LEGAL RESERVATION FROM MECHANICS.

Basic and methodological guide to laboratory work in mechanics

Navchalno-methodical guide for appointments for students in the current forms of apprenticeship. To give a short introduction to the theory and description of the order of laboratory work in the "Mechanics" section.

Officers: Leibert B.M., Associate Professor, Candidate of Technical Sciences Shestakova R.G., Associate Professor, Candidate of Chemical Sciences

Gusmanova G.M., Associate Professor, Candidate of Chemical Sciences

Ufa sovereign naftovy technical university, 2010

Meta robots: derivation of the moment of inertia of Maxwell's pendulum from the law of conservation of energy.

Attachment and attachment: Maxwell's pendulum, caliper.

When the wrapping of the overwhelming movement takes place, the understanding of the "masa" corrosives the understanding of the "moment of inertia". The moment of inertia of a material point, as the axis of wrapping, is called the value, equal to the additional masi i-ї dots on the square in the middle of the square dots to the axis wrap

Solid body and succulence n material points, so that the moment of inertia is good for the axis of wrapping

In times of uninterrupted expansion, the sum of the sums is reduced to the integral

deintegration is carried out from the bottom of the body.

Vіdpovіdno up to (3) otrimani moments of inertia tіl be-any form. For example, the moment of inertia of a uniform cylinder (disk)

de R - radius of the cylinder, internal radius R 1

m is the mass, and the moment of inertia of the empty cylinder with the outer radius R 2 is about the axis of the cylinder

I 1 m R 1 2 R 2 2 .

Designated moment of inertia

next, that the moment of inertia is solid

Dogo body is an additive quantity. Addi-

resistance to the moment of inertia means that

moment of inertia of the system tіl dorivnyuє sum-

me moments of inertia of all bodies,

sneeze at the system. Yak butt op-

we can determine the moment of inertia of Maxwell's pendulum, which is composed of three elements.

tov: axis, roller and ring (Fig. 1). Vіs - a strong cylinder, for which

Ring and roller - empty cylinders, for those

m K D K 2 D P 2

m P D P 2 D 0 2 .

According to the power of additivity, the moment of inertia of Maxwell's pendulum is equal to the sum of the moments in the inertia of the axis, the roller of that ring.

Here m 0 , m r, m to, D 0 , D r, D k - it is clear that the masses and ovn_shn_ diameters of the axis of the roller and ring.

The moment of inertia of Maxwell's pendulum is significant experimentally according to the law of conservation of energy (Fig. 2). McSwell's pendulum is a disk, all of which is suspended on two threads that are wound on it. Turning the pendulum, mi

we ourselves raise yoga to a height h above the pochatkovy camp and help it to its potential energy

We let the pendulum collapse under the force of gravity. When the thread is untwisted, the pendulum immediately zdіysnyuє overtall and forward movement. Diyashovshi to the lower position, the pendulum again rises uphill again with that cob swidkistyu, like a wine reach at the lower point. If you fight against the forces of rubbing, then on the basis of

Conservation of mechanical energy The potential energy of Maxwell's pendulum is transformed into the lower point into the kinetic energy of the translational and wraparound swing

mgh mV 2 I 2 , 2 2

de V - the speed of the forward movement to the center of the mass of the pendulum; - kutova shvidkіst wrapping ruhu;

I - the moment of inertia of the pendulum, which is about the axis of the wrap. Vikoristovuyuchi zv'azok mizh liniynoy and kutovoy shvidkistyu

de r - radius of the axis of the pendulum, we know s (10)

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MAXWELL PENDULUM
Navchalno-methodical help

to laboratory work No. 1.10

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 the movement of a solid body is called a movement, when it is straight, it is 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 infusion. 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 in 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

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 .

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. In which direction for the point Δm i another Newton's law in scalar

(7)

. (8)

. (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:

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:

(13 *); (14 *); (15 *); (16 *).

dL=Mdt(17); (17*).

Vіdpovіdno to vіnnyannja momentіv schodo osі (17) - change the moment of impulse

(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).

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.

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.

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.

For a working formula, we can look at the forces that make Maxwell's pendulum (Fig. 5).

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)

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.

10

a thread of thread d o .

Behind the table data:

a) using the formula (25) to find the average value of the moment of inertia of the wheel of the Maxwell pendulum, to know the error and the visible pardon to the result;

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.