For a new accelerating point m dorivnyuє. Designated trajectory, speed and acceleration of the point with the vector method of setting the movement. Determining the speed and speed of the point with the coordinate method of setting the speed
The main formulas of the kinematics of the material point are given, their development and development of the theory.
ZmistDiv. also: Butt of solving problems (coordinate method of setting the point movement)
Basic formulas for the kinematics of a material point
We introduce the main formulas of the kinematics of a material point. After that, ladies of their visnovoks and a great theory of theory.
Radius-vector of the material point M in the rectangular coordinate system Oxyz:
,
de - single vectors (orthy) along the x, y, z axes.
Width of point:
;
.
.
A single vector for the direct point to the point trajectory:
.
Quick points:
;
;
;
;
;
Tangential (dotichne) accelerated:
;
;
.
Normal speed:
;
;
.
A single vector, straightening to the center of curvature of the point trajectory (pushing the head normal):
.
.
Radius vector and point trajectory
Let's look at the rigid material point M. Let's choose a non-permanent right-angle coordinate system Oxyz with the center at the non-permanent point O . The same positions of the point M are uniquely assigned by її coordinates (x, y, z). Qi coordinates are components of the radius-vector of a material point.
The radius vector of the point M is the vector of drawings from the cob of the non-violent coordinate system O to the point M .
,
de - Alone vectors in a straight line x, y, z axes.
In Russian, the coordinate points change from hour to hour. Tobto stinks є functions in the hour. Todi system rivnyan
(1)
it is possible to align a curve given by parametric aligners. Such a curve is the trajectory of a point.
The trajectory of the material point is the whole line, which is the point of movement.
If the ruh points are seen in the plane, you can choose the axes and coordinate systems so that the stench lies in this plane. The same trajectory is marked by two equals
You can turn off the hour at certain times of the day. To the same level of trajectory matimum deposition mind:
,
de – day function. Tsya staleness to revenge is less than change. Vaughn do not revenge the parameter.
Width of the material point
The speed of a material point is costly її radius-vector per hour.Vіdpovіdno to vyznachennya shvidkostі and vznáchennya pokhіdnoї:
Pokhіdni in hours, in mechanics, signify a dot above the symbol. Let's imagine this for the radius-vector:
,
de mi clearly christened the staleness of the coordinates at the hour. We take:
,
de
,
,
- Projections of speed on the axes of coordinates. The stench of the differentiation over the hour is the component of the radius vector
.
Such a rank
.
Speed module:
.
Chodo trajectory
From the mathematical point of view, the system of alignments (1) can be seen as alignment lines (curves) given by parametric alignments. Hour, at a glance, plays the role of a parameter. 3 course mathematical analysis it seems that the direct vector for dotichnoї up to tsієї curve ї maє components:
.
Alece of the components of the vector of the sharpness of the point. Tobto the flexibility of the material point is straightened in a way that is accurate to the trajectory.
Everything can be demonstrated without intermediary. Let the point be at the moment of the hour in position with the radius-vector (div. little ones). And at the moment of the hour - at the position with the radius vector. Through the specks and we will draw a straight line. For the purpose, dotichna - it’s so straight, like a pragne straight at.
Let's introduce the notation:
;
;
.
Then the vector of straight lines is straight.
When pragnenny straight pragne to dot, and the vector - to the speed of the point at the moment of the hour:
.
Oskіlki the vector of straightening of the uzdovzh is straight, and the straight line is the vector of the straightening of the straightening of the uzdovzh dotichny.
That is the vector of flexibility of the material point of straightening of the uzdovzh trajectory.
Introduced direct vector of dotic single dougini:
.
It will be shown that the length of this vector is the most valuable one. True, shards
, then:
.
The same vector of speed of a point can be given at a glance:
.
Accelerated material point
The quickening of the material point is costly її quickness by the hour.Similarly to the front one, we take the component of the acceleration (projections of the acceleration on the coordinate axes):
;
;
;
.
Acceleration module:
.
Tangential (dotichne) and normally accelerated
Now let's take a look at the nutrition about the direct vector of acceleration along the direction to the trajectory. For whom we need the formula:
.
Differentiation by the hour, zastosovuyuchi the rule of differentiation to creation:
.
The vector of straightening along the trajectory. Is yoga straightened in the right direction for an hour?
Shchab v_dpovisti on the chain of food, we will speedily, that the vector's life is stable and the most expensive one. Todi square yogo dozhini tezh dorіvnyuє odinі:
.
Here and there two vectors in round arcs denote the scalar complement of vectors. Differentially stay equal by the hour:
;
;
.
Oskіlki scalar dobutok vektor_v i dorіvnyuє zero, і vectors and perpendicular to one to one. Since the vector of straight lines can be dotic to trajectory, then the vector of perpendiculars to dot.
The first component is called the tangential or dotifical acceleration:
.
The other component is called normal scaling:
.
Todі povene prikorennya:
(2)
.
Tsya formula є razkladannya accelerated on two mutually perpendicular components - dotichna to trajectory and perpendicular to dotika.
Oscilki, then
(3)
.
Tangential (dotichne) accelerated
Let's multiply the hurt parts of jealousy (2)
scalar to:
.
Shards, then. Todi
;
.
Here we put:
.
It can be seen that the tangentially accelerated projections of the total acceleration are straight up to the trajectory of the chi, which is itself, directly the sharpness of the point.
Tangential (dotichne) accelerating the material point is the projection of the total accelerating directly to the trajectory (or directly to the speed).
The symbol means the vector of tangential acceleration, directing the bridle to the trajectory. Todi - tse scalar value, which is a good projection of the total acceleration on a direct dot. It can be both positive and negative.
Submitting, maybe:
.
Let's put the formula:
.
Todi:
.
Tobto tangentially accelerated the speed of the hourly view of the module of the speed of the point. in such a manner, tangentially accelerate to change the absolute value of the point's widness. With an increase in the speed, the tangential acceleration is positive (otherwise, the increase in the speed is straightened out). With a change in speed, the tangential acceleration is negative (or, in the opposite direction, speed is straightened).
Now doslijuemo vector.
Let's look at a single vector of a random trajectory. Place the cob on the cob of the coordinate system. Then the end of the vector will be on the sphere of a single radius. With Russian material points, the end of the vector will move around the sphere. Tobto wine wrap around your cob. Come on - mitteva kutova shvidk_st wrapping of the vector at the moment of the hour. Todi yogo is pokhіdna - tse shvidkіst ruhu kіntsya vector. Vaughn is straightened perpendicular to the vector. Zastosuєmo formula for ruhu, scho turns around. Module vector:
.
Now we can look at the position of the point for two close moments in an hour. Let the point be in position at the moment of the hour, and at the position at the moment of the hour. Go ahead and - single vectors, directing random trajectories at these points. Through points i we draw planes perpendicular to vectors i . Come on - it's straight, illuminated by the peretina of these flats. 3 points we drop the perpendicular on the line. If the position of the point is close, then the point of the point can be seen as a wrap around the stake of the radius on the axis, as if it were a mitt of the wrapping of the material point. Scattered vectors are perpendicular to the planes i , then cut between these planes and the cut between vectors i . Todi mitteva swidkost wrapping of the point on the axis of the dot vnuyu mitteva swidkost wrapping of the vector:
.
Here - stand between the dots and .
In this way, we knew the module of the hourly vector:
.
As we pointed out earlier, the vector is perpendicular to the vector. From the guidance of the mirror, it is clear that the faults are straightened from the side of the mitt to the center of the curvature of the trajectory. Such a straight line is called the head normal.
Normally quick
Normally quick
straightened sigh vector. Yak mi z'yasuvali, tsey straightening vector is perpendicular to dotichnyy at mittevy center of curvature of the trajectory.
Move a single vector, directing from the material point to the center of curvature of the trajectory (vertical head normal). Todi
;
.
Shards of resentment are vectors and may still be straight - to the center of the curvature of the trajectory, then
.
3 formulas (2)
maybe:
(4)
.
3 formulas (3)
we know the module of normal acceleration:
.
Let's multiply the hurt parts of jealousy (2)
scalar to:
(2)
.
.
Shards, then. Todi
;
.
It can be seen that the modulus of the normal acceleration is more advanced than the projection of the total acceleration directly of the head normal.
Normally accelerating a material point is the projection of a total acceleration directly, perpendicular to dotichno to the trajectory.
Imagine. Todi
.
Tobto normal priskrennya viklikaє zamіnu svіnu svіdnostі point, and it's connected with the radius of curvature of the trajectory.
Zvіdsi you can know the radius of curvature of the trajectory:
.
For example, respectfully, the formula (4)
can be rewritten in the step-by-step look:
.
Here we zastosu formula for vector creative three vectors:
,
they put it in a yak
.
Father, we took away:
;
.
We compare the modules of the left and right parts:
.
Ale vectors and mutually perpendicular. Tom
.
Todi
.
Tse vіdoma formula of differential geometry for the curvature of a curve.
Let me see the function now. On fig. 5.10 
і 
vector and speed of the point that collapses at the moment t that t. To remove the increase in the speed vector 
portable parallel vector 
exactly M:
Average quickening of specks for an hour t
is called the increase in the speed vector 
until the end of the hour t:
Otzhe, quickening of a point at a given moment to an hour is the first slow by the hour in the direction of the speed vector of the point or another slow radius-vector by the hour
.
(5.11)
Quick points This is a vector quantity that characterizes the speed of change of the speed vector per hour.
Let's have a speed hodograph (Fig. 5.11). p align="justify"> The smoothness hodograph for the assigned є curve, so that the end of the vector of the smoothness at the Russian points, so the vector of the smoothness is included in one and the same points.
Determining the sharpness of a point with the coordinate method
Let's move the task points in the coordinate way in the Cartesian coordinate system
X = x(t), y = y(t), z = z(t)
Radius-vector of the road point
.
So alone vectors 
fast, then for the appointed
. (5.12)
Significantly, the projections of the velocity vector on the axis Oh, OUі Oz through V x , V y , V z
(5.13)
Comparing equalities (5.12) and (5.13) are taken away
(5.14)
Nadali pokhіdnu hour by hour is signified by the dot of the beast, tobto.
.
The modulus of point stiffness is determined by the formula
. (5.15)
The direction of the velocity vector is indicated by direct cosines:
Designation of the accelerated point of the coordinate method
Speed vector in the Cartesian coordinate system
.
For appointment
Significantly projections of the acceleration vector on the axis Oh, OUі Oz through a x , a y , a z clearly and arranging the vector of the speed along the axes:
.
(5.17)
Equivalence (5.16) and (5.17) are taken away
The module of the point acceleration vector is calculated similarly to the module of the point speed vector:
, (5.19)
and directly the acceleration vectors - by direct cosines:
Designation of speed and quickening of the point in the natural way
|
|
With this method, the natural axis with the cob is twisted at the flow position of the point M on the trajectory (Fig.5.12) and single vectors |
single vector 
directions along dotichnіy to traektorії y bіk positive vіdlіku arc, single vector 
straightening along the head normal of the trajectory of the bik її curvature, a single vector 
directing along the binormal to the trajectory at the point M.Orti 
і 
lie by flats that stick, orti 
і 
in normal plane, orti 
і 
in straight flat.
The subtracted trihedron is called natural.
Let the tasks go to the law of the dot s = s(t).
radius vector 
specks M so that a fixed point will be a collapsible function of the hour 
.
From differential geometry in the Serre-Fresnet formulas, which establish links between single vectors of natural axes and the vector function of the curve
de radius of curvature of the trajectory.
Vikoristovuyuchi designing swidkostі that formula Serre-Fresnet, we take:
. (5.20)
Meaning the projection of swidkosti on dotichna 
that vrakhovuychi, sho
. (5.21)
According to the equalities (5.20) and (5.21), we take the formulas for assigning the vector of uniformity to the value of that directly
Value 
positive, like a point M collapsing in a positive direction in the direction of the arc s i is negative in the proliferative type.
Vikoristovuyuchi vyznachennja priskrennya that Serre-Fresnet formula, we take:
Significantly the projection of the accelerated point 
to dotichnu 
, main normal and binormal 
obviously.
Todі prikorennya one
From formulas (5.23) and (5.24) it is obvious that the vector of acceleration lies near the plane, that it sticks, and spreads behind the straight lines 
і 
:
(5.25)
Projection of accelerated onto dotica 
called dotic or tangential acceleration. Vono characterizes the change in the magnitude of the speed.
Projection of accelerated head normal 
called normal squats. Vono characterizes the change in the speed vector directly.
Acceleration vector module 
.
Yakscho 
і 
one sign, we will speed up the ruh of the point.
Yakscho 
і 
different signs, then the rest of the points will be componable.
The butt of rozv'yazannya tasks is looked at with a folding hand of a point. The speck collapses along the straight edge of the plate. The plate wraps around a non-destructive axis. It shows absolute swidkіst that absolutely accelerated point.
ZmistUmov's tasks
A rectangular plate wraps around a non-destructive axis according to the law φ = 6 t 2 - 3 t 3. A positive direction to the kuta is shown on the little ones with an arc arrow. All wrapping OO 1 to lie near the flat of the plate (the plate wraps around the open space).
Point M is collapsing along the straight line plate BD. 40(t - 2 t 3) - 40(s is in centimeters, t is in seconds). Come b = 20 cm. In the small picture, the point M is shown at the position, at which s = AM > 0 (for s< 0 point M is located on the lower side of point A).
Find the absolute speed and absolute acceleration of the point M at the time t 1 = 1 s.
Vkazivki. Tse zavdannya - on folding points. For її vyshennya it is necessary to speed up by theorems about the folding of quicknesses and quick folding (theorem of Corioles). The first work of all the developments, following the minds of the manager, determine where the point M is located on the plate at the time t 1 = 1 s, and draw a point at the same station (and not in the right one, shown by the little plant).
Problem solving
Given: b= 20 cm, φ = 6 t 2 - 3 t 3, S = | AM | = 40(t - 2 t 3) - 40, t 1 = 1 s.
Know: v abs, a absPoint position definition
Significant position of the point at the time t = t 1 = 1 s.
s= 40(t 1 - 2 t 1 3) - 40 = 40 (1 - 2 1 3) - 40 \u003d -80 cm.
Oskilki s< 0
then point M is closer to point B, lower to D.
|AM| = |-80| = 80 div.
Robimo little ones.
Vіdpovіdno up to the theorem about the folding of odds, the absolute pliability of the point is more vector sumi portable and portable:
.
Appointment of the viable smoothness of the point
We can see the swedishness. For whom, it is important that the plate is not broken, and the point M is to break the tasks. So the point M collapses along the straight line BD . Differentiating s by hour t, we know the projection of straight line speed BD:
.
At the moment t = t 1 = 1 s,
cm/s.
Oskіlki , then the vector of straightening of the straight line BD . That way from point M to point B.
v vіd = 200 cm/s.
Designated figurative point sharpness
Significantly portable swidk_st. For whom it is important that the point M is tightly tied from the plate, and the plate is responsible for the tasks. So the plate wraps around the axis OO1. Differentiation φ over the hour t is known to the apex of the plate wrapping:
.
At the moment t = t 1 = 1 s,
.
Oskіlki vector kutovoy svidkostі straightening at bіk positive kuta turn φ , that is from the point O to the point O 1 . Module of top slickness:
ω = 3 w -1.
The vector of the apex shvidkost of the plate is depicted.
From the point M we drop the perpendicular HM to the entire OO1.
In figurative Russian, the point M collapses near the radius |HM| centered at point H .
|HM| = | hk | + | KM | = 3 b + | AM | sin 30° = 60 + 80 0.5 = 100 cm;
Portable security:
v lane = ω | HM | = 3 100 = 300 cm/s.
The vector of straightening by extension to the stake at the bik wrap.
Designation of the absolute smoothness of the point
Significantly absolute swidk_st. The absolute speed of the point is more expensive than the vector sum of the carrying capacity and the figurative speed:
.
Draw the axis of the non-moving coordinate system Oxyz. Everything z is directed to the axis of the wrapping of the plate. Let at a given moment all x be perpendicular to the plate, all y lie in the plane of the plate. Then the vector of the water tightness lies near the plane yz. The portable vector of the straightness of the straightening is proportional to the x-axis. If the vector is perpendicular to the vector, then according to the Pythagorean theorem, the modulus of absolute flexibility:
.
Appointment of the absolute acceleration of the point
Appropriate to the theorem about the folding of the acceleration (theorem of Corioles), the absolute acceleration of the point of the vector sum of the visual, figurative and coriole accelerations:
,
de
- Korіolisov priskrennya.
Appointment of a prominent accelerant
It is evidently evidently quickened. For whom, it is important that the plate is not broken, and the point M is to break the tasks. So the point M collapses along the straight line BD . Two differentiating s by hour t, we know the projection of the acceleration on the straight line BD:
.
At the moment t = t 1 = 1 s,
cm/s 2 .
Oskіlki , then the vector of straightening of the straight line BD . Tobto from point M to point B. The module of acceleration
a vіd = 480 cm/s 2.
We represent the vector on the little one.
Designation of a portable bait
It seems to be portable. In figurative Russian, the point M is tightly tied to the plate, so that it collapses around the radius |HM| centered at point H . Rozlademo portable priskornnya on dotichne to the stake that normally prikorennya:
.
Two differentials φ per hour t is known to be the projection of the apex acceleration of the plate on the entire OO 1
:
.
At the moment t = t 1 = 1 s,
h -2.
Oskіlki is the vector of the corner acceleration of straightening y bіk, the length of the positive corner of the turn φ, that is from the point O 1 to the point O. The module of the corner acceleration:
ε = 6 h -2.
The vector of the apex of the plate is shown.
Portable dotichno more quickly:
a τ lane = ε | HM | \u003d 6 100 \u003d 600 cm / s 2.
Vector of straightening by extension to stake. Oskіlki is the vector of the corner acceleration of straightening y bіk, prolonging to the positive kuta turn φ , then straightening y bіk, prolonging the positive straight turn φ . Tobto straightening at bіk osі x.
Tolerably normal speed:
a n lane = ω 2 |HM| = 3 2 100 = 900 cm/s 2.
Straightening vector to the center of the stake. Tobto y bik, protilene axis y.
Appointment of Coriole Acceleration
Korіolisov (turning) speedy:
.
The vector of the apex straightness of the straightening of the z-axis. vector db | . Kut mizh tsimi vectors dorіvnyuє 150°. For the quality of vector creation,
.
The direction of the vector follows the rule of the drill. If you turn the handle of the drill from position to position, then the screw of the drill will move in a straight line, opposite to the x axis.
Appointment of absolute repentance
Absolutely humbly:
.
We design the vector alignment on the xyz axis of the coordinate system.
;
;
.
Absolute acceleration module:
.
Absolute swidkist;
absolutely hastened.
Formulas of fastness (sharpness) are the point of a solid body, expressed through swidkity (suspension) of the pole and the top speed (suspense). Vysnovok tsikh formulas іz principle, scho vіdstanі mіzh be-like points of the body in yogo rusі become permanent.
ZmistBasic formulas
The speed and acceleration of a point of a solid body with the radius vector are determined by the formulas:
;
.
de - Kutov shvidkіst wrapping, - Kutov priskorennya. The stench is equal to all points of the body and can change from hour to hour.
і - quickness and speeding up the point A with the radius vector. Such a point is often called a pole.
Here and far create vectors in square arms mean vector create.
Visnovok formula for swidkost
Let's choose a non-rigid coordinate system Oxyz. Take two full points of a solid body A and B. Come on (x A, y A, z A)і (x B, y B, z B)- Coordinate points. At the time of solid body, it functions at the hour t. Їхні pokhіdnі for the hour t
,
.
Hurry up, scho pіd an hour to the collapse of a solid body, vіdstan | AB | between dots is filled with the constant, so it does not change with the hour t. So postiynym є square vіdstani
.
Prodifferentiation by hour t, zastosovuyuchi the rule of differentiation folding function.
Fast on 2
.
(1)
We introduce vectors
,
.
Todi river (1)
you can apply to the scalar creation of vectors:
(2)
.
Zvіdsi viplivaє that the vector is perpendicular to the vector. Hurry up to the power of vector creation. Todi can be seen in the sight:
(3)
.
de - deaky vector, which mi is introduced less in order for the Umov to automatically win (2)
.
Let's write down (3)
at the sight:
(4)
,
Now let's take a look at vector powers. For whom the storage is equal, it is not possible to avenge the swidkost point. Let's take three full points of solid body A, B and C. Let's write down for dermal steam and dots equalization (4)
:
;
;
.
Warehouse qі vnyannya:
.
Soon the sum of the swedes in the left and right parts. As a result, we will take away the vector equalization, which should be avenged only after further vectors:
(5)
.
It's easy to remember that it's equal (5)
my solution:
,
de - yakys vector, scho maє equal value for any pairs of points of a solid body. Todi river (4)
for swidkost, the dot of the body will look in the future:
(6)
.
Now perceptibly equal (5) from a mathematical point of view. If you write the vector alignment for the components on the x, y, z coordinate axes, then the vector alignment (5)
є linear system, which is added up from 3 equals with 9 changes:
BAx , BAy , BAz , CBx , CBy , CBz ,ωACx , ωACy , ωACz .
How equal is the system (5)
linearly not fallow 9 - 3 = 6
quite fast. So we didn’t know all the solutions. Іsnuyut more yakіs. In order to know, it is important to know that the solution has been found to determine the swidkost vector. This additional decision is not to blame, leading to a change in speed. Respectfully, that the vectorial addition of two equal vectors is equal to zero. Todi, yakscho in (6)
add a proportional member to the vector, then the speed will not change:
.
Other solutions of the system (5)
may look:
;
;
,
de C BA , C CB , C AC - constant.
Vipishemo heating system solution (5) have a clear look.
ω BAx = ω x + C BA (x B - x A)
ω BAy = ω y + C BA (y B - y A )
ω BAz = ω z + C BA (zB - zA)
ω CBx = ω x + C CB (xC-xB)
ω CBy = ω y + C CB (y C - y B)
ω CBz = ω z + C CB (z C - z B)
ω ACx = ω x + C AC (x A - x C)
ω ACy = ω y + C AC (y A - y C )
ω ACz = ω z + C AC (z A - z C)
Tse decision to avenge 6 good fasts:
ω x , ω y , ω z , C BA , C CB , C AC.
Yak and can buti. In this rank, we knew all the members of the infamous solution of the system (5)
.
Physical zmist vector
Yak was conceived, the members of the mind are poured into the meaning of the speed of the dot. That їх can be omitted. Todі shvidkostі point of solid body pov'yazanі spіvvіdnostnyam:
(6)
.
Tse vector of the apex stiffness of a solid body
Z'yasuemo physical sense of the vector .
For which v A = 0
. It’s always possible to work like a vibrate system for yourself, like at the moment of the hour, when you look at it, it’s possible to collapse a viably indestructible system from swidkistyu. The cob of the system in line with O can be moved to point A. Todi r A = 0
. І formula (6)
I will look:
.
The z-axis of the coordinate system is redirectively vector.
For the power of the vector creation, the vector of flexibility is perpendicular to the vectors i . Tobto vin parallel to the plane xy. Speed vector modulus:
v B = ω r B sin θ = ω | HB |,
de θ - tse cut between vectors ta ,
|HB| - Price of the perpendicular dropped from point B to all z.
If the vector does not change over time, then point B collapses around the radius |HB| zі shvidkіstyu
v B = | HB | ω.
That is why ω is the wrapping of the point B around the point H.
In this rank, we come to Visnovka, what vector.
Shvidkist point of a solid body
Later, we showed that the stability of a sufficient point B of a solid body is assigned to the formula:
(6)
.
It's worth the sum of two members. Point A is often called pole. Like a pole, sound to choose a non-violent point or a point that creates a ruh with a given swidkistyu. The other term is the wrapping point of the body around the pole A.
If point B is an adequate point, then the formula (6)
you can create a substitution. The exactness and speed of a point of a solid body with the radius vector are determined by the formula:
.
The widness of the dovilnoy point of the hard body is more equal to the sum of the widness of the progressive movement of the pole A and the widness of the obertal ruch of the pole A.
Accelerating point of the hard body
Now we will show the formula for accelerating the points of a solid body. Quickly - tse pokhіdna shvidkіst by the hour. Differentiation formula for firmness
,
zastosovuyuchi rules of differentiation sum that dobutku:
.
Input acceleration point A
;
that kutove crouched body
.
Dali respectfully, scho
.
Todi
.
Abo
.
So the vector of the accelerated point of a solid body can be given by looking at the sum of three vectors:
,
de
- quick enough points, which are often called pole;
- overt;
- zagostrennya quick.
Although the top speed changes only after the value and does not change directly, then the vectors of the top speed and the speedy directing of the air are straight. Go straight ahead overweight zbіgaєtsya chi in the opposite direction of the sharpness of the point. If the top swedishness changes directly, then the overtly accelerated swedishness can be the mother of a direct change.
Gostryuvalne sooner zavzhdi is directed to the bіk mittєvoї axis of the wrapping so that it goes over її under a straight cut.
Sharpness of the point.
Let's move on to the top of another main task of the kinematics of the point - the assignment of speed and acceleration for the already given vector, coordinate, or natural way of movement.
1. The speed of a point is called a vector quantity that characterizes the speed and direction of movement of a point. In system SI, speed is reduced by m / s.
a) Designation of speed with the vector method .
Let's move the task points in a vector way, tobto. in the house of vector alignment (2.1): .
Rice. 2.6. To the point
Come on in an hour Dt point radius vector M change in size. Todi medium swedishness specks M in an hour Dt called a vector quantity
Guessing the appointment of a pokhіdnoy, we put:
Here, and with a sign, we will signify differentiation by the hour. When exercising Dt to zero vector , а, later, i vector , rotate around the point M and in between they move from a dotty trajectory to the tsіy point. in such a manner, the speed vector is the first turn of the radius vector by the hour and the start of directing along the trajectory to the drop point.
b) The speed of the point with the coordinate method of setting the movement.
We will show the formula for determining the speed with the coordinate method of setting the speed. Vidpovidno to virazu (2.5), maybe:
So it’s like pokhіdnі vіd vіd іnіh vіdіnіh by the value of that directly single vectorіv vіvnyuyuyut zero, otrimuєmo
A vector, like and be a vector, can be expressed through its projections:
Porіvnyuyuchi virazi (2.6) and (2.7) Bachimo, scho pokhіdnі coordinates for an hour to mayut as a whole geometrical shift - є projections of the vector of swidkosti on the coordinate axes. Knowing the projections, it is easy to calculate the modulus and directly of the speed vector (Fig. 2.7):
Rice. 2.7. Up to the specified value and straightening of the speed
c) Appointment of speed for the natural way of zavdannya rush.
Rice. 2.8. Swiftness of the point in the natural way
Zgidno (2.4) ,
de is a single dot vector. in such a manner,
Value V=dS/dt called algebraic swidkistyu. Yakscho dS/dt>0, then the function S = S(t) growing and the point is collapsing at the edge of the arc coordinate S, tobto. the point is collapsing in a positive direction dS/dt<0 the point collapses straight ahead.
2. Quick points
The speed is called a vector quantity, which characterizes the speed of the change in the modulus and the direction of the speed vector. In the system CI hurry up in m/s 2 .
a) Appointment accelerated with the vector method .
Come on speck M at the moment t change in position M(t) and maє swidkіst V(t), and at the moment t + Dt change in position M(t + Dt) and maє swidkіst V(t + Dt)(Div. Fig. 2.9).
Rice. 2.9. Accelerating points with the vector method
Average quickening for an hour Dt is called change of speed up to Dt, tobto.
Mezha at Dt ® 0 called mittevim (or just quicken) dots M at the moment t
Zgidno (2.11), accelerated with the vector method, the order of the road is more expensive, the vector speed is up by the hour.
b). At speed with coordinate method .
Substituting (2.6) for (2.11) and differentially creating for the arms, we know:
Vrahovyuchi, scho similar to single vectors equal to zero, we take:
The vector can be rotated through its projections:
Por_vnyannya (2.12) and (2.13) shows that schooevery coordinates for an hour can make a whole geometric shift: they are equal to the projections of a pohіdnі podskorennya on the coordinate axes, tobto.
Knowing the projections, it is easy to calculate the modulus of the total acceleration and the direct cosines, which directly indicate it:
in). Accelerating points with a natural method
Let's put some effort into differential geometry, the necessary speed up with the natural way of driving traffic.
Come on speck M crumble like a spacious curve. Three mutually orthogonal straight lines (dotichna, normal and bionormal) are connected with the skin point of the curve, which unambiguously characterize the spatial orientation of an infinitely small element of the curve near the given point. Below is a description of the process of assigning direct appointments.
In order to draw dotichna to the curve at the point M, draw through it and adjoin the point M 1 sichnu MM 1.
Rice. 2.10. Assignment of a dot to the trajectory of a point
Hundreds of crooked to the point M vynachaetsya as a borderline situation MM 1 at the right point M 1 to the point M(Figure 2.10). A single dot vector is usually denoted by a Greek letter.
Let's carry out one by one vectors, scho trajectory in points. Mі M 1. Transferable vector u mottle M(Fig. 2.11) and it is possible to create a plane that can pass through the qiu point and vector. Repeating the process of making similar planes at the right point M 1 to the point M, we take it in between the plane, I call sticking flat.
Rice. 2.11. Appointment of the area that sticks
It is obvious that for a flat curve the plane that sticks, bends with the plane, in which the curve itself lies. The area that passes through a point M i is perpendicular to dotichny at tsіy point, called normal flat. Peretin sticks to that normal flatness straight, calling head normal (Figure 2.12).
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