MexLekcii2010prn

										3
	a	=		v	;					(9)
				t				d2x
				dv					&&
		a = dt					= dt2			(9 )
									≡ x .

.

, ,

: ,

( , ,

).

: = f1(t) y	= f2(t).
	= f( ).

( ) :

vx = x ;	vy		v		≡ v = vx2 + vy2 ,						(10)
		= y – .
&		&
,
. : cos(v^ vx ) =						v	x	, cos(v^ vy ) =		vy	.
										v
						v

: ax = v&x = x&& , ay = v&y = y&&;

( )

a =

ax2 + ay2 .

v –

, .

S=f(t), :

S

dS

dv

d2S

&&

v =

v =

t

,

dt

; a =

dt

=

dt2

≡ S .

(11)

( . (1) .1).

§3.

	( . 2).
	,
	: R= S/Δϕ,
	ρ=Δϕ/		S=1/R (ρ – :
. 2						S =
	1 )	{		S/Δϕ=2 R/2 = R }.

21

					3
		.			t1
		v1,	t2	– 1
		v2 ( . 3).
		, ,
		,		.	,
		,
	. 3	.
		?
	v2 .

= v2 - v1 = Dv. Dv : 1 -

( , v1),

Dv , 2 – ( ),

Dv . v1

Dt, Dv , v2, ACD .

1. , ,

, , ( )

. . 3 , CD=BK=Dv , ( ) 1= v1×Dt.

(Dt dt) 1

1. :

CD

=

AA1

Þ

Dv

=

v1Dt

.

Dv

=

v1v2

.

v2

R

v2

R

Dt

R

dv

v2

= an ,

an

v2

(wR)2

2

R

Dt = dt

v1 = v2 = v

=

=

=

= w

.

dt

R

R

R

( ) n

,

.

,

( . 4): τ –

,

n.

: | τ|=| v |.

&

,

. 4

:

v2

é

æ dy ö2 ù3 / 2

d2y

an =

, R = ê1+

ç

÷ ú

–

(12)

R

dx2

ê

è dx ø

ú

ë

û

= y( ).

			a		= aτ2 + an2 .	(13)

22

3

,

, :

cos(a^ an ) =	an	,	cos(a^ aτ ) =	aτ	.
	a			a

,

:

a = a × i =

a ×v

=

xx + yy

+ zz

,

v = v ,

τ

&&&

&&&

&&&

v

x

2

+ y

2

+ z

2

&

&

&

&&&

&&&

2

&&&

&&&

2

&&&

&&&

2

a

=

a ´ i

=

a´v

=

(yx

- xy)

+ (zy

- yz)

+ (xz

- zx)

,

n

v

&2

&2

&2

x

+ y

+ z

v

2

&

2

&2

&

2

)

3 / 2

R =

=

(x

+ y

+ z

.

a

&&&

&&&

2

&&&

&&&

2

+

&&&

&&&

2

n

(yx

- xy)

+ (zy

- yz)

(xz

- zx)

. ,

, ( . 5).

t=0

t. j ,

.

j=f(t). ,

Dt

Dj, , Dj .

w e.

w

=

Dj

,

e

=

Dw

,

(14)

Dt

Dt

:

. 5

Dj

dj

&

dw

d2j

&&

w = lim

=

e =

=

(15)

Dt

dt

º j ,

dt

dt2

º j.

t →0

(dj=wdt i dw=edt), :

–(w=w0=const):

–:

w = w + et	( ),	j = j	0	+ w t +	et2
0				0	2

j = j0 + w0t; (16)

( ), (17)

j0 w0 – .

(17, ), t = (w–w0)/e,

(17, ),

:

23

								3
		w =		w2 + 2e(j - j )		.		(18)
			0		0
								:
w	= (w0 + w) / 2 ,	e = ±const.					w=const, e=0.
( ,							S=2 R).
– :							n=N/t = 1/T,	N –
t.

w =

Dj

=

2p

= 2p

1

= 2pn ,

v =

DS

=

2pR

=

2p

R = wR ,

w =

v

. (19)

Dt T

T

Dt T T

R

, .

	: a =	dv	=	d(wR)	=	dw	R = eR ,	e =	a	.	(20)
		dt		dt
						dt			R

v , w e (

). w e

. w ,

“ ”: ,

( . 6).

. 6, R

r a

z. : v = wR = wr sin a Þ v = ´ r ,

-

. ( )

.

. 6 (20),

:

a = dv = d [ × r] = d ´ r + ´ dr = ´ r + ´ v .

dt dt

dt

dt

,

– . ,

(20), , ,

:

a = [e×r] – , n = [w×v] –

( an =	v2		v × v		v	× v = w × v ).
		=		=
			R		R
	R

: a = a2 + an2 = e2R2 + w2v2 = w& 2R2 + w4R2 , R – .

24

3

:

S

ϕ

S = ϕR

v =

dS

ω =

dϕ

v = ωR

dt

dt

aτ

=

dv

ε =

dω

τ= εR

dt

dt

an

=

v2

n= ω2R

an = ωv

R

(v=const)

(ω=const)

S=S0 + vt

ϕ = ϕ0 + ωt

(a=const)

(ε=const)

v = v0 + a t

ω = ω0 + ωt

S = S0 + v0t + a t2/2

ϕ = ϕ0 + ω0t + εt2/2

v2

− v2

= 2a(S

2

− S )

ω2

− ω2

= 2ε(ϕ − ϕ

2

)

2

1

1

2

1

1

:

n = 0	a = 0
n = 0	a = +const
n = 0	a = -const
n = 0	a = f(t)
n = const	a = 0
n 0	a 0	:
n 0	a = f(t)
n 0	a = ±const

25

4

4.

. . .

. . .

– ,

.

,

. , ,

.

,

: – , – .

.

, ,

.

– . 3: x = f1(t), = f2(t), z = f3(t).

N ,

, 3N ( ).

( ),

, .

,

( 3

). ? ,

. 6 : x1, y1, z1; x2,

y2, z2. , ,

L, 6 ,

L2 = (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2

,		, 6-
L 5 , .
	5
– 5 .
			5
		( .1): 3
		) – 3
		– 2 ϕ –
				,
	. 1	′ ( ,
		).

.

26

4

( . 2)

, 3 ,

( , , ), 9 , 3

), , (

).

L2AB = (xA − xB )2 + (yA − yB )2 + (zA − zB )2

L2CB = (xC − xB )2 + (yC − yB )2 + (zC − zB )2

L2AC = (xA − xC )2 + (yA − yC )2 + (zA − zC )2 .

	,
	6 ( ),
. 2	6 .

? 3

, , 3 –

. , ,

. , (

– ).

.

'

', ', z'. ( , , z),

, .

', ',z' , ,z.

′Y′,

'

'

',

XY 0ρ ( ) ( . 3).

:

:

ϕ = <ρ0x'

( 0 ≤ ϕ ≤ 2 ) – ,

0Z'

= <x0ρ ( 0 ≤ ≤ 2 ) – ,

,

= <z0z'

( 0 ≤ ≤ ) – .

, .

0Z.

,

, 3 :

,

ϕ = ϕ(t), = (t),

= (t).

6

0Z

, :

d /dt, - 0Z'

x = (t),

= (t),

z = z(t),

( )

0Z.

ϕ = ϕ(t),

= (t), = (t).

:

(

)

0Z'

.

.

27

4

. 3

, -

, ,

.

– ( , ),

. ,

,	,
3 : – .
	4 .
	', ', Z' – ',
	, . , ,Z –
	.
	, ' r', ,
	'0, '0, z'0 (
	¢ ,
. 4	), –
	r . rc – , = (t),

= (t), zc=zc(t) ( ).

:

r = rc + r¢							(t) = (t) + x¢0,		(t)= (t) + y¢0, z(t)=zc(t) + z¢0
	dr	=	drc	+	dr¢	=	drc	+ 0 Þ v = v		, a = a
	dt dt				dt		dt

, , ,

3- = (t), = (t), zc=zc(t), 3 .

– ,

.

,

, , –

.

28

4

,

z f(t) = const.

, 3-

: Z

(

f(t) = const, f(t) = const).

,

. 5 ( ), ( S),

,

( . 5). ,

( ), .

S,

.

,

.

3- : ( , ) j

. : x = (t), = (t), j = j(t).

j=const, ; x=const y=const

,

( ).

,

.

S

, ,

( . 6). r¢ –

¢.

r = r + r¢,

	dr		drp		dr¢		¢	= vp	+ w´ r	¢
											.
		= +				Þ
. 6	dt				dt		v = vp + v
			dt

,

vp

v'=[w×r] ,

.

.

,

.

29

4

: ( . 7).

t = T j=2

= L=2 R=jR. dt dx= dj×R = w¢×dt×R, w¢ – ',

. , (

¢) :	v0	=	dx	¢
			dt	= w R .

' ( )

v¢=w¢R ( R).

:

v = v' + v0,

( ). ,

( . . 7) v =0.

. v

.

. 7

– ,

. , ,

. , , .

,

.

.

,

.

( )

w1 w2,

( –“ ”): w = w1 + w2.

30