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Ужгородский национальный университет

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m1v1+ m2v2 = m1v1′+ m2v2′

m1(v1– v1′) = m2(v2′– v2).

(2) (3), :

(v1+ v1′) = (v2′+ v2)

(v1 - v2) = (v2′ - v1′) = –(v1′ - v2′)

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, (4 ) ,

( , k=1).

(4) 2 (3):

2 (v1+ v1′) – m1(v1– v1′) = 2 (v2′+ v2) – m2(v2′– v2)

→

2v1+ 2v1′ – m1v1+ m1v1′ = 2v2′+ 2v2 – m2v2′+ 2v2 →

2 2v2 = ( 2 – 1)v1 + ( 2 + 1)v1′ , v1′:

′

2m2v2 +

(m1 − m2 )v1

v1 =

(5)

m1 + m2

2m2v2 + (m1 − m2 )v1 +

1v1 − (m1 + m2 )v1

m1v1 − m1v1

2m2v2 + 2m

m1 + m2

′

m1v1

+ m2v2

v1 = −v1

+ 2

= −v1 + 2VC

(5 )

+ m2

V – .

, v2′, (4) 1 (3):

1 (v1+ v1′) + m1(v1– v1′) = 1 (v2′+ v2) + m2(v2′– v2)

2 1v1 = ( 1 – 2)v2 + ( 2 + 1)v2′ , v2':

′

2m1v1 −

(m1 − m2 )v2

(6)

2m1v1 − (m1 − m2 )v2

m1 + m2

+ m2v2 − m2v2

2m1v1 + 2m2v2 − (m1 + m2 )v2

m1 + m2

′

m1v1

+ m2v2

= −v2

+ 2VC

(6 )

+ m2

1. v2 = 0 (1- 2- ):

′

− m2

′

2m1

(5) (6)

;

(7)

+ m2

′

(5 ) (6 )

= −v1 + 2VC ,

= 2VC .

(7 )

					′	2m1
)	1 = 2 :	v1′=0,			v2 =		v1	= v1 – .
						2m1
)	m1<<m2 ( , ); (7) :
	′		m2			′		2m1
	v1	≈ −		v1	= −v1 ,	v2	≈		v1 ≈ 0
			m2					m2

111

– ( )

, , .

2m1

) m1>>m2 :

» -

= v1 ,

= 2v1

–

, .

2. v1 v2 , m1<<m2 . (6) 2 ( “ ” –

, ) :

v¢2 = 2m1v1 + m2v2 = 2m1 v1 + v2 » v2

m2 m2

–.

(5) :

	2m v		+ m			(	m1		-1)v		2m v				- m v
	2m v		+ m			(			-1)v
2		2		2			m	1						2
v1¢ =							2			=	2			2		2	1	= -v1 + 2v2 .	(8)
v1¢ =		m2			m1		+1)			=			m					= -v1 + 2v2 .	(8)
		m2		(m			+1)								2
					2							v2			2
					2
(8)				:				v1¢ = -v1(1 - 2							)	,
(8)				:				v1¢ = -v1(1 - 2							)	,
													v1			v1>v2 –			( 1)
																v1>v2 –			( 1)
( 2).

v1=2v2, v1¢=0 ( ).

		v2<v1<2v2, v1¢<v1,
, v2 ( v1).
		v1>2v2, v1¢= –av1 ( 0<a<1),

v2 ( v1) .

( )

v»107 .

~3×103 ,

( ). ,

, ( )

– ,

, 2=2 .

“ ”, D2O,

= 0.

( 7) v1¢ =	m	- 2m	v = -	m	v = -	1	v .
	m	+ 2m		3m
					3

, 2-

112

3 v =

v , n-

v .

3 v1

( ):

v	= 3n Þ lg	v	= lg 3n	Þ n =	lg(v / v¢n )	=	lg(107 / 3 ×103 )	=	3,4771	= 7,3 » 7 .
¢		¢
					lg3		lg 3	0,4771
vn		vn

§3.

, ,

, .

,

m1v1+m2v2=(m1+m2)v¢,	(9)

(m1+m2) – ,

), v¢– :

¢		m1v1	+ m2v2
v	=			= vc .	(10)
v	=	m1	+ m2	= vc .	(10)
		m1	+ m2

, v¢ – ( ) ,

, –

( ).

, Q

( ).

						E				m v2					m v2
:								=			1 1		+			2 2	.
:								=					+				.
											2					2
											2					2
										( (10)):
						¢2			m1 + m2							(m1v1 + m2v2 )					2		(m1v1		+ m2v2 )				2
E		=	(m1 + m2 )v				=		m1 + m2					×		(m1v1 + m2v2 )						=	(m1v1		+ m2v2 )					;
E		=					=							×								=								;
				2							2					(m		+ m		)2			2(m + m				)
																1		2							1	2
Q = E	-				m v2	+ m v2						(m v + m v						)2		1 m m								2			m	2
				=	1 1		2 2			-			1 1			2 2			=			1 2		(v		- v	)		=			(Dv)	,
				=						-									=					(v		- v	)		=			(Dv)	,
						2						2(m1 + m2 )								2 (m1 + m2 )					1	2					2
						2						2(m1 + m2 )								2 (m1 + m2 )											2

µ – .

. ,

, .

113

, , ,

ì(m + m )v

= m v

+ m v

¢2 ,

(m1 + m2 )v

¢2

U +

m1v1

m2v2

U – , , ,

( , ),

, , , .

2- 2 , v2=0, 1

v1. ( )

Q = DE =	1	m1m2		v2	=	m2		E	= a	,	(11)

	2 (m + m ) 1					(m + m ) 1			1
	1		2			1	2

a<1 , 1

( ). (1-a) 1 –

. a,

. , 2

1 ( . 11).

) m2>>m1, a»1 Q»D –

: , ,

) m2<<m1, a»0 Q»0, » –

( ) :

§4.

–

) :

m0c	2	¢ 2	,	hn	¢ ¢	(11)

		+ hn = m c		c	= m v

= m0 +

chn

» c

2 .

(12)

m c

m0c +

m0c

+ hn

m0c

114

hv<<m0c2 ( ),

(12) .

( 0)

E	=	m0 v¢2	=	m0	(	hn	)2	=	(hn)2	.	(13)

	2			2 m0c					2m0c 2

, ,

( , ,

, )

– .

(

), – (1960 .)

, , ,

. ,

. 1)

–

, , . 2)

. 1-

, . –

, ,

. – .

s, , ,

s =	1	ln	I0	,	([s] =	1	=	1	= 2 )
						[n]×[x]
	nx I							− 3

n – , ( ),0 – =0 ( ), –

115

15.

. .

, .

. .

§1.

( ),

	dvc		dN			åmi vi			1
m		= F;		= M ,	vc =	i	vc	=		òvdm . (1)

						åmi

	dt		dt						m V
						i

: 1) vi 2) r

. , ,

3 . 6

, .

§2. .

) Z . v =0, (1)

. .

Z ,

, ,

ri ( . 1).

F ,

. 1 . F

3 ( . ): Fn

( ri), F|| ,

Z , ,

F, F . F =Fi

= ri×Fi ( , sin900=1)

, (“ ”) :

116

	15
dAi =Fidli = Fi×rida = Mida A = åMi da = Mda .
dAi =Fidli = Fi×rida = Mida A = åMi da = Mda .	\|M\|
i	α
, ,

, ( 2- ,	Fi=miai):
M = åriFi = r1m1a1+ r2m2a2+ … = {a = e×r} = e×(m1r12 + m2r22 + …) =
= e×(I1+I2+ …) = e×I.	(2)

: I =m ×r 2

. – .

( ) , ,

I= åIi = åmiri2 .

(2) .

, , .

(2), .1, z,

Z, :

z = e×Iz

(2 )

e w,

Z :

Mz	= Iz = I	d	.	(3)
		dt

(3)

. 2-

( ) , .

) ( “ ”),

( const),

r . 2- (3)

, :

M =	d	(I × ) Mdt = d(I × ) –	(4)

	dt

( , : Fdt=dp).

=0, ×w = const,

( )

1w1 = 2w2.

(4 )

( : )

117

		15
	, w=N ( )	(4)
(1),		,

r v=wr:
Iw = mr 2	v	= r ×mv = r × p = N .	(5)

	r

1.	I = åmiri2 – ,
	,
	. , e.
2.	I = åIi	– : =
	.
3.	I = åmiri2 ,
	,
4.			I = òr2dm	.	(6)
	.

, (

, ).

0, r –

0, w -

( 0) ( .2).

v =w×r ,

N = å i ´ mivi

= åmi i ´ ( ´ i ) =

= åmiri2 - åmi i ( i × ) .

A×(B×C)=B(A×C)-C(A×B)

. 2

ìNx

= wx åmiri2 - åmi xi ( i

× )

= wy åmiri2 - åmi yi ( i

× ) ;

(r ×w) w w +z wz ,

íNy

= w

m r2

m z (

× )

ïN

z å

= xi

+ yi

+ zi

i i

ìNx = åmi (ri2 - xi2 ) × wx - åmi xi yi ×wy - åmi xi zi × wz

= -åmi yi xi

×wx

+ åmi (ri2 - yi2 ) ×wy - åmi yi zi ×wz .

(7)

íNy

= -

m z x

×w

m z

×w

m (r2

- z2 ) ×w

ïN

i i i

i i

å i i

118

N – , ( ) w –

(N=Iw). :

Ixx = åmi (r2i - xi2 ) = åmi (xi2 + yi2 + zi2 - xi2 ) = åmi (yi2 + zi2 ) ;

Iyy = åmi (xi2 + zi2 ) ,	Izz = åmi (xi2 + yi2 ) ;	(8)
.
Ixy = Iyx = -åmi xi yi ;	Ixz = Izx = -åmi xi zi ; Iyz = Izy = -åmi yi zi .	(9)

(7) :

ìNx = Ixx ×wx + Ixy ×wy + Ixz × wz

= Iyx ×wx + Iyy ×wy + Iyz ×wz .

(7 )

íNy

= I

×w

+ I

×w

+ I

×w

ïN

: N

( )

j ,

	æI	xx	I	xy	I	ö
	ç	xx		xy		xz ÷
:	I = çIyx		Iyy		Iyz ÷.		(10)
	ç		Izy			÷
	è Izx		Izy		Izz ø

– : ,

, ,

. (

), :

( ) ( ).

=0, ,

, .

( 1, 2, 3).

(

), .

. ,

3 ( 1 2 3) –

( ,

). 2 , 3- – ( 1 2 3), ( ,

). 3 ( 1 2 3),

119

( ) (

. ,

, .

§3. .

.3). , .

zz. dm r

Z. (6) :

Izz = òr 2dm =ò(x2 + y2 )dm =ò(x2 + y2 )r(x, y, z)dV = rò(x2 + y2 )dxdydz . (11)

(dm)-

dxdy.

dV=2cdxdy,

	-
dm=rdV=r2cdxdy,
	Z.

dIzz =2 r( 2+ 2)dxdy.

. 3 zz 2 ,

2 dy:

æ a

dIzz = 2rc ò(x

+ y

dx + òy

)dxdy = 2rcdyç

òx

dx ÷

− a

æ a

ç x

a ÷

+ y x

+ y a - (-y a)

= (

rca

+ 4rcay )dy .

= 2rcdyç

− a ÷

= 2rcdyç

è 3

− a

è 3

Izz

rca3 + 4rcay2 )dy =

rc(a2a)2b +

rc2a(b2b) =

= ò(

− b

r ×(2a)(2b)(2c) ×(a2 + b2 )

Þ Izz

m(a2 + b2 ) .

(11 )

Ixx

m(c2 + b2 ) ,

Iyy

m(a2 + c2 ) .

(11 )

120

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