Васильев / Vasilev_mechanics_ru[1]

( ) ). , F = 0 , ' p = const.

' .

( , N ). "

" " Fij, " i - ' ), j -

"; - 6 Fi , i -

). . * ,

' . (

II & ) :

						= F		+ F	+......+F + F
			dp1
							12	13	1N	1
			dt
		dp2
						=	F21	+ F23	+......+F2 N + F2
			dt
	............................................
dpN = F							+ F +......+F			+ F
						N1		N2	NN−1	N
dt

* . 7, &

Fij + Fji = 0. # " '

): P = p1 + p2 + ... + pN. ( "

dP	N			.
	= F		= F
dt	i=1	i	&,8 &.

: ) " . ! '

:

.

" ' . $

						,
									'
P01			p02		pΣ ( .(( " ') ,
						6 ). $
							p01, - p02.
						$ 6 "
						pΣ. 7 ",
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,

	p01 + p02 = pΣ
	1.10 /
		. (, ,
	V	" " , ,
		, . , ' ,
		« + "» '.
		, ,
		: dp = F dt. dt
	v#$%	. $ m(t) - , v(t) -
		t, " p = m v. 2

dm (dm < 0),

dv. ( " p(t+dt) = (m + dm) (v + dv). 2

, ",

dt: dm3.!v3.!. ( "

dp = ( m + dm ) ( v + dv) + dm3.!v3.!- m v = F dt.

( ) , dm + dm3.!= 0 ,

mdv + dm( v − v#$%) = F dt
m	dv		dm				μ =	dm	- ' ,
	dt	= F +	dt	(v	3.!	− v) .		dt

u = v#$% - v - " . ( "

+ ":

m ddtv = F + μ u

μu .

% 9 ".

, , " F = 0. ( "

+ " ,

m dv	= −μu		;	1	dv = − dm	;ln m = −	v	+ C .
				u
dt					m		u

% ' C: t=0 m=m0 v=v0 .

			v − v0	; m = m0e−	v−v0				m0
	$ ln m − ln m0	= −			u	; v = v0	+ u	ln
									m
			u

1.11*

' ! F = mg.

" ). # ? & '

,

" . ( , - ,

. + . 0 "

, ' ' , " . $ '

, " ' ' . , ,

II & 0 = N + mg, III & P + N = 0 , " N - ) ,

x

N

mg

P - . P = mg ,

.

,

a, II &

ma = N + mg; P + N = 0 P0 = m ( g - a ).

. / :

1)',

P− = m (g + a) - ".

2),

- P↓ = m (g - a). , " g = a,

.

3)" , II &

) ' x y :

OX: 0 = Px - mg OY: ma = Py

( " Px = mg Py = ma ; P→ = Px2 + Py2 = mg2 + a2 . ( ,

, .

1.12 / . +.

$ ) F 1 - 2.

F ) ) ,

. /

2

dr

,

'

"

F

Fs

.

0

F dr

dr

'

,

αααα

F

F dr,

F

dr. ,

" F dr = F Cosads = Fs ds, "

1

a

-

"

F

dr,

ds =

½dr½-

, Fs - )

( ). %, dA = Fs ds . , " a ,

, " a , ). , " a p/2,

. * (") dA

1 2, F :

2 2

A= Fdr = Fsds

11

" " . % " Fs

Fs						) ) . $ ,
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						. % ,
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0					s	", " , 1 2
		1	2			s. $ " s
						(
						), " s -

( ) ). /

.

.) / " F = - k r , " r - - ) + .

	+	dr	$ ) +, ,
1		2	1 2. &
	r		F
				dr:	dA = F dr = - k rdr. *
r1	r2
			rdr = r (dr )r = rdr ,
0			dA = -k r dr = - d ( kr2/2 ). (
					, "

1 2:

2	kr 2				kr	2		kr	2
A = - d				=	1		-	2
		2			2			2
1

:) / " ) ( ) .

$ ' ) - ,

) + F, " ) ",

" F = (α/r2) er, " α -

( - γ m1m2 " ) kq1q2 ), r - ) +, er - -

( ).

dr dA = F dr = ( α/r2 ) er dr = α dr/r2 = - d ( α/r ). /

																1												2
			2		α	kq		q		(r			); A			2		α		γm	1	m		2	(r			)
A		= −		d	=		1		2			− r			= −		d		= −								− r
																				r r
					r	r r		2			2	1		"				r					2			2	1
			1			1										1				1
) / F = m g .
!						F = − m g ez , " ez									- ,

'. dr dA = F dr= - mg ez dr = - mgdz = - d ( mgz ) .

/ 1 2

2

A = − d(mgz) = mg(z1 − z2 )

1

, ) ) ,

' F = F1 + F2 + …, ,

" , '

									.	0,
A = (F1 + F2 +......)dr = F1dr + F2 dr+.....= A1 + A2 +.......
, ) *% 0: 1 0= 1 & * 1
	+- , ) ,
	d A							t 2
			F d r			d r		N d t
N =		=			= F		= F v ; A =
	d t		d t			d t
								t 1
*% ': 1 = 1 0* 1 -1 .
	1.13 # .
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, " ' , ( "). $ ,

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) 1 2, ,

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, ' , ) ,

1 2. *, , .

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	a					:
	1	2	,
			. 2 ,
			: 1 2
	b		2b1 ( ). ( " .
			A = A1a2 + A2b1. & ,

)

, dr1a= - dra1 , , A2b1 = - A1b2, A = A1a2 - A1b2 . $

' , ' A1a2 = A1b2 . !

. = 0. ( ' ' .

1.14 $) " ). * ) "

* ' ) Ep

(r), .			$
A12	2		) Ep0, .
		!	) " 1
1		Ep0 A10,
	A20	) 1 0 Ep1 =
A01		Ep0 + A10. 0 2 Ep2 = Ep0 + A20. ( " Ep1
	0	- Ep2 = A10 - A20. ( '
		, A01 + A12 + A20 = 0

Ep1 - Ep2 = A12 (DEp = - A12). ( , ) ' " '

) .

( F ) " " Ep(r).

, Ep(x,y,z), F(r). $ )

', " dA = F dr = Fx dsx, dsy = dsz = 0. ( " dA = Fx dx. dEp = - dA Fx dx = - dEp. ) ,

Fx = - dEp/dx dsy = dsz = 0.

:

F x	= -	∂ E	p
		¶ x

% ¶ d ,

,

. ." ) zy. ( "

¶Ep

ex +

¶Ep

ey +

¶Ep

F = Fx ex + Fy ey + Fz ez = -

¶y

¶z

ez .

¶x

0

) " ,

) "

F = - grad Ep.

"

grad º ex

∂

+ ey

∂

+ ez

∂

¶z

¶x

¶y

& gradEp l, ) Ep

, dEp/dl ,

) . & "

) " - grad º Ñ , " F = - grad Ep = - Ñ Ep .

1.15 # " )

$ ) m F (

F ' ). & ,

F ) dr. * '

F = mdv/dt dr = v dt,

dA = F dr = m v dv

* v dv = v ( dv )v, " (dv)v - ) dv

v. ) dv - . $ v dv = v dv

mv2

dA = m v dv = d . , F

2

	mv2
, ": E =	2 .

( , " )

dEk = dA , 1 2

Ek2 - Ek1 = A12

. $ ).

# " ' " ), '

. &

) , .

, ), )

(' ). $ .12 ' ,

', ', ' ). *,

& , .

1.16 ! ' " '.

' " ' . *

" ) ( ),

' ). % " ,

" ) F ' ,

' ). 2 ? , ) '

) ' , F

. # ", ) " " ,

'. & ' F . % , "

) . ( ,	F ' ,
' ), F = F + F . / ' '
" ):
E = . + . . $ ) ",
) " ) : . = -	E $

,p , Ek + Ep = ( Ek + Ep ) = A .

, ' Ek + Ep.

- ) " - '

" ) E = Ek + Ep. %, E2 - E1 = A .

" ): ' " )

' . '

, , , ,

' " . ,

, )

) ".

* ' " ):

, &

,

E = Ek + Ep = Const.

(' ' ' " ,

N ). / "

). $ , " '

1 2 , ,

) . / , ,

- . / ' '

' , ' )

" A = - Ep . , ) '

) ' , ,

) , " (

) , (

). * A '

" ' ' ' : . = . +

. . / ' )

" . = - , , " . = - , + . . *, " ) '

1 2 , = - , + . - , + . . , (, + , + , ) = . + . . ' "

), ' . , = , + ,

' " , ,’ = , + , + ,

' " ) ' . *

" "

' " :

E ©2 − E 1© = A + .

" ' " ,

; "

( " "). $ , '

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, ,

. $ , '

.

: " ) ' " " "

" '

":

' ,

( & .

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1.17 9 "

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) " . 9 ' ,

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. " , ' "

' '. ) "

' ) " " ). !

, " ". " )

" " ) '

, - ' ".

, -

. + m1 m2,

v1 , v2 u1 , u2 .

! '		m1v1 + m2v2 = m1u1 + m2u2
! '	"	m1 v12 + m2 v22 = m1 u12 + m2 u22
%		A2 - B2 = ( A - B ) ( A + B ) ,

AB = CD A = C , B = D ,

m1 ( v1 - u1 ) = − m2 ( v2 - u2 )

m1 ( v1 - u1 ) ( v1 + u1 ) = m2 ( u2 - v2 ) ( u2 + v2 )

v1 + u1 = u2 + v2 .

7 m2 " ,

2m

2

v

2

+ (m

1

− m

)v

1

2m

1

v

1

+ (m

2

− m

)v

2

u1

=

2

; u2

=

1

m1 + m2

m1 + m2

/ ' :

1)+ , " , u1 = v2; u2 = v1 .

2)( m2 , " " m1. ( "

m1 << m2

u1 = - v1 ; u2 = 0.

1.18 ( ) . ( # ".

$ N ' , '

r1, m1; r2, m2; .............. 9 ( )

)) , - rc

- :

									N
										m
		m		+	m		2 + . . . . . . .				i r i
	=		1 r1			2 r		=	i = 1
r c
			m 1	+ m 2			+ . . . . . .			N
									m i
									i	= 1

$ )

) ' :

m

+ m

+ . . .

=

d r

c

=

1

v

1

2

v

2

=

v c

d t

m 1 + m 2 + . . .

,

P

M

) :

		=	d P = F
M d v c
	d t		d t

( ) :

) ,

, -

, . $ :

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" .

( ". 5, ,

. & ' ,

" ' ' %* ". *

) mi

vi # #’,

' V . ( " 3 vi’ = vi - V

" ) # :

m

m

+ V)2

m

v2

m

V2

v2

(v

i

E

ki

=

i

i

=

i

=

i

i

+ m

i

v,

V +

i

2

2

2

2

i

Eki = Eki’ + 0.5 mi V2

+ ( pi’ V ) (*) , " pi’ - #’.

( N ' . 0

(*), ' . ) #’

P’ = m1 v1’ + m2 v2’ +...... = M vc’,

			1				1
E	k	= E′ +		MV2	+ P′V = E′	+		MV2	+ Mv′ V
		k	2		k		2			c

, #’ ) ) ( vc),

vc’ = 0 Ek = Ek’ + 0.5 MV2 (V = vc).

( # ": # " '

" , )

, "

) .

1.19 + , .

5, " ", . % " , , " -

, ' (

). 0					. +
					- *, "
*					M = F l = F r Sinα, "
					, r
					F. $ ,
O					"	-
		r		F	(" ,
		l	A	α	)
					M = [r , F]. ! "
					, ,
					r F
" . $ ( ).
					$ "
L					,
					. L
O					L = [r , p] = m [r , v]. ," L = m v
				p	r Sinα,	,
					. $
l			r	α
			A		. 2 ) ,
					,
					. /

' :

1). 2 )

p

m

r

l

O L

m

r

L v

. & ", " ' )

L . +

L = m v r Sinα= m v l .

l .

2). 2 ) r. + )

L = mvr. & L

) -

.

1.20 ! '

, '

L . 0 " )

:

	d L
		=	d r				+			d p
					, p				r ,
	d t			d t						d t

( ( ,

" 1.19),	dr/dt v	),
p ,

. 0, " &, dp/dt = F, " F -

' , ' . *,

dL / dt = [r , F]. , , -

, . %,

L )

M F

d L = M

d t

).

' ) . , -

) , ", )

'.

( , N ),

. & )

' ).

:

	d L	=		d L i
	d t			d t
			i

Fij

ri

O

N	N
=
i = 1	j= 1

rij

rj

	+
M ij & 7 (/		M i & ,8 &

(

III &, Fij = − Fji

Mij* = Mij + Mji = [ri , Fij] + [rj , Fji] = [ri - rj , Fij] =

Fji - [rij , Fij] = 0 ,

. ( "

: