Mechanika by Valdemar Dushinin

. .

1999

531 . . :

.–

. .

, 1999. – 132 . . ,

, .

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, .

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. , .

: . .

– , -

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;

–

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1999 ©

, 1999

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r r r e1 , e2 , e3 .

1,2,3 O(

. 1, a).

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O. (

. 1, b).

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. 1, b . (

. 2).

. X,

1 Y,

Z. .

r r r i , j ,k

. X

Z ,

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. (1564 - 1642) (1643 - 1727).

, .

, . , 1783),

(1717 - 1783), (1736 - 1813).

(1707 -

1862),

. (1847 - 1921),

. .

. . (1851 - 1918).

. .

(1801 ,

. . . . (1906 - 1966).

(1859 - 1935), . .

(1857 - 1935), . . .

. -

. ,

1. ยง 1.

, ,

. . .

, .

. .

. ,

, ,

. ,

: 1)

; 3) , -

; 2) .

, ,

. .

r F,

.3 A.

AB .

. -

B . ,

LM, . .3

, ).

. (1

1 ,

, ,

. . ,

, . . 3)

OXYZ ( XA, Y A, Z A

, -

, .

Fx, Fy, Fz . A, : F=

(1)

Fx2 + Fy2 + Fz2 ;

r r Fy r r r r F F cos(i ∧ F ) = x ; cos( j ∧ F ) = ; cos( k ∧ F ) = z . F F F (2)

(2) -

, . , . .

r r r ( F1 , F2 ,..., Fn ) , , , . r r r ( F1 , F2 ,..., Fn )

. ,

r r r ( P1 , P2 ,..., Pk )

, .

r r r r r r ( P1 , P2 ,.., Pk ) ~ ( F1 , F2 ,.., Fn ) .

, r r r ( F1 , F2 ,.., Fn ) ~ 0 . . .

. , . ,

r R* ,

r r r ( F1 , F2 ,..., Fn ) .

. ,

, ,

, . .

. . , ,

. . . , . , ,

. .

. , . ,

. r4. r ( F1 , F2 ) ~ 0 . )

, (

. )

( ,

. .

, . . r5 FA .

r Fr , FA

B. , FA. -

r FB '

FB’

. ( .5), r r r ' r ( F A , FB , FB ) ~ F A . , ,

, ,

r FA

r FB '

r r ( F A , FB ' ) ,

r r F A ~ FB . B.

r F,

r r r (FA , FB , FB ' )

r FB

r F

B . .

. , ,

r r r ( F1 , F2 ) ~ R * ( ,

, 6).

. -

, ,

.6 r r F12 + F22 + 2 F1 F2 cos( F1∧ F2 ) .

. , r* r r R = F1 + F1 ; R* =

. .

, (

r R*

. 6),

, , . ,

, . 7.

(3-

). .

, .

, ,

. . 8

, ,

r, F1 -

. , 1

r F2 -

2. .

(

). , (

, ,

. .

, .

, ,

, .

(

. ;

;

). ;

;

, . ,

. ,

. . , .

. ,

. . 1.

, ,

. ,

. 2. 3.

. ,

. . .

, -

, ,

. . 9, a

. ,

. .

r N,

. , .

. 9, b .

, .

. .

. 9, c,

. . .

.9 A .

B D

. .

. . .

(

)

( , , .

. 9,

, (r r r N A, NB, ND , . 9, d), , .

). -

. . -

. ( ,

. 10, a)

(

. 10, b)

. , ,

A. ,

. .

A ,

, - A.

r RA r r X A , YA , 10, a.

. -

. 10 ,

. , A. .

, r r r X A , YA , Z A . A , -

, ,

(

. 10, b)

. ,

ยง2.

. 3. .

r r F1 , F2 . 11, a).

r . F3 ,

r r F1 , F2

A (

. 11, b). , -

, r ,r r R = F1 + F2 . ,

r R

r F3 (

. 11 . 11, c).

r R

, .

r F3 -

, .

. ,

4. .

, . ,

. r r r ( F1 , F2 ,..., Fn )

( ,

. . 12, a).

A ,

(

. 12, b).

14 . 12

, ,

r F2 ,

r Fn ,

(

r R* (

. -

. 12, b)

. 12, c),

. , (

: r r r r* r* r r r ( F1 , F2 ,..., Fn ) ~ R ; R = F1 + F2 + ⋅⋅⋅ + Fn . . .

(1) ,

, : n r r r r F1 + F2 + ⋅ ⋅ ⋅ + Fn = ∑ Fi = 0 .

(2)

i =1

(1). r r r ( F1 , F2 ,..., Fn ) ~ 0 .

r R* = 0 ,

(2) .

(2)

, .

. , -

. (2)

. 13), . -

( r R*

r R* = 0 ,

, ,

. 13

, , .

. . OXYZ,

. (2),

F1x + F2 x + ⋅ ⋅ ⋅ + Fnx = 0 ; F1Y + F2Y + ⋅ ⋅ ⋅ + Fny = 0 ; F1z + F2 z + ⋅ ⋅ ⋅ + Fnz = 0 .

, n

∑ Fix

∑ Fiy

= 0;

i =1

∑ Fiz

= 0;

= 0.

(3)

i =1

. ,

XOY,

0 ≡ 0.

(3)

: n

∑ Fix

∑ Fiy

= 0;

i =1

(4)

= 0.

i =1

α,

O. T

. , . 1.

, .

. 2.

r P,

. -

O. r N.

. .

, ,

. . 14 , .

, 3. . .

r r T N . 14, c

(

( . 14, a) . . 14, b),

, ,

r Q.

r - T/ T/ = T ,

r N,

r T N.

, Q= N.

, .

, . 14, a. ,

. (4)

( )

. 5.

. ,

, -

. ,

r N

r T,

r P.

, (

. 14, d).

r N

, . .

. 14, r b r P T

r T

α.

P = cosα ; T

N = tgα ; P

P ; cosα

Q = N = P ⋅ tgα .

OY (

. 14, a). ,

OX (4)

OXY, n

∑ Fix = 0 , − N + T ⋅ sin α = 0 ;

i =1

∑ Fiy

= 0 , − P + T ⋅ cosα = 0 .

i =1

, : T=

P sin α ; N = T sin α = P = Ptgα . cosα cosα

( . 15) P,

, B.

1. A

r D. P.

. .

. 15 ,

. ,

D .

, , r P

r ND

C. A.

, C.

r RA 3. 4.

, .

r RA

, ,

r ND . .

. 5. ,

(

. 15).

, ,

. 15.

P ; cos α CD a 1 ADC tgα = = = . AD 2a 2 AD 2 . AC = a 2 + 4a 2 = a 5 cos α = = AC 5 ,

N = Ptgα .

r RA

RA =

P P 5 . ; ND = 2 2

: RA =

. AXY,

(4) ∑ Fix = 0 , R A cosα + P = 0 ;

∑ Fiy

AY -

: = 0 , R A sin α + N D = 0 .

RA = −

N D = Ptgα . RA = −

P 5 P . ; ND = 2 2

”- ”

P cos α

;

α,

. 15.

§ 3.

. 5.

(

. (

)

(

. 16):

r r r r mO ( F ) = r × F . -

(1) LM, .

. ,

, .

r r r mO ( F ) = rF sin( r ∧ F ) .

, r r , h = r sin(r ∧ F ) ( r mO ( F ) h= . F ,

r F

. 16 . 16), (2)

r r mO (F ) ,

. -

. . ,

, ,

. .

. 16 -

. ,

, O1 AB .

r r r r mO1 ( F ) ≠ mO ( F ) . 6.

(

, (

. 17, a).

”+”,

, , ”-” .

( . 17, b), ”+” , -

. 17 , .

”-”

, ,

r m A ( F ) = ± Fh . ,

(3) -

, .

, h,

. ,

, ,

XOY, ,

. , ,

, -

, ( r . 18). OZ, F -

, .

, r r mZ ( F ) = mO ( F ) = ± F h .

(4) -

, ”+”.

”-”. .

. .

. 18

. OZ (

. 19).

h=h ,

F =F /. r r r mZ ( F ) = mO ( F ) = mO / ( F / ) ,

r F1

OZ ( ,

r F2

. 19).

, .

r mZ ( F1 ) = 0 . .

, . OZ.

r F2 , r mZ ( F2 ) = 0 .

h2 = 0 .

, .

. 19

. ,

. . 19

r F3 ,

. r r mO (F3 ) = mZ (F3 ) .

. , r r mOZ ( F ) = mZ ( F ) , r r mO ( F ) Z.

r mOZ ( F ) -

Z, (5)

.r . 20 A F r r Z. mO ( F ) h, OAB , , r . mO ( F ) = Fh = 2 S Δ OAB , r r mOZ ( F ) = mO ( F ) cos α = 2 S Δ OAB ⋅ cos α , α Z ( . 20), OAB. r / / / mZ ( F ) = F h = 2 S Δ OA / B / . , Δ OA B .

Δ OAB

β -

r r

Z (6) S Δ OAB -

S Δ OA / B / = S Δ OAB ⋅ cos β ,

h/ .

. 20

, r mZ ( F ) = 2 S Δ OAB ⋅ cos β . r r mO ( F )⊥h , OZ⊥h / . α=β . (6) (7), r r , mOZ ( F ) = mZ ( F ) , , . OZ O. OX OY, -

(7) -

r r r r r r mOX ( F ) = m X ( F ) ; mOY ( F ) = mY ( F ) ; mOZ ( F ) = mZ ( F ) .

(8)

. -

, .

, , ,r r r r ( FX , FY , FZ , m X ( F ), mY ( F ), mY ( F ), mZ ( F )) .

, .

, . ,

, -

. ,

. 9. .

. (1). -

OXYZ.

, r i r r r r mO ( F ) = r × F = x FX

, : r k r r r z = ( yFZ − zFY )i + ( zFX − xFZ ) j + ( xFY − yFX ) k . FZ

r j y FY

(8),

OXYZ: r r r r (9) m X ( F ) = yFZ − zFY ; mY ( F ) = zFX − xFZ ; mZ ( F ) = xFY − yFX . , , , , . , , , , , , . 1. a, b, c F, . OXYZ. , . . 21 r , , r F / / OX , mX ( F ) = 0 . , , OY, . 21

r r mY ( F ) = mO1 ( F ) = ± Fh1 = − Fc .

r mY ( F )

”-”

OY .

r F,

OZ, r r mZ ( F ) = mO2 ( F ) = ± Fh2 = Fb . , r r r m X ( F ) = 0 ; mY ( F ) = − Fc ; mZ ( F ) = Fb .

”+”.

x = a , y = b , z = c.

, FZ = 0 .

OZ, OX r F XY

XOY,

r F XY

OZ ,

OY.

, F XY = F ; FX = − F ; FY = 0 . r : mX ( F ) = b ⋅ 0 − c ⋅ 0 = 0 ;

. 21

(9), r r mY ( F ) = c ⋅ ( − F ) − a ⋅ 0 = − Fc ; mZ ( F ) = a ⋅ 0 − b ⋅ ( − F ) = Fb . § 4.

. 10.

. -

(

. 22).

h, ,

, .

. ,

. 22

. . ,

. .

(

)

. , .

, ,

. . ,

(

. 23).

r r ( F1 , F2 ) , r , r R1 R2 (

r r (Q1 , Q2 ) ~ 0 . B,

A . 23, a).

, b).

r C( R2 r r (Q1 , Q2 ) . , . 23, d). , -

r R1

, (

. 23, -

. 23, c), ,

r r r r ( F1 , F2 ) ~ ( F1 , F2 ) C ( , , ( . 23, c): r* r* r r r r * ( F1 , F2 ) ~ R ; R = F1 + F2 ; R = F2 − F1 .

(1 ) ,

D AB.

, ,

. 23 R * = F1 + F2 ,

AB. n

, . . .

. 22, a

. AC

F1 → F2 BC

(1)

R* → 0 ,

, C. ,

. . -

, , , .

. 24

, .

. 24)

, .

. ,

rr F ′ = −F ,

r r r r r r r r r r r M (F , F ′ ) = mO (F ) + mO (F ′ ) = r1 × F + r2 × F ′ = (r1 − r2 ) × F .

(2)

BA , . 24

r r r1 , r2

r r r1 = r2 + BA

(2)

r r r1 − r2 = BA .

B, ,

r r r r M ( F , F ′ ) = BA × F .

(3)

. O

(

r F)

, (

). ,

(

. 24. (3),

r BA ⋅ sin( BA F ) = h : r r ∧ r M ( F , F ′) = BA ⋅ F sin( BA F ) = Fh . O

. 24),

∧

(2)

r F -

(4) A

r r r r r r r M ( F , F ′ ) = mB ( F ) = m A ( F ′) .

(5) ,

. ,

, . ( ,

, . . 1.

. 2.

, . ,

, ,

. .

, . .

, , .

, .

, ,

r r M ( F , F ′) = ± Fh .

(6) ,

, . r r r r ( F , F ′) ~ ( P, P ′) ,

r r r r r r M ( F , F ′) = M ( P, P ′) . . , .

F ≠ P,

, . .

, 2-

11.

. ,

. . .

r r r r, r r r r : (( F1 , F1′),( F2 , F2′ ),...,( Fn , Fn′ )) ~ ( F , F ′) ; r r r r r r r M ( F , F ′) = M = M1 + M 2 + ⋅ ⋅ ⋅ + M n , r r r r r r r r r r r r M1 = M ( F1 , F1′) , M 2 = M ( F2 , F2′ ) ,..., M n = M ( Fn , Fn′ ) .

(8)

. 25 . 25, a

. , (

, -

. 25, b). r r ( F , F ′) ,

r M

r r r r(8). M = M ( F , F ′) (

. 25, c).

. r r r r r r r r r r (( F1 , F1′),( F2 , F2′ ),...,( Fn , Fn′ )) ~ ( F , F ′) ; M ( F , F ′) = M = M 1 + M 2 + ⋅⋅⋅ + M n . . , .

(9)

: n r r M = ∑ Mi = 0 .

(10)

i =1

r r r r r r , (( F1 , F1′),( F2 , F2′ ),...,( Fn , Fn′ )) ~ 0 .

, .

r r ( F , F ′)

r M = 0,

(8).

r r ( F , F ′) ~ 0

r M ≠ 0, , -

(10) . ,

(10)

. , , : n

M = ∑ Mi = 0 .

(11)

i =1

, ,

§5.

, , .

. 12.

. , ,

. 26 ,

, .

r F1 ,

(

A B,

(

r F1

r F1′

r r mB ( F1 ) :

r r ( F2 , F1′) ~ 0 ,

r F1

. 26, a). r F2

. 26, b). r r r r r F1 ~ ( F1 , F2 , F1′) . F1

(5)

. 10

r r r r r r r r r F1 ~ ( F2 ,( F1 , F1′)) ; M ( F1 , F1′) = mB ( F1 ) , . ( . 26, c), B.

. .,

(1)

13. (

. , : n r r r r r R = F1 + F2 + ⋅ ⋅ ⋅ + Fn = ∑ Fi .

(2)

i =1

, : n r r r r r r r r r M O = mO ( F1 ) + mO ( F2 ) + ⋅ ⋅ ⋅ + m( Fn ) = ∑ mO ( Fi ) .

(3)

i =1

. , ,

, ,

, :

n r r r r r n r r r r r r r r r ( F1 , F2 ,..., Fn ) ~ ( R,( F , F ′)) ; R = ∑ Fi ; M ( F , F ′) = M O = ∑ mO ( Fi ) . i =1

(4)

i =1

r r r ( F1 , F2 ,..., Fn ) . O.

, .

. ,

r R. O,

r MO.

r r ( F , F ′) ,

. -

, .

)

(

. 14.

r MO

r M O′

, r - M O′′ -

, (

. 27, a). ,

(

. 27, b).

, ( )

. 27 ( AL

. ) .

. ,

r r M O′ 1 = M O′ ,

(

r M O′ , -

, . 27, c d). , r M O′

O1 , -

, . .

15. .

r, R′

, ,

r r ( F , F ′) , r - R ′′ -

r r ( F , F ′) ,

r P′

. 28 (

r R′ (

. 28, a).

r r ( P, P′)

. 28, b). ,

(

.r28,r c), r. . r r ( F1 , F2 ,..., Fn ) ~ ( R ′′, P) .

16.

(5) (

, . ,

, -

, .

, . 29

, r r ( F , F ′) ,

r R *,

, r r ( R*, R′) ,

r R′

(

r R(

r MO

. 29, a). -

. 29, b).

. 29, c). ,

n r r r r r r r r mO ( R*) = M ( F , F ′ ) = M O = ∑ mO ( Fi ) .

(6)

i =1

, (6)

, . (6)

. , r r m A ( R*) = ∑ m A ( Fi ) , n

(7)

i =1

A-

O Z,

: r r mZ ( R*) = ∑ mZ ( Fi ) . n

(8)

i =1

, . , .

, ,

, -

. §6.

, . 17.

: n r r r r r n r r r r r r r r r ( F1 , F2 ,..., Fn ) ~ ( R,( F , F ′)) ; R = ∑ Fi ; M ( F , F ′ ) = M O = ∑ mO ( Fi ) . i =1

i =1

(1)

, :

r r r ( R ,( F , F ′)) ~ 0

r r R = 0 ; MO = 0 . r r r (2) ( F1 , F2 ,..., Fn ) ~ 0 .

(2) (1).

(2)

. ,

, .

(2),

. (2) -

. OXYZ. -

(2).

, .

: n

R X = ∑ Fix = 0 ;

RY = ∑ Fix = 0 ;

RZ = ∑ Fiz = 0 ;

r M X = ∑ m X ( Fi ) = 0 ;

r M Y = ∑ mY ( Fi ) = 0 ;

r M Z = ∑ mZ ( Fi ) = 0 .

i =1

(3)

. , .

(3)

, r r , r r M O1 = mO1 ( R ) + M O .

(2), r M O1 = 0 .

r R=0

O1 ,

18.

, (3) .

(

. . 30, a),

, ,

OZ (

(0 ≡ 0) ,

.30, b).

, ,

(3),

n r r RZ = ∑ Fiz = 0 ; M X = ∑ m X ( Fi ) = 0 ; M Y = ∑ mY ( Fi ) = 0 . n

i =1

(4)

i =1

. -

. 10

, -

. . 30 .

r MO = 0,

, ,

(3)

, : n

i =1

R X = ∑ Fix = 0 ; RY = ∑ Fix = 0 ; RZ = ∑ Fiz = 0 .

(5) . -

, (

XAY ,

, .31, a).

AZ -

, XAY ( . 13). , -

(3) . 31

. (3)

( .

13) n r r m F ( ) = ∑ Z i ∑ m A ( Fi ) . n

i =1

∑ Fix

= 0;

i =1

∑ Fiy

= 0;

i =1

∑ m A ( Fi ) = 0 .

(6)

i =1

, .

(6) .

, .

I-

. , , :

r ∑ m A ( Fi ) = 0; n

i =1

II-

r ∑ mB ( Fi ) = 0 ; n

∑ mC ( Fi ) = 0 .

i =1

(7)

i =1

, ,

, r

: n

∑ m A ( Fi ) = 0 ; ∑ mB ( Fi ) = 0 ; ∑ Fix

i =1

= 0.

(8)

i =1

(

). .

. (

.31, b).

(6)

, :

∑ Fiy

= 0;

i =1

r ( m F ∑ A i ) = 0. n

(9)

i =1

. (

. 31, c).

(6)

( ) : n

∑ Fix

i =1

= 0;

∑ Fiy

= 0.

(10)

i =1

, (

)

, .

, -

, . ,

. . , (

19.

. ,

, :

1. ,

, .

, 2.

, , , .

I-

II-

. , , . , ,

. .

, .

. , -

, ,

. 24

, -

, .

(

) ,

, ,

. ,

, . ,

, . . . , (

, -

) ,

. ,

. , . AD,

Q. CD α = 60°. .

, A

, . r A Q;

( .8);

C r RA

Br NB , r RC

(

( ).

: -

( .8) ( . 32, a). , -

, A

. . 32 . .

T = T′ .

(a)

(

. 32, b) r YA ,

r XA

AXY, A

r NB .

AX . -

, ,

A: n

∑ Fix

i =1

∑ Fiy

i =1

= 0 , X A + T cosα = 0 ;

(b)

= 0 , YA + N B − T sin α = 0 ;

( )

∑ m A ( Fi ) = 0 ,

N B ⋅ AB − T ⋅ AD ⋅ sin α = 0 .

, (

(d)

(d). . 32, c)

CXY, C

r T′.

CY ,

T′,

r XC

r YC . ,

r (e) ( m F ∑ C i ) = 0 ; T ′r − Qr = 0 . (e) r . , , r (a) r -r (e) r r r r X A , YA , N B , T , T ′ , X C , YC ( . 32), 7 . 7 . , . T′ = Q , , (a), T = Q. (e) , , . , , , , . . . , , , , , . T=Q (b) - (d), , AD = 2 ⋅ AB α = 60°. , Q 3 N B = 2Q sin α = Q 3 ; X A = − Q cosα = − ; . YA = Q sin α − Q = − Q 2 2 “-“ XA YA , . , . 32, b

RA =

X 2A + YA2 = Q .

, .

I-

II-

r T

r r r X A , YA , N B ( ). A B. r r YA NB . II, : r m ( F ∑ A i ) = 0 , N B ⋅ AB − T ⋅ AD sin α = 0 ; r ∑ mB ( Fi ) = 0 , − YA ⋅ AB − T ⋅ BD sin α = 0 ; ∑ Fix = 0 , X A + T cosα = 0 . , Q). , BD = AB . , . , . , , , , A E r . 33 NB ( . 33). , ∠AEB = α , ABE BDE : , ∑ Fix = 0 , R A cosα + T cosα = 0 ; ∑ Fiy = 0 , R A sin α + N B − T sin α = 0 .

, R A = −T = −Q . , 2.

“-“

. 33. ABD A.

P= 300 ,

α 1 ,

r RA

D 30°. 50

, , A, ,

1000

AB = 2 , BD =

. .

r Q,

, . AB

C2 ( ,

. 34). P1 = 2/3P = 200

BD, ,

P2 = 1/3P =

100 .

, . 34. , -

, . , .

. 34

, . ,

. ,

r XA

r YA , ,

, MA ( A

. 34).

, -

AZ,

. , .

. 34 . , . 34,

. ,

. X A , YA

M A,

. ,

: ∑ Fix = 0 , X A + Q sin α = 0 ;

∑ Fiy r XA

r YA .

. QY = Q cosα .

(a)

= 0 , YA − P1 − P2 + Q cosα = 0 .

A, r Q .

r Q

(b)

r QX

r r r m A ( Q) = m A (Q X ) + m A (QY ) .

r QY , Q X = Q sin α ; hX

. 34.

: hX = BD = 1 :

;

hY = AB = 1 . , r ∑ m A ( Fi ) = 0 ; − P1 ⋅ AC2 − P2 ⋅ AB − Q sin α ⋅ BD + Q cosα ⋅ AB − M + M A = 0 . ( ) ( ) , , , , . , , α =30°, AC2 = BD = AB / 2 , X A = − Q sin α = −1000 ⋅ 0,5 = −500 ; YA = P1 + P2 − Q cosα = 200 + 100 − 1000 ⋅ 0,87 = −570 ; AB AB M A = P1 + P2 ⋅ AB + Q sin α ⋅ − Q sin α ⋅ AB + M = 200 ⋅ 1 + 100 ⋅ 2 + 1000 ⋅ 0,5 2 2 − 1000 ⋅ 0,87 ⋅ 2 + 1000 = 160 ⋅ . , “-“ X A YA , . 34. , , A , , . α = 30° 3. F1 = 100 . F2 , , . , l ( ) 0,1 . , . 35, a . .

r N1

r N1 ,

. -

. 35, a r F1

r F2 .

OXY.

, r r F1 , F2

r N1 .

. ,

. -

. 35

r X (

. XOY.

. 35

M -

). -

(

. 35

OZ,

). . ,

. ,

(

, . 35, a: ∑ Fix = 0 ; F1 + X = 0 .

( )

. 35, b).

N 2 − N 2′ = 0 . ,

N 2′ = N 2

(b)

. 35, b: ∑ Fix = 0 , N 2 sin α + X = 0 ;

( )

∑ Fiy = 0 , N 2 cosα − F2 = 0 ; ∑ m A ( Fi ) = 0 , − X ⋅ l + M = 0 . .

r N 2′ .

(e)

r N1 .

r F1

(d)

. 35, b

: ∑ Fiy = 0 ; N1 − N 2′ cosα = 0 .

(f) . -

. (a) - (f) (

) : X = − F1 ; M = − F1 ⋅ l ; N 1 = F1ctgα ; F2 = F1ctgα . , -

N 2 = N 2′ = F1 / sin α ,

X = −100

, ; M = −100 ⋅ 01 . = −10

: ; N 1 = 100 ⋅ 173 . = 173 . , . 35. .

F2 > F1 . 1.73

, F1 .

, , 1 > sin α ⋅ cosα

1 / sin α > ctgα α, α = 30°

F1 .

; F2 = 100 ⋅ 173 . = 173 . N 2 = N 2′ = 200 . ″- ″ X M , , , α < 45° α = 30° F2 ,

. , ,

. 36 ,

l. .

r X′

, F = F′ =

M l = F1 , d d

. 36r F

r F′ ,

l >> d

. F1 ,

. . . . 37, a

r r ( F1 , F2 ).

. 37, b. . , r YA′

r YB′ .

, (

37, c). : (

); ).

( (

)

, , , . 37 -

. . . ,

, .

P = 100

B (

A CE,

38). C ∠ECA = ∠BAC = 30°.

A, . ABCD.

r P,

. A.

AB.

r P,

, -

A r r r X A , YA , Z A .

AY),

BY,

( r r X B , YB .

, BX

. 38. , r ,r X A, XB

AY XAY

AY OZ

B. A,

. 38

r r Z A , ZB .

r T

EC,

, ,

, .

, . .

, r TZ .

r TXY ,

r TXY -

r TXY

r T

. r T

XAY, TZ = T sin 30 ° TXY = T cos30 °. r r TX TY .

TX = TXY sin 30 ° = T cos30 ° sin 30 ° TY = TXY cos30 ° = T cos2 30 °. , : (a) ∑ Fix = 0 , X A + X B − TX = 0 ; (b) ∑ Fiy = 0 , YA − TY = 0 ;

∑ Fiz = 0 ,

Z A + Z B + TZ − P = 0 .

( ) , .

r r, r r X A , YA , Z A , TY ( r TY (

r ZB

r. r X B , TX (

) r r r r r r ). AY - X A , YA , Z A , X B , Z B , TX ( r r r r ). AZ - X A , YA , Z A , T ( ). , AX: r ∑ m X ( Fi ) = 0 ; Z B ⋅ AB + TZ ⋅ CD − P ⋅ KM = 0 . YAZ,

, CD

)

AX,

r D. P

“+“. AX ( KM). M

XAY

. 38 -

AX.

“-“. ,

AY : r ∑ mY ( Fi ) =r 0 , − TZ ⋅ BC + P ⋅ KL = 0 ; ∑ mZ ( Fi ) = 0 , − X B ⋅ AB = 0 .

(e) (f)

r T

(f) AZ. XAY . 38

, AY

r P,

CD = AB

AD, AB = 2KM AD = 2KL. XB = 0. ( ) (e) (f) , − TZ ⋅ 2 KL + P ⋅ KL = 0 ; Z B ⋅ 2 KM + TZ ⋅ 2 KM − P ⋅ KM = 0 . ZB = 0. (c) Z A = 0.5 P . TZ = 0.5T ; TY = 0.75T ; TX = 0.43T , T = P. X A = 0.43 P , YA = 0.75 P . : T = 100 ; X A = 43 ; YA = 75 ; Z A = 50 , ,

(a) ,

, . .

BC =

: TZ = 0.5 P ; r T, : (b) , , -

; X B = ZB = 0 .

KL.

(a-f)

§7.

r ZB

KM,

)

(d)

r P

r r ZB , P (

AX).

CD,

AX,

r “+“ ( Z B

AB,

AX -

, . 20.

16, , r r r r ( F1 , F2 ,..., Fn ) ~ R ∗ . , .

, ,

O1 -

. O1 , r Fi (

r r -

r ri -

. 39).

O1 . -

ρi ,

r Fi .

r r r r m ( F ) = ρ × F ∑ O i ∑ i i. r r r 1r r r ri = r + ρi ρi = ri − r ,

. 39

n r r r r r ( ) ( ) = − × m F r r F ∑ O1 i ∑ i i. n

i =1

r e,

(1)

i =1

r r Fi = Fi′e (i = 1, n) ,

Fi′ -

(2) ,

, F1′ (1)

. 39,

F2′

(2), ,

. .

r e

n n r r r r r r r r m F r r F e ( ) = ( − ) × = [ ∑ O1 i ∑ i ∑ Fi′(ri − r )] × e . i′

i =1

(3)

i =1

r R∗ :

n r r r r ∑ mO1 ( Fi ) = ∑ mO1 ( R ∗ ) = 0 . n

i =1

(4)

(3)

r r r [ ∑ Fi′( ri − r )] × e = 0 .

(5)

i =1

e ≠ 0,

r e

: n r r r rn − = F r r F r ( ) ′ ′ ∑ i i ∑ i i − r ∑ Fi′ = 0 . n

i =1

(6)

i =1

(6)

r r,

, . C(

. 39),

(6) n

r rC =

r rC :

∑ Fi′ri

i =1 n

(7)

∑ Fi′

i =1

OXYZ

(7): n

∑ Fi′xi

XC =

i =1 n

∑ Fi′yi

; YC =

∑ F′

i =1

i =1 n

∑ Fi′

∑ Fi′zi

; ZC =

i =1 n

(8)

∑ Fi′

i =1

(8) xi , yi , zi ,

r e,

, ( X C , YC , Z C ) . .

21.

(8). r pi ,

. -

. , r r r ( p1 , p2 ,..., pn )

, (

. 40

, . 40,

r P

, n

P = p1 + p2 + ⋅ ⋅ ⋅ + pn = ∑ pi .

(9)

i =1

r pi

, .

r P

. 26 C,

. ,

, .

(7), (8) , p i′ = p i

r e

r rC =

i =1 n

i =1

∑ pi

i =1 n

i =1

∑ pi

i =1 n

∑ pi

i =1

, pi = γvi

. γ ∑ pi = γ ∑ vi = γV , (10) (11),

: r 1 n r rC = ∑ vi ri ; V i =1 XC =

(

(11)

i =1

V-

∑ pi zi ∑ pi zi

; ZC =

∑ pi yi

(

(10)

∑ pi

i =1 n

∑ pi ri

∑ pi yi

; YC =

r ∑ pi ri

i =1 n

∑ pi xi ∑ pi xi

XC =

. 40)

: n

(

1 n 1 n 1 n v x Y v y Z = ; = ; ∑ i i C V ∑ i i C V ∑ vi zi . V i =1 i =1 i =1 , . . ( , , , ) . (12) (13) .

(12) (13) C

. .)

, :

(13) n

1 1 1 si xi ; YC = ∑ si yi ; Z C = ∑ si zi . ∑ S i =1 S i =1 S i =1 , (14) XC YC zi = 0 ZC = 0 , . , li L , : 1 n 1 n 1 n X C = ∑ li xi ; YC = ∑ li yi ; Z C = ∑ li zi . L i =1 L i =1 L i =1 , pi = γ i vi , γi , vi (11), . . XC =

(13)

22.

(14)

OXY , -

(15)

. . 21

, .

. .

. . 21

, .

, ,

n → ∞,

, ,

γ,

γi

, ,

, -

. . 21, :

XC =

1 1 1 γ ( x , y , z ) xdv ; YC = γ ( x , y , z ) ydv ; Z C = γ ( x , y , z ) zdv . ∫ ∫ P (V ) P (V ) P (V∫ )

( )

: XC =

1 1 1 xdv ; YC = ydv ; Z C = zdv ; ∫ ∫ V (V ) V (V ) V (V∫ )

(17)

1 1 1 xds ; YC = ∫ yds ; Z C = ∫ zds ; ∫ S (S) S (S ) S (S)

(18)

XC =

(16)

1 1 xds ; YC = ∫ yds ; ∫ S (S) S (S)

(19)

1 1 1 xdl ; YC = ydl ; Z C = zdl . ∫ ∫ L ( L) L ( ∫L ) L ( L)

(20)

XC = XC =

(17-20) dv , ds, dl -

x, y, z -

(16)

∫ γ (x , y , z)dv .

(21)

(V )

. l ( OX. (20)

. 41, a) dl dx,

l 1 1 = ∫ xdl = ∫ xdx = . l ( L) l0 2

, a, ds XC =

, YC ,

b( x a

. 41, b). y,

(19)

1 1 1 1 a 2b = = = ⋅ . xds xdxdy xdx dy ∫ S ( ∫S ) S ∫∫ S ∫0 S 2 0

. 41 XC = a / 2 .

S = ab , b / 2. ,

ds = ρdϕ ⋅ dρ ,

OY dxdy,

ρ dρ ; dϕ -

R OY

(19)

(

. 41, c).

ϕ. x = ρ cosϕ , y = ρ sin ϕ ,

(19):

π /2

1 1 1 2 = ∫∫ ρ 2 dρ cosϕdϕ = ∫ ρ 2 dρ ∫ cosϕdϕ = ⋅ R 3 ; S S0 S 3 −π / 2 π /2

1 1 YC = ∫∫ ρ 2 dρ sin ϕdϕ = ∫ ρ 2 dρ ∫ sin ϕdϕ = 0. S S0 −π / 2

, = 4 R / 3π = 0.42 R ,

πR 2 / 2 ,

YC = 0 .

, ,

, . , 0 γ m , . . γ = kx ,

k =γm /l.

. 42

r pm (

. 42).

(21) P=

∫ γ ( x)dl = ∫ kxdx = ( L)

kl 2 γ m l = , 2 2

(16) XC

l 1 1 1 kl 3 1 γ m l 2 2 2 = = = ⋅ = ⋅ = l. γ ( x ) xdl kx dx ∫ ∫ P ( L) P0 P 3 P 3 3

, .

, H. 1,

( 2,

. 43). AA

. 43

C, AA.

(

, H /2

. 43). ,

. .

, .

. AD (

ABD . 44).

BC1 ,

DC2 , .

C CC1 = BC1 / 3 ,

. CC2 = DC2 / 3 .

. 44

, R r H (

. 45). -

. OZ

OZ -

OXYZ

. ,

. 45 . 45 z1 = H / 2 ,

z 2 = 3H / 2 ,

X C = YC = 0 . ZC , (13) z1 , z2 -

v1 = πR 2 ⋅ H

v2 = πr 2 ⋅ H .

. ,

R = 2r ,

V: v1 = 4πr ⋅ H ; v 2 = πr ⋅ H ; V = v1 + v2 = 5πr 2 ⋅ H . (13), 2

v1z1 + v2 z2 2πr 2 ⋅ H 2 + 3πr 2 ⋅ H 2 / 2 7 = H. ZC = = V 10 5πr 2 ⋅ H ZC < H , . . , , , . , . ( , -

. .). ,

(10 - 14)

, “-“. -

R = 2r (

. 46).

C1 X ,

C1 -

. 46

. .

C1 X ,

YC = 0 .

(14),

XC s2 :

“-“ XC =

s1 x1 − s2 x 2 . s1 − s2

s1 = πR 2 = 4πr 2 ; s2 = πr 2 ; x1 = 0 ; x 2 = r . XC =

0 − πr 2 ⋅ r

r R = − = − . 3 6 4πr 2 − πr 2 C ,

(

. .)

. ,

, .

. , .

XC (

. 47).

AXY (

AB r = l )r N1 N2 , N1 N2

N2 ,

P = N1 + N 2 , = N 2 l / ( N1 + N 2 ) . .

AY C

N1 .

. 47

. 23. ,

. ,

, ,

. . ,

. 48

(

. 48, b). ,

(

. 48, c),

(

. 48, a).

, . . . ,

, ,

, . ,

: qi =

Fi F F F F F , q = lim i ; qi = i , q = lim i ; qi = i , q = lim i . vi si li vi →0 vi si →0 si li →0 li

(22)

(22) qi -

, q , vi , si , li -

Fi -

. ,

c, ,

. 48, a ,

, q = q( x, y, z) .

( (

); );

/ /

: ,

(

). ,

, -

. ,

. .

r R∗ . ,

, ,

, . ,

, , (16)

(21) -

XC =

R∗

∗

∫ q ( x, y , z) xdv ; YC (V )

1 R∗

R∗ =

∫ q ( x, y , z) ydv ;

ZC =

(V )

1 R∗

∫ q ( x, y , z) zdv ; (V )

∫ q ( x, y, z)dv .

(23) (24)

(V )

(23)

(24)

. , -

, , . ,

. , ,

r N

, (

. 48, c)

N = qab .

b, ,

, .

r R1∗

q (

. 49, a)

R1∗ = ql ,

, (

)

r R2∗

. 49 q max ,

0 R2∗ = q max l / 2 ,

l/3

(

. 49, b).

, .

, ,

. 49, c.

, , .

, (

. 50).

, ∠BOD = ∠AOD = α ,

R, ,

O. OD -

OX.

r ,r r Qi = Qix + Qiy .

ρ :

. . -

ϕ,

Qix = Qi cosϕ ;

Qiy = Qi sin ϕ . r Q

li = ρϕ i = Rϕ i ,

q, ϕi -

. 50 (

i =1 n

i =1

. 50):

Qx = ∑ Qix = ∑ qi li cosϕ = qR ∑ cosϕ ⋅ ϕ i ;

Q y = ∑ Qiy = ∑ qi li sin ϕ = qR ∑ sin ϕ ⋅ ϕ i .

n → ∞,

ϕ0

Qx = qR ∫ cosϕ ⋅ dϕ ; Q y = qR ∫ sin ϕ ⋅ dϕ .

, ϕ 0 = −α , r r , Q = Qx . ,

: Qx = 2qR sin α = qAB ; Q y = 0 .

ϕ =α,

, ,

. 1.

(

. 51, a),

, ,

. . , . 51, a.

. 51

r r R1∗ , R2∗

r R3∗ .

R1∗ = (4 ⋅ 4.5) / 2 = 9 R2∗ = (2 ⋅ 3) / 2 = 3

R3∗ = 2 ⋅ 3 = 6

, ,

r X

r Y, M. -

. , ,

(

. 51, b), -

, . ,

r X

r Y:

∑ Fix = 0 , F cos30o − R1∗ − R2∗ − R3∗ + X = 0 ; ∑ Fiy = 0 , − F sin 30o + Y = 0 ; ∑ m A ( Fi ) = 0 , − F ⋅ 7.5 cos 30o + R1∗ ⋅ 4.5 + R2∗ ⋅ 2 + R3∗ ⋅ 15. + M1 + M = 0 . : X = 13.7

; Y = 2.5

; M = −27

§8.

. . ( . 2).

. , , .

. , ,

. .

. 24.

. . .

, .

1781 ., ,

1. ,

. 2.

r Fl :

0 ≤ F ≤ Fl .

(1)

3. , (

)

Fl = f 0 N .

(2)

f0 . .

0.15 ÷ 0.25 ,

f 0 = 0.4 ÷ 0.7 ,

(1)

f 0 = 0.027 .

(2) , 0 ≤ F ≤ f0 N .

(3) -

. 52

r N,

r F.

, r, Fl .

, ,

. 52, a.

. ,

, .

r N

ϕ0 , 60

r Fl .

(

. 52, b). 53, b tgϕ 0 = Fl / N ,

. (2) Fl = f 0 N , tgϕ 0 = f 0 . ,

, (4) ,

. ,

(

c). , ,

. 52,

2ϕ 0 .

. .

r Fl .

, -

r - N

Fl .

(2).

. r R, ϕ0 .

, f0 .

αl ,

F = Fl (

r P.

). , OXY,

∑

. 53

. 53.

, Fix = 0,Fl − P sin α l = 0 ; tgα l = f 0 .

∑

αl α < αl ,

f0 ,

, ,

tgα ≤ f 0 .

, f0 ,

(2): Fiy = 0,N − P cos α l = 0 ; Fl = f 0 N .

αl

, -

α > αl ,

P P.

r P

G, . 1.

f0 .

P .

r Fl

r G,

, . 54.

∑

, Fix = 0, − Fl − P sin α + G = 0;

∑ ,

G = P (sin α + f 0 cosα ) .

, 2.

OXY (2): Fiy = 0, N − P cos α = 0; Fl = f 0 N .

G ,

. 54

= 0 , Fl − P sin α + G = 0 ;

rP Fl

∑ Fix

∑ Fiy

= 0 , N − P cosα = 0 ; Fl = f 0 N .

G = P (sin α − f 0 cosα ) . P , G ,

, -

, , P (sin α − f 0 cosα ) ≤ G ≤ P (sin α + f 0 cosα ) . , , -

. 54 .

: 1)

; 2)

;

F = fN .

(5)

f .

f f0 .

. 25.

. -

, . -

(

. 55, a),

r r ( Q, F ) ,

r Q

r P, r Q.

r F

r N

r Q

Ql .

. 55 (

r Q

. r r (Ql , F ) δ -

. . 55, b

). .

r r ( P, N ), r r ( P, N )

Ql ⋅ r

N ⋅δ ,

. Ql =

Q < Ql ,

δ r

(6)

Q > Ql

. 55, b

, ,

. 55, a

. 55, c.

r r (P, N ) :

r r m f = M ( P, N ) = δ ⋅ N .

(6)

(7)

. . (

δ = 0.0005 ÷ 0.0008 ; 0.00005;

) ( ) - 0.00001.

( (6)

δ /r

f0 .

, (

. .).

. 56)

, ,

δ.

r r P, N

α = αl .

r F,

, , . A AY,

∑ m A ( Fi ) = 0 , − PR sin α l + m f ,

δ,

α (

: ) -

αl

. 56

: = 0;

∑ Fiy

= 0 , N − P cosα l = 0 ; m f = δ ⋅ N .

tgα l = δ / R .

αl

α < αl .

. -

. , .

ml ,

ml = λN , 64

(8)

r N

. , ,

位

(8) .

位

未)

( ,

( ) .

位.

, ,

2. ยง 9. ,

, . .,

. -

, . ,

, ,

, .

, , , .

, .

. : ;

. ,

. , ,

. ,

. ยง 10. , . ,

. ,

, ,

26.

. .

, 3)

: 1)

. O (

. 57). M

r r

r r,

r r r = r (t ) .

(1)

(1) .

M -

, -

, .

. ,

. . 57 .

Oxyz.

x, y, z (

M . 57). M

x = x (t ) ; y = y (t ) ; z = z(t ) .

(2)

(2) , x = x (ϕ ( z )) ; y = y (ϕ ( z )) .

(2), : t = ϕ ( z) . : (3) . .

, .

. ,

O1 ; 2)

: 1) +;

M s(

. 57). M

s = s( t ) .

(4)

(4) . . s

. 57, ,

. ,

, ,

. M

2s. .

. .

. ,

: r r r r r = xi + yj + zk ,

r r r i , j,k

(

(5) )

(

. 57). -

. (

) ds (

. 57).

dx, dy, dz, (

ds ds = (dx ) 2 + (dy ) 2 + (dz ) 2 .

(6)

dx dy dz = x& ; = y& ; = z& , dt dt dt & ; dz = zdt & ; dy = ydt & . : dx = xdt (6), ds =

x& 2 + y& 2 + z& 2 dt .

(7)

t=0

(7)

(s=0), 0

: t

∫

x& 2 + y& 2 + z& 2 dt .

(8)

Oxy x = a cosωt ; y = a sin ωt , .

ω .

t = [arcsin( y / a )] / ω ,

t .

: x y = cosωt ; = sin ωt . a a , : x2 / a2 + y2 / a2 = 1. , a( . 58). : x& = − aω ⋅ sin ωt ; :

y& = aω ⋅ cosωt .

s = aω ⋅ ∫ sin 2 ωt + cos2 ωt ⋅ dt = aωt . 0

t=0 . Ox (

, O1

Δt ,

. 58

(

t = 0.

x (0) = a ; y(0) = 0 . . 58). ,

x ( Δt ) > 0

y ( Δt ) > 0 , I ,

. 58).

§11.

: ,

, -

. ,

. 27.

r r (t ).

r r

t: r r r r drr r ( ) (t ) Δ + Δ − r r t t r . V = = r& = lim = lim Δt Δt →0 Δt Δt →0 dt . 59

Δt -

r Δr ,

. M

r Δr . MM 1 ,

. ,

(1)

Δt → 0 ,

r V

M1 → M ,

(1), M ,

Δt

r Δr / Δt . ), / ( ) . .

( (

. 59

, ,

r r r = re ,

r e.

Δϕ -

, , r r e& ⊥e .

r r. (1) r r r dr dr r de r r & + re& . V = = e +r = re dt dt dt (10) r e

(2) -

r r Δe Δϕ ⋅ e dϕ de = lim = lim = , Δ⋅t →0 Δt Δ⋅t →0 Δt dt dt r r r (t ) r (t + Δt ) ( . 59). r Δe .

. r r& 2e ⋅ e = 0 . r e

r r e& ⊥r .

M1 ,

r r e ⋅e = 1

r e ≠ const ,

r n, r dϕ r de r n = rϕ& ⋅ n , =r dt dt

π /2 r e

r r,

, (3)

r r n ⊥e . r r n ⊥e

(3) (2), r drr dr r dϕ r r r & + rϕ& ⋅ n , V= = e +r n = re dt dt dt

r r n ⊥r .

(4) ,

(4)

r VR

r VT

r r r V = V R + VT ,

(5)

r r r r & ; VT = rϕ& ⋅ n . V R = re

(6)

, .

, -

, ,

, . 60

. -

, r , b,

(4)

r e

r e r b. ,

. 60

. ,

(4) ,

, u.

r db db r dϕ r = e +b n, du du du r r b, n , r (7) db dϕ r =b n. du du

(7) , -

(8)

28.

Oxyz,

, x = x (t ) ; y = y (t ) ; z = z (t ) . (5) . 26, (1) r r r r r drr r r di dj dk & +x & + y + zk & +z V = = xi + yj . dt dt dt dt Oxyz , ( , ), ,

: : r r r i , j,k

r r r r & + yj & + zk & . V = xi

(9)

, V x = x& ; V y = x& ; Vz = z& .

: (10)

V = Vx2 + V y2 + Vz2 = x& 2 + y& 2 + z& 2

: r r V r r V r r Vy ; cos( k ∧V ) = z . cos(i ∧V ) = x ; cos( j ∧V ) = V V V (9) .

(11) ,

r r r r V = V x + V y + Vz ,

(12)

r r r r r r & ; Vz = zk & . & ; V y = yj V x = xi

(13)

(21)

: 1)

, ; 2)

. 29.

, (1)

Δs → 0

61)

, r

r r r = r ( s(t )) .

: r r drr drr ds dr V= = ⋅ = s& . dt ds dt ds r (14), dr / ds . , Δs . r r dr Δr r = lim =τ , ds Δs→0 Δs

(14) , MM1 = Δr ( ,

(15)

τ 61

Δs > 0 (

M ).

r Δr / Δs

r Δr ,

. Δs < 0 ,

M1 , ,

, ),

r Δr M, r Δr / Δs r Δr ( Δs ,

. 61

. (15) r

τ,

τ.

(14), r r V = s& ⋅ τ . V = | s&| .

(16) s& > 0 , ,

s& < 0 ,

. § 12.

. ,

. 30.

r r r r ΔV V (t + Δt ) − V (t ) r dV a= = lim = lim . Δt dt Δt →0 Δt Δt →0 Δt → 0 , M1 → M ; ,

r ΔV ,

(1) r r V (t ) , V (t + Δt ) M

r a

. 62),

. -

Δt ,

/ ).

M1 ( M;

r ΔV / Δt .

(

, -

. 62

rΔV

31.

Oxyz , , : x = x (t ) ; y = y (t ) ; z = z(t ) . (1), , r r r r && + && a = xi yj + && zk .

(17)

. 27,

: (2) -

(2), z. a x = && x ; a y = && y ; a z = &&

, (3)

: (4)

a = a x2 + a 2y + a z2 = x&&2 + && y 2 + && z2 .

, , r r r r r r a a a cos(i ∧ a ) = x ; cos( j ∧ a ) = x ; cos( k ∧ a ) = z . a a a (2)

(5)

r r r r a = a x + a y + az ,

(6)

r r r r r r && ; a y = && a x = xi yj ; a z = && zk ,

(7)

(7) .

, ,

, .

32.

. -

r r V = s& ⋅ τ .

τ s.

, ,

τ = τ ( s(t )) . -

, r

r r r r 2 dτ dτ ds r dV = &&s τ + s& ⋅ = &&sτ + s& a= . ds ds dt dt r τ ( τ = const ), r dτ dϕ r dϕ r dϕ Δϕ . =τ = lim n= n, ds ds ds Δs→0 Δs ds

(8) (16)

. 27

R (

dϕ / ds = Δϕ / Δs = 1 / R . dϕ / ds = 1 / ρ ,

. 63), -

M. ,

. , , dϕ / ds ,

r r n⊥τ , ,

. 63

ρ = ∞.

r dτ 1 r = n, ds ρ

(9)

r r n⊥V .

V = s& ,

(9)

(8),

r s& 2 r r V2 r r a = &&s τ + n = &&s τ + n.

(10) r aτ

(10)

r an

, ,

r r r a = aτ + an ,

(11)

r r s& 2 r V 2 r r aτ = &&s τ ; a n = n= n.

aτ = &&s ; an =

s& 2

(12)

(13) (

) &&s > 0 ,

τ, .

&&s < 0 , s& &&s

(

), , . ,

, ,

(10) . ,

r aτ .

r an

, ,

r a

, .

74 . 64

s& 2 / ρ

r n

. . 64

r an ,

r a

r V,

&&s < 0 . , r r r r a cos(V ∧ a ) = V ⋅ a / V . r r V ⋅a /V

r r r r V ⋅ a = Va cos(V ∧ a ) , 64 ,

r aτ

s& > 0 . , .

. -

, ,

r r r r V ⋅a V ⋅a r aτ = aτ = = . V V

(14) ,

. 27

(2) ,

(17) . (14), -

. 29 (18)

. 27 ,

, .

“+“ ;

&&& + yy &&& + zz &&& xx r . aτ = aτ = 2 2 2 x& + y& + z& r r V ⋅ra / V r &&& + yy &&& + zz &&& V ⋅ a = xx ,

“-“

(15)

r aτ

r V

. aτ = dV / dt .

(15)

, -

V 2 = x& 2 + y& 2 + z& 2 ,

. ).

(

, .

r b,

(

r r r b =τ ×n, . 65).

r n, -

, , . ,

, . 65

. , ,

r V

r a

r r r r r V = Vτ τ + 0 ⋅ n + 0 ⋅ b = Vτ τ ; r r r r r r a = aτ τ + a n n + 0 ⋅ b = aτ τ + a n n ,

: (16) (17)

Vτ = s& ; aτ = &&s ; an = s& 2 / ρ = V 2 / ρ . (16) Vτ (17) aτ an 1. s = 20sin πt (t

(18) ,

R = 20 ,s-

). -

t=1 . ,

. ,

(

. 66). -

r r r r r r V = s&τ ; aτ = &&s τ ; an = ( s& 2 / ρ )n . : s& = 20π cos πt ; &&s = −20π 2 sin πt . ρ = R = const , , r r r r r r V = 20π cos πt ⋅ τ ; aτ = − 20π 2 sin π t ⋅ τ ; a n = 10π 2 (1 + cos 2πt ) ⋅ n .

, Vτ = 20π cos πt ; aτ = −20π sin πt ; a n = 10π (1 + cos 2πt ) . t =1 , , s(1) = 20 sin π = 0 . . 66 M, , r t r= 1 , τ n . : . 66 2 s&(1) = 20π cos π = −20π < 0 ; &&s = −20π sin π = 0 . 2

V (1) = s&(1) = 20π

/ ,

τ (

s&(1) < 0 .

r r r r a (1) = aτ (1) + a n (1) = a n (1) , r aτ (1) = 0 .

&&s (1) = 0

, ,

r n (

. 67 r V

20π 2

. 67, a

a n (t ) .

: 20 , 20π 2 / , ( s(1) = 0) ,

(Vτ (1) = −20π )

), ,

1 . /

s( t )

Vτ (t ) ,

, s(t ) , Vτ (t ) aτ (t ) T = 2π / π = 2 , an (t ) s(t ),Vτ (t )

. 68, a),

ϕ = 0.

aτ ( t )

20π 2 / 2. - 0. t =1 , (aτ (1) = 0) ,

(

/c2. -

, an (t )

a (1) = a n (1) =

s& 2 (1) / R = 20π 2

r a.

aτ (t )

. 67, b -

. 66).

(an = 20π 2 ) . M AB OA = AB = 60 , BM = 20 , ϕ = 4πt (t ,

M Oxy, x y

Ox B. ΔOAB M, , x = OA cosϕ + AM cosϕ ; y = MB sin ϕ . ϕ AM, OA MB, : x = 100 cos 4πt ; y = 20 sin 4πt .

: -

. x / 100 = cos 4πt ; y / 20 = sin 4πt ,

M x / 100 + y / 20 = 1 . , 2

100

: . .

68, b. : -

r r r r r r . 68 & + yj & ; a = && xi + && yj . V = xi : x& = −400π sin 4πt ; y& = 80π cos 4πt ; && x = −1600π 2 cos 4πt ; && y = −320π 2 sin 4πt . , ϕ = 0, , t= 0. : 2 x& (0) = 0 ; y& (0) = 80π ; x&&(0) = −1600π ; && y (0) = 0 . -

r r r r : V (0) = 80πj ; a (0) = −1600π 2 i . , .

M M , t=0, : x (0) = 100 ; y (0) = 0 . r r V (0) = V y (0) , Ox ( . 68, b). , r r a ( 0) = a x (0) , V (0) = V y (0) = y& (0) = 80π /c, r a (0) = a x ( 0) = x&&(0) = 1600π 2 /c2. . 68, b V (0) , r V y (0) , Oy r a(0) , y ( y&(0) > 0) , r a x (0) , Ox x ( x&&(0) < 0) . r r a (0) ⊥V (0) , . 68, b r r a (0) = a n (0) , r aτ (0) = 0 . , t = 0, (13) , 2 2 2 ρ (0) = V (0) / an (0) = 6400π / 1600π = 4 . (15), aτ (0) = x& (0) ⋅ && x (0) + y& (0) ⋅ && y (0) / V ( 0) = 0 ,

an (0) = a 2 ( 0) − aτ2 ( 0) = a (0) .

§ 13.

, . 33.

, . , ,

. ,

. -

. ,

. .

, ,

, . ,

, -

. , ,

, ,

. -

, . , .

, .

Oxyz

. 69

ΔABC . ,

A, B, C, .

, -

: ( AB ) = ( x2 − x1 ) + ( y2 − y1 ) + ( z2 − z1 ) 2 ; 2

( BC ) 2 = ( x3 − x2 ) 2 + ( y3 − y2 ) 2 + ( z3 − z2 ) 2 ; ( AC ) 2 = ( x3 − x1 ) 2 + ( y3 − y1 ) 2 + ( z3 − z1 ) 2 .

, .

. .

: 1.

, ,

, .

2. -

, ,

. 69

. 3.

, .

4. ,

, .

34.

( ,

. ,

: r AB V A =

r V AB B . .

(1) AB (

. A

, . 70)

(1) -

AB = var ,

, ,

(1),

. , , . ยง 14.

. 70

, . 35.

. 71 A, B, C -

r r rA , rB

, A′ , B ′ , C ′ (

r rC .

Δt

. 68).

ABC A ′B ′ / / AB , A ′C ′ / / AC -

B ′C ′ / / BC .

, ,

, , . 71

r r r r ΔrA = ΔrB = ΔrC = Δr . ,

(1)

. (1),

r r r r V A = VB = VC = V . (2)

r r Δr , V = lim Δt → 0 Δt

(2)

r r r r a A = a B = aC = a .

(3)

, ,

. .

, ,

. ,

. 36.

(

, 72, a).

(

B C

, ,

OC = R ,

(

. 72, a),

. , ,

, ,

. 72

, C. AC .

, -

BC.

, , . ,

C , (

Oxyz,

. 72, b).

Ox1y1z1 , ,

Ox1

Oz1

R (

. ABC,

Ox1 y1 AB.

Oz R = const ,

: (4)

xOy). ,

ϕ , ).

ϕ = ϕ (t ) . (

. 72, b),

ϕ, .

, . 72, b -

C. -

. . (

. 72, b Oz,

. ,

ω=

dϕ Δϕ = ϕ& = lim . Δt →0 Δt dt

(5) -

ω~ = ϕ& .

(6) .

ω.

ϕ& < 0 ,

ϕ& > 0 ,

. . 72, b ,

ϕ& > 0 .

(5), Δϕ / Δt

t. Δt . ,

. . ,

ε=

Δω dω = ω& = ϕ&& = lim . Δt →0 Δt dt

(7)

ε~ = ω& = ϕ&& . ϕ& ϕ&& ,

(8) -

. (

), ;

, ,

ε. ,

ϕ& > 0 ,

, t.

ϕ&& < 0 . (7) Δω / Δt Δt .

. 72, b

, . .

. . 73 . 72, b C R

Ox, ϕ,

C s = Rϕ ,

. 73 (

r r r & ; V = s&τ = Rϕτ

(9)

r s& 2 r r r r r r && + Rϕ& 2 n . a = aτ + a n = &&sτ + n = Rϕτ R (9) (10), V = R ϕ& = ω~R ; aτ = R ϕ&& = ε~R ; a = Rϕ& 2 = ω~ 2 R .

(10) (11) (12) (13)

r τ ⊥R ,

r n

r an

( s& 2 / R = Rϕ& 2 > 0) .

r V , ( R > 0) .

r aτ

(14)

. 73 , s& > 0 , &&s < 0 .

r V

: r r V⊥R ; aτ ⊥R . ,

r aτ -

) -

ϕ& > 0 , ϕ&& < 0 ,

: 1. , ; . 2. ;

3. ; . ,

, (15)

a = aτ2 + an2 = R ε~ 2 + ω~ 4 .

β,

tgβ = aτ / a n = ε~ / ω~ 2 .

. ,

, , (

. 74). -

ω = ω = ω~ , . :

. 74 r dω r& ε= =ω . dt r

(16) ,

, . , , .

. 74

. . 75

, , .

. 75

. ,

r V.

r r r V =ω ×r . r r r r , ω × r = ωr sin(ω ∧ r ) = ωR = ω~R , , (17)

(17)

ω ×r , ,

, .

r r r dω r r dr r dV d r r = (ω × r ) = ×r +ω × a= . dt dt dt dt r r r r ω& = ε ; r& = V , , r r r r r a = ε × r + ω ×V . (18) , : r r r r r r r r r aτ = ε × r ; a n = ω × V = ω × (ω × r ) . (19) , r r r∧ r : ε × r = εr sin(ε r ) = εR = ε~R , r r r r r r ; ω × V = ωV sin(ω ∧V ) = ωV = ω 2 R , V⊥ω ,

(19)

V = ωR ,

r, b,

, (b = const): r db r r =ω ×b , dt

r .r V = dr / dt . r = const, .

r r r V =ω ×r .

(18)

(20)

r b.

. r i

r j

r V = ωx

ωy

(17) , r r r i j k r r ω z = 0 0 ω z = − yω z i + xω z j , z x y z r k

ω x , ω y , ωz ωx = ωy = 0, ; x, y, z -

(21) ,

ω z = ±ω

(21)

Oxyz V x = − yω z ; V y = xω z ; Vz = 0 .

(22)

(ω

(22)

Oz)

. r i

r j

r a = εx

εy

r k

r i

r j

r k

r i

r j

(18) r k

εz + ω x ω y ωz = 0 0 εz + z

r i

r j

r k

ωz ,

− yω z

xω z

. : a τx = − yε z , a τy = xε z , a zτ = 0 ; a xn = − xω z2 , a ny = − yω z2 , a zn = 0 .

(23)

: a x = − yε z − xω z2 ; a y = xε z − yω z2 ; a z = 0 .

(24)

, v

ω = ω (1) r

r r r r ,ω , ε r r r r r r = ρ = x1i1 + y1 j1 + z1k1 ; r r r r r r r r = ω x1 i1 + ω y1 j1 + ω z1 k1 ; ε = ε (1) = ε x1 i1 + ε y1 j1 + ε z1 k1 ,

ρ , ω (1) , ε (1)

ϕ = ϕ 0 sin kt , , ϕ = 0. t1 = T / 4 , T-

. l

ϕ0 .

ω = kϕ 0 cos kt ; ε = −k 2ϕ 0 sin kt . -

V = ω~l = kϕ 0 cos kt l ,

aτ = ε~l

Ox1 y1z1 :

− k 2ϕ 0 sin kt l ,

an = ω 2 l = ( kϕ 0 cos kt ) 2 l . t1 ,

T = (2π ) / k ,

ϕ (t1 ) = ϕ 0 sin

π 2

α = kt

, 2π ,

, α = 2π , t = T . t1 = T / 4 = π / ( 2 k ) . , , t = t1 ,

= ϕ 0 ; ω ( t1 ) = kϕ 0 cos

t = t1 = T / 4 ,

= 0 ; ε (t1 ) = − k 2ϕ 0 sin

2 2 2 : V ( t1 ) = 0 ; aτ (t1 ) = k ϕ 0 l ; an (t1 ) = 0 .

= − k 2ϕ 0 .

, ,

. ,

. 2. (

r2 = 15 ,

r1 = 10

. 76). 1,

ϕ = 30t 2

, (

ϕ1,

. 76). ,

1 ϕ1 = ϕ = 30t . ϕ, 2

. 76

ϕ2

. 76

. S 2 = ϕ 2 r2 ,

S1 = ϕ1r1 ,

ϕ 2 = ϕ1 (r1 / r2 ) = 20t 2 . ω1 = ϕ&1 = 60t ; ε1 = ϕ&&1 = 60 ; ω 2 = ϕ& 2 = 40t ; ε 2 = ϕ&&2 = 40 . t > 0, ϕ&1 > 0 , ϕ&&1 > 0 , ϕ& 2 > 0 , ϕ&&2 > 0 ω~1 = 60t ϕ2 ,

/ ; ε~1 = 60

/ 2; ω~2 = 40t

/ ; ε~2 = 40

1 . 76

/ 2.

ϕ1 2-

ε1

ω1

ε2

. 76) ,

ω2 . ,

1 V1 = ω~1r1 = 600t V2 = ω~2 r2 = 600t / .

2 (

/ ,

r r V2 = V1 .

a1 = ε~1r1 = 600 a2τ = ε~2 r2 = 600 r r a2τ = a1τ .

/ ,

/ 2.

, :

a1n

= ω12 r1

= 36000t

/ ;

a2n

= ω 22 r2

= 24000t ( :

a1 = (a1τ ) 2 + (a1n ) 2 = 600 1 + 600t 4

/ 2. . 76).

/ 2;

/ 2. a2 = (a 2τ ) 2 + (a 2n ) 2 = 600 1 + 400t 4 ( . 76

). , ,

, . §15.

. . ,

, .

. 37.

. ,

D, E

, xOy

Oxyz,

C, -

. ,

Oxy , xOy.

AB, CDE

(

. 77).

. 77

Oxy,

AB, (

. 78).

Q (

. 77), .

B x A , y A , xB , yB ,

. 78

( x B − x A ) 2 + ( y B − y A ) 2 = ( AB ) 2 .

AB: -

AB Oxy.

, . 79).

yA (

∗ ∗

- Ax y , . .

A . ,

Ax1

AB = const ,

Ax1 y1 . 79).

. 79

B ϕ -

Az1 , (

. 79

∗

Az ,

). ,

x A = x A (t ) ; y A = y A (t ) ; ϕ = ϕ (t )

(1)

, . .

. -

(1)

, Ax ∗ y ∗ ,

. 79

x A′ ≠ x A

y A′ ≠ y A , .

∗

Az1 ,

. A′ (

, . 80).

B′ ,

A ′x1′ y1′ (

. 80)

. 80

ϕ′ = ϕ

( )

. 38.

, . ,

: r r r r r r V B = V A + ω × ρ = V A + VBA , r r , VA , V BA -

r VB -

ω~ , ω -

(2)

r V BA = ω~ ⋅ AB = ω ⋅ AB ; V BA ⊥AB ,

(3)

; AB -

. r rB (

ρ,

. 81 ,

r rA .

r r r rB = rA + ρ .

B . 81).

r r r drB drA dρ = + . dt dt dt r r r r (4) drB / dt = VB , drA / dt = V A

(4)

(4) B

. Az1

Az ∗

. ,

(

ω . 81).

r r r dρ / dt = ω × ρ .

Az1 ρ = const , r r ω×ρ x ∗Oy ∗ ,

. 81 -

r r

ωρ sin(ω ∧ ρ ) = ωρ = ω ⋅ AB . xOy,

. B

Az1 ,

A, r V BA ,

, ,

. -

. , . .

. , .

. 82 r VA .

, r V PA

AP = ω / V A . r r r P. V P = V A + V PA .

V PA = ω ⋅ AP = V A , r VA (

, r). VA

r V PA (

. 82),

. 82

AP , r VP = 0 , .

. A

83),

.r r Pr r r r V A = V P + V AP ; V B = V P + V BP , r r r r V A = V AP ; V B = VBP ,

(6)

. .

, . , (

r VP = 0 , (5)

r r V A = V AP = ω ⋅ AP ; V B = V BP = ω ⋅ BP ; V A ⊥AP ; V B ⊥BP . . 83 (5) (6)

. 83

, . . -

. . r r (6): V A ⊥AP ; V B ⊥BP . ,

,B(

, . 84).

A -

B, P,

. AP

, r . VA

, 84 -

84 -

r ( V B ⊥BP ) ).

r VB

. 84

(

. (

(6):

ω=

VA BP . ; V B = ω ⋅ BP = V A AP AP ,

. 85.

. 85

. 85, a

. ω = V A / AP = 0 .

AP = ∞

V BA = ω ⋅ AB .

ω = 0,

, , , r r r V B = V A + V BA ,

r r VB = V A .

, .

. 85, b

, .

, (

). ,

r VB

. r VA

r VB . ,

ω = V A / AP = V B / BP .

. 85, c

. ,

. 35.

, , P

. -

, ;

; ,

1. (

. -

. 86).

VC .

. 86 A, B, D, E

. , .

, , .

r. VC

, (

r VC

. 86). C

, .

r VE = 0 .

r r r r 0 = VC + VEC V EC = −VC ( V EC = ω ⋅ EC = ωR ,

. 86a).

R , , / R = VC / R

ω = V EC r VEC

V BC = ω ⋅ BC = VC ; V DC AC, BC, DC . 86, a.

C.r r r r r r r r r : V A = VC + V AC ; V B = VC + V BC ; V D = VC + V DC . : V AC = ω ⋅ AC = (VC / R ) ⋅ R = VC ; = ω ⋅ DC = VC . , A D , 2 2 V A = VC2 + V AC = VC 2 ; V D = VC2 + V DC = VC 2 . AD , π / 4. r r VC V BC , , , V B = 2VC . . P E ( . 86, b), ). ω = VC / CP = VC / R , -

, B , VE = 0 .

, P( -

r VC

, P.

AP = R 2 + R 2 = R 2 ; BP = 2 R ; DP = R 2 + R 2 = R 2 . V A = ω ⋅ AP = (VC / R ) ⋅ R 2 = VC 2 ; V B = ω ⋅ BP = 2VC ; V D = ω ⋅ DP = VC 2 . AP, BP, DP , . 86b. , . , ,

: -

. r AB V B ; (

86, c). r CD VC =

r CD V D .

AC V A

AC VC ;

AB V A

V A cos45o = VC ; V A = V B cos45o ; VC = VD cos45o .

V A = VC / cos45o = VC 2 ; V B = V A / cos45o = 2VC ; V D = VC / cos45o = VC 2 . , , .

, . OA

0.5 ω0 = 1

DB = 2 ,

/ .

E . 87,

O1B = O1B . . ; 2)

: 1) ,

, ; 3)

, . .

DB O1 .

. AB

E )

DE .

(

AB,

A, OA. . 87

AB, D,

O1 B .

DE, E,

O1 D . DE,

(

r VA ,

. -

. 87).

r VA

PAB

r VA

r VD . 87). = ∞,

O1 , ( DPDE

r VB

, D ,

. ,

VE = VD .

O1 D = O1 B ,

V A = ω 0 ⋅ OA = 0.5

BA, V E = V D = V B = 0.58

r VB . 87). -

rVB

E, ,

PAB . (

VD = VE ,

r r = 0 , VE = VD ( , VE = VB . .

ω DE

, . 87)

/c.

V B cos30o = V A , /c.

V B = V A / cos 30o = 0.58

AOB

BPAB .

OBPAB

/ APAB :

. ; BPAB = OB / tg 30o = 173

; ω AB = V B / BPAB = 0.34 ω DB = VB / O1 B = 0.58 /c. V E = V D = V B = 0.58 / ; ω AB = 0.34 / ; ω DB = 0.58 /c; ω DE = 0 . , . 87 , , AB , , B , D , . , 1 2 , , . ,

OB = 2OA = 1

A -

/ . ,

. 39. ,

, . .

. -

(2), r r r r r r r r r r r r r dV dV A dω r r dρ r r + × ρ +ω × = a A + ε × ρ + ω × ω × ρ = a A + ε × ρ + ω × V BA . aB = B = dt dt dt dt r

∗

Az1 ,

(

88). , .

. 88

, ,

r aA

. (

. . 33)

r r r a τBA = ε × ρ , AB

(7)

r r a τBA = ερ sin(ε ∧ ρ ) = ερ = ε ⋅ AB .

(8)

r r r r r rn a BA = ω × ω × ρ = ω × V BA , AB B

(9) A,

r r n a BA = ωV BA sin(ω ∧V BA ) = ωV BA = ω 2 ⋅ AB . (8) (10) r r , ω ε

r ρ V BA .88). r

(

. 88

(10) ,

r (7) (9) aB , rτ rn r r r r + a BA = a A + a BA , a B = a A + a BA

(11)

. , . , (

: 1) )

; 2) .

A, B, D, E

1, . 35,

aC .

R, , C,

CP = const .

ω = VC / CP .

, P(

. 87) -

ε=

d ⎛ 1 ⎞ dω dVC 1 = ⋅ + VC ⋅ ⎜ ⎟, dt dt CP dt ⎝ CP ⎠

dVC / dt aCτ ;

CP = const .

, ,

, : ε = a cτ / CP .

r rτ aC = aC ,

r aCn = 0

ε = a C / CP = a C / R .

r aC

, P( . 90). C , r r rτ rn r r rτ rn r r rτ rn A, B, D, E: a A = aC + a AC + a AC ; a B = aC + a BC + a BC ; a D = aC + a DC + a DC ; r r rτ rn a E = a C + a EC + a EC . , A : n τ 2 2 2 2 a AC = ε ⋅ AC = (aC / R) ⋅ R = a C ; a AC = ω ⋅ AC = ω R = (VC / R) ⋅ R = VC / R , n n n : a τBC = a τDC = a τEC = a C ; a BC = a DC = a EC = VC2 / R .

, AC, BC, DC

. 89. .

r aC

rτ , a EC

, rn a EC .

, ,

r rn a E = a EC

n a E = a EC = VC2 / R .

, , Axy ,

. 89

, A(

. 89), .

n A: a Ax = aC + a CA = a C + V 2 / R ; a Ay = a τA = a C ,

1 2aC2 R 2 + 2aC2 VC2 R + VC4 . R τ n = a BC + a C = 2a C ; a By = − a BC = −VC2 / R ;

a A = a 2Ax + a 2Ay = (a C + VC2 / R ) 2 + aC2 =

a Bx

a Dx = a C − a τDC = aC − VC2 / R ; a Dy = − a C , 2 2 a B = a Bx + a By = ( 2aC ) 2 + ( −VC2 / R ) 2 = 2 2 a D = a Dx + a Dy = (aC − VC2 / R ) 2 + ( − aC ) 2 =

1 4a C2 R 2 + VC4 ; R

1 2aC2 R 2 − 2aCVC2 R + VC4 . R

2, . 35

DE = 4 .

(

2, . 35) : 4)

; 5) ,

. . 90 : ω AB

ω 0 = const ,

A. a nA = ω 02 ⋅ OA = 1 ⋅ 0.5 = 0.5

ω 0 = ω OA = const = 1 = 0.34 / ; ω DB = 0.58

/ 2.

r a nA

/ , / ; ω DE = 0 . r r ε 0 = dω / dt = 0 a A = a An , OA

O, A .

r r r r r rτ rn + a BA a B = a A + a BA = a τA + a nA + a BA .

(a)

O1 , r r r a B = a τB + a Bn . ,

(b),

r a τA = 0 ,

. 90).

100

A (

rn rτ r n rτ r n rτ aB + a B = a A + a A + a BA + a BA . ( ) rτ r n r n rτ rn aB + a B = a A + a BA + a BA .

. (b) (a) ( )

(d)

rn a BA

n . a BA = ω 2AB ⋅ AB = ω 2AB ⋅ OA / tg 30o = (0.34) 2 ⋅ 0.87 = 01 rn B A. aB

2 a Bn = ω DB ⋅ O1 B = (0.58) 2 ⋅ 1 = 0.34

r a Bτ

O1 .

, r : a Bτ ⊥O1 B ,

(

/ 2,

rτ a BA ⊥AB .

(d)

rτ a BA

(d) rτ aB .

. 90), . 90.

/ 2,

Bx (

. 90), (d)

rτ a BA . ,

By,

n n Bx − a τB cos 30o − a Bn cos 60o = a BA ; By − a Bn = − a nA − a τBA cos 30o + a BA cos 60o .

a τB = −

n a Bn cos 60o + a BA

cos 30o

a τBA = −013 . / 2. r rτ a Bτ a BA . 90,

″-″

a τB

n a n + a BA cos 60o − a nA ; a τBA = B . cos 30o , a τB = −0.31

a τBA

, , : ε AB

r a Bτ

/ 2; ε DB = a τB / O1 B = 0.31 / 1 = 0.31 rτ , a BA A,

. = a τBA / AB = 013 . / 0.87 = 015 .

/ 2.

O1 ( B

/ 2;

, . 90).

a B = (a τB ) 2 + (a Bn ) 2 = (0.31) 2 + (0.34) 2 = 0.46 / 2. r rτ r n D DE + aD . aD = aD a τD = ε DB ⋅ O1 D = 0.31 ⋅ 1 = 0.31

/ 2,

O1D (

91).

n 2 aD = ω DB ⋅ O1 D = (0.58) 2 ⋅ 1 = 0.34 , 2 / , O1 D = O1 B . D ,

/ 2,

O1D

D a D = a B = 0.46

O1 . E .

, . ( ),

, -

101

r r rτ r n rτ rn . a Eτ + a En = a D + a D + a ED + a ED r r r , a En = 0 a Eτ = a E .

rn a ED = 0.

ω DE = 0 :

r aE ,

90.

(e) -

(e), r rτ r n rτ aE = aD + a D + a ED . rτ rn aD aD , r . aE rτ , a ED ⊥DE . ,

(f)

DE ε ED

(f)

rτ a ED

r aE

, (f) rτ a ED .

Ex1 ,

(f)

Ey1 ,

/ 2.

n − a τED . Ey1 | a E = a τD ; Ex1 | 0 = a D

n a E = a τD = 0.31 / 2; a τED = a D = 0.34 E D, 2 τ = a ED / DE = 0.085 / . rτ , a ED .

D. -

. Q

, .

. , (

Q). ,

. 91

. . 91, , , . rn r rτ rn r rτ a A = a AQ + a AQ ; a B = a BQ + a BQ , (12)

n a τAQ = ε ⋅ AQ ; a nAQ = ω 2 ⋅ AQ ; a τBQ = ε ⋅ BQ ; a BQ = ω 2 ⋅ BQ .

102

(12) : (13)

a A = AQ ε 2 + ω 4 ; a B = BQ ε 2 + ω 4 .

(14)

n = ⋅ ⋅ ⋅ = a τMQ / a nMQ = ε / ω 2 . arctgα = a τAQ / a nAQ = a τBQ / a BQ

(15)

. ,

3.r VC

C . ( . 92). r r VC = const , aC = 0 Q. P. , : ω = VC / R = const ; ε = dω / dt = 0 ; tgα = ε / ω = 0 ; (12) (13)

α = 0.

a M = a nM = ω 2 ⋅ MQ = ω 2 R = VC2 / R . r VC = const , ,

(

α = 0.

VC2 / R

, . . 92

, Q . §16.

, . . 93

Oxyz ,

O A

OB.

O ,

( -

. 93

)

. 40.

OAB,

103

OAB

Oxyz .

B ,

∠AOB = π / 2 . Ox1 Oy1 , Oz1 Ox1 y1z1

, , .

, OAB . r r i1 , j1

Ox1 , Oy1

Oz1 ,

r k1 ,

. 93

Ox1 y1z1

Oxyz . -

, . , -

. r r r∧r r∧ r r r (i i1 ) , (i j1 ) , (i ∧ k1 ) ,.., ( k ∧ k1 )

, ,

a22 a33

r r a11 = cos(i ∧ i1 ) ; r r = cos( j ∧ j1 ) ; a23 r∧ r = cos( k k1 ) . A r r r i1 , j1 , k1 r Ox1 i1 ; A Oxyz . .

104

⎡a11 a12 A = ⎢a 21 a 22 ⎢ ⎢⎣a 31 a 32 r r a12 = cos(i ∧ j1 ) ; r r = cos( j ∧ k1 ) ;

a13 ⎤ a23 ⎥ , ⎥ a33 ⎥⎦

(1)

r r a13 = cos(i ∧ k1 ) ; r r a31 = cos( k ∧ i1 ) ;

r r a21 = cos( j ∧ i1 ) ; r r a32 = cos( k ∧ j1 ) ;

Ox1 , Oy1 , Oz1 Oy1

r j1 ;

Oxyz :

Oz1

r k1 . r r r i1 , j1 , k1 A

Oxyz

Ox1 y1z1

: ;

r r r r j1 = a 21i + a22 j + a23 k ;

. ,

r r r r i1 = a11i + a12 j + a13 k ;

r r r r k1 = a 31i + a32 j + a33 k ;

r r j1 ⋅ k1 = 0 ;

, r r i1 ⋅ i1 = 1 ;

Ox1 y1z1 r r i1 ⋅ j1 = 0 ;

r r k1 ⋅ k1 = 1 ;

r r j1 ⋅ j1 = 1 ;

r r i1 ⋅ k1 = 0 ;

r r r r r r r r r r r r i ⋅ i = j ⋅ j = k ⋅ k = 1; i ⋅ j = i ⋅ k = j ⋅ k = 0 . A

Oxyz : , ,

, .

x1 , y1 , z1 x, y, z

(

. 94) -

ρ,

r r r ρ = x1i1 + y1 j1 + z1k1 , r r r r r = xi + yj + zk . r

r r,

⎡ x1 ⎤ ⎡a11 a12 a13 ⎤ ⎡ x1 ⎤ ⎡x⎤ r ⎢ ⎥ r r = Aρ ; ⎢ y ⎥ = A ⎢⎢ y1 ⎥⎥ = ⎢⎢a21 a 22 a 23 ⎥⎥ ⎢⎢ y1 ⎥⎥ . ⎢⎣ z1 ⎥⎦ ⎢⎣a31 a 32 a 33 ⎥⎦ ⎢⎣ z1 ⎥⎦ ⎢⎣ z ⎥⎦ Ox1 y1z1 Oxyz A −1 , ,

⎡ x ⎤ ⎡ a11 ⎡ x1 ⎤ r ρ = A T r ; ⎢⎢ y1 ⎥⎥ = A T ⎢⎢ y ⎥⎥ = ⎢⎢a12 ⎢⎣ z ⎥⎦ ⎢⎣a13 ⎢⎣ z1 ⎥⎦ r

a 21 a 22 a 23

r r r i1 , j1 , k1

(2)

A, a 31 ⎤ ⎡ x ⎤ a 32 ⎥⎥ ⎢⎢ y ⎥⎥ . a 33 ⎥⎦ ⎢⎣ z ⎥⎦

−1

=A , T

(3)

Ox1 , Oy1 , Oz1

, .

(2) : x = a11 x1 + a12 y1 + a13 z1 ; y = a21 x1 + a22 y1 + a 23 z1 ; z = a31 x1 + a 32 y1 + a33 z1 . , :

. M

x1 , y1 , z1

(

); ,

(4)

105

, . -

. ,

. ( ) Oxyz (

. 94). Oz

ψ.

θ,

Ox1 y1z1 . Ox1

Ox2 y2 z2 . Oz2

φ,

Ox3 y3 z3

(

. 94

. 94). ,

, : ψ = ψ (t ) ; θ = θ (t ) ; φ = φ (t ) . , . -

(5)

. , ,

. , . . 95

106

, ,

, . , ,

Oxyz,

Ox Oy ( . 95).

Ox2 . Ox3 y3 z3 (

ϑ -

Oz,

Oy1 ,

. 95). ,

, -

ψ = ψ (t ) ; ϑ = ϑ (t ) ; ϕ = ϕ (t ) . ,

(

(6) , ),

) ,

(

. A

r r r i3 , j3 , k 3 Oxyz.

. -

. 96 .

107

. 96 . 96, b) Oz (

(

α x ,α y

. 96, c)

Ox ( ,

αz

. 96, a), Oy

. Oxyz

Ox1 y1z1 ⎡x⎤ ⎢ y⎥ = ⎢ ⎥ ⎢⎣ z ⎥⎦

0 ⎡1 ⎢0 cos α x ⎢ ⎢⎣0 sin α x

⎡x⎤ ⎢ y⎥ = ⎢ ⎥ ⎢⎣ z ⎥⎦

⎡ cosα y ⎢ ⎢ 0 ⎢− sin α y ⎣

⎡x⎤ ⎢ y⎥ = ⎢ ⎥ ⎢⎣ z ⎥⎦

⎡cos α z ⎢ sin α z ⎢ ⎢⎣ 0

⎤ ⎡ x1 ⎤ − sin α x ⎥ ⎢ y1 ⎥⎥ ; ⎥⎢ cos α x ⎥⎦ ⎢⎣ z1 ⎥⎦ Oy: 0 sin α y ⎤ ⎡ x1 ⎤ ⎥ 1 0 ⎥ ⎢ y1 ⎥ ; ⎢ ⎥ 0 cosα y ⎥⎦ ⎢⎣ z1 ⎥⎦ 0

: ⎡ x1 ⎤ ⎢y ⎥ = ⎢ 1⎥ ⎢⎣ z1 ⎥⎦

0 ⎡1 ⎢0 cos α x ⎢ ⎢⎣0 − sin α x

⎡ x1 ⎤ ⎢y ⎥ = ⎢ 1⎥ ⎢⎣ z1 ⎥⎦

⎡cos α y ⎢ ⎢ 0 ⎢ sin α y ⎣

Oz:

− sin α z cos α z

0⎤ ⎡ x1 ⎤ ⎡ x1 ⎤ 0⎥ ⎢ y1 ⎥ ; ⎢ y1 ⎥ = ⎥⎢ ⎥ ⎢ ⎥ 1⎥⎦ ⎢⎣ z1 ⎥⎦ ⎢⎣ z1 ⎥⎦

Ox 0 ⎤⎡ x⎤ sin α x ⎥ ⎢ y ⎥ , ⎥⎢ ⎥ cos α x ⎥⎦ ⎢⎣ z ⎥⎦

(7)

0 − sin α y ⎤ ⎡ x ⎤ ⎥ 1 0 ⎥⎢ y⎥ ⎢ ⎥ 0 cos α y ⎥⎦ ⎢⎣ z ⎥⎦

⎡ cos α z ⎢− sin α z ⎢ ⎢⎣ 0

sin α z cos α z

(8)

0⎤ ⎡ x ⎤ 0⎥ ⎢ y ⎥ . ⎥⎢ ⎥ 1⎥⎦ ⎢⎣ z ⎥⎦

(9) -

ρ M r r r r ρ = x3i3 + y3 j3 + z3 k 3 ,

r r r r r = xi + yj + zk

: [ x2 y2 z2 ] = Az (φ ) [ x3 y3 z3 ] ; [ x1 y1z1 ] = Ax (θ ) [ x2 y2 z2 ] = Ax (θ ) Az (φ ) [ x3 y3 z3 ] ; T

[ xyz ]T = Az (ψ ) [ x1 y1z1 ]T = Az (ψ ) Ax (θ ) Az (φ ) [ x3 y3 z3 ] .

⎡cosψ A = Az (ψ ) Ax (θ ) Az (φ ) = ⎢ sin ψ ⎢ ⎢⎣ 0

− sin ψ cosψ 0

0 0⎤ ⎡1 ⎥ ⎢ 0 0 cosθ ⎥⎢ 1⎥⎦ ⎢⎣0 sin θ

r r

0 ⎤ ⎡cos φ − sin θ ⎥ ⎢ sin φ ⎥⎢ cosθ ⎥⎦ ⎢⎣ 0

− sin φ 0⎤ cos φ 0⎥ .(10) ⎥ 0 1⎥⎦

M [ x1 y1z1 ] = AzT(ψ ) [ xyz ]T ; [ x2 y2 z2 ]T = AxT(θ ) [ x1 y1z1 ]T = AxT(θ ) AzT(ψ ) [ xyz]T ;

[ x 3 y 3 z3 ]T = AzT(φ ) [ x 2 y 2 z2 ]T = AzT(φ ) AxT(θ ) AzT(ψ ) [ xyz ]T . 0 ⎡ cos φ sin φ 0⎤ ⎡1 ⎢ ⎥ ⎢ A T = AzT(φ ) AxT(θ ) AzT(ψ ) = − sin φ cos φ 0 0 cosθ ⎢ ⎥⎢ ⎢⎣ 0 0 1⎥⎦ ⎢⎣0 − sin θ (10) (11) ,

108

0 ⎤ ⎡ cosψ sin θ ⎥ ⎢− sin ψ ⎥⎢ cosθ ⎥⎦ ⎢⎣ 0

sin ψ cosψ 0

0⎤ 0⎥ (11) ⎥ 1⎥⎦ .

, ,

A .

. -

, , ,

⎡cosψ A = Az (ψ ) A y (ϑ ) Ax (ϕ ) = ⎢ sin ψ ⎢ ⎢⎣ 0

− sin ψ cosψ

0⎤ ⎡ cos ϑ 0⎥ ⎢ 0 ⎥⎢ 1⎦⎥ ⎣⎢− sin ϑ

0 ⎡1 A T = AxT(ϕ ) A yT(ϑ ) AzT(ψ ) = ⎢0 cosϕ ⎢ ⎢⎣0 − sin ϕ

0 0 sin ϑ ⎤ ⎡1 ⎥ ⎢ 0 cos ϕ 1 0 ⎥⎢ 0 cosϑ ⎦⎥ ⎣⎢0 sin ϕ

0 ⎤ ⎡cosϑ 0 − sin ϑ ⎤ ⎡ cosψ 1 0 ⎥ ⎢− sin ψ sin ϕ ⎥ ⎢ 0 ⎥⎢ ⎥⎢ cos ϕ ⎥⎦ ⎢⎣ sin ϑ 0 cosϑ ⎥⎦ ⎢⎣ 0 (12) (13), . ,

0 ⎤ − sin ϕ ⎥ (12) ⎥ cos ϕ ⎦⎥ sin ψ cosψ 0

0⎤ 0⎥ (13) ⎥ 1⎥⎦ -

, ,

, .

, . ,

OL, , Oxyz

(

. 97).

β : α = α (t ) ; β = β (t ) ; φ = φ (t ) .

φ,

. 97

(14) l, m, n

φ,

A . ,

A ,

109

41.

(20)

r b

. 33

r db r r =ω ×b , dt

ω -

. , ,

. Ox1 y1z1 ,

, r i1

r j1

r dρr r r V = = ω × ρ = ω x1 dt x1

r k1

(15)

ω y1 ω z1 , y1

ω x1 , ω y1 , ω z1 ,

V x1 = ω y1 z1 − ω z1 y1 ; V y1

, = ω z1 x1 − ω x1 z1 ; Vz1 = ω x1 y1 − ω y1 x1 .

(16) Oxyz,

r i

r j

r drr r r = ω × r = ωx V = dt x

r k

: (17)

ω y ωz ; y

V x = ω y z − ω z y ; V y = ω z x − ω x z ; Vz = ω x y − ω y x .

(16)

(18)

: ⎡ 0 ⎡V ⎤ ⎡ 0 − ω z1 ω y1 ⎤ ⎡ x1 ⎤ r ⎢ x1 ⎥ ⎢ ⎥ ⎢ ⎥ r ⎢ V = ⎢V y1 ⎥ = ⎢ ω z1 0 − ω x1 ⎥ ⋅ y1 ; V = ⎢ ω z ⎢ ⎥ ⎢− ω y ⎥ ⎢Vz ⎥ ⎢− ω y 0 ω x1 ⎣ ⎦ ⎢⎣ z1 ⎥⎦ ⎣ 1⎦ ⎣ 1 r r r (1) r (1) Ω Ω V = Ω ρ ; V = Ωr ,

− ωz 0

ωx

ω y ⎤ ⎡x⎤ ⎥ − ω x ⎥ ⋅ ⎢ y⎥ ⎢ ⎥

(19)

0 ⎥⎦ ⎢⎣ z ⎥⎦

. .

110

, , .

, -

, ,

. .

P, O P

, ,

. (16)

x1 P , y1 P , z1 P

, , (18),

xP , yP ,zP

: x1 P

ω x1 (20) .r

y1 P

ω y1

z1 P

ω z1

;

ωx

ωy

ωz

(20)

. ,

, , .

(

. 98). -

. 98

t2 .

t 2 − t1

t 2 − t1 = Δt → 0 ,

. 99

, -

OL OP,

111

dφ = ωdt

(

. 99, a). ,

OP dφ1 ,

OP1

. . -

(

. 99, b).

. ,

ϕ, ,

ω.

, , ,

αω

βω ,

, r .

. 100 , . , .

(15)

(17)

r r r r r V =ω ×ρ =ω ×r .

(21)

r r r r V = ωρ sin(ω ∧ ρ ) = ωr sin(ω ∧ r ) = ωh ,

(22)

h -

. . 100). (21), , r& r r r& r& r r r r a = (ω × ρ ) + (ω × ρ ) = (ω × r ) + (ω × r& ) . r r r r r ω& = ε , ρ& = r& = V , r r r r r r r r r a = (ε × ρ ) + (ω × V ) = (ε × r ) + (ω × V ) . r r r r r aε = ε × ρ = ε × r r r r . aω = ω × V r r r a = aε + aω . MOP

(

(23) ,

, (24) .

112

, OE (

r aε

. 101).

OME, r∧ r r r daε = ερ sin(ε ρ ) = εr sin(ε ∧ r ) = ε ⋅ d , r . aω , h r r aω = ωV sin(ω ∧V ) = ωV sin(π / 2) = ω 2 h , , ,

, r V

(

. 102), V = ω ⋅h.

, r r r( r r V =ω ×ρ =ω ×r , r r r r r ) aε = ε × ρ = ε × r , r r r aω = ω × V . .

. 101

§17.

( . 30),

, .

42.

Oxyz .

A (

xA, yA,zA

, . 102).

Ax ∗ y ∗ z ∗ ,

A .

ψ ,θ , φ ,

, Ax1 y1z1 ,

ABC , -

, .

, . 102 ,

113

: (1)

x A = x A (t ) ; y A = y A (t ) ; z A = z A (t ) ; ψ = ψ (t ) ; θ = θ (t ) ; φ = φ (t ) . (1) . A . .

, -

. ,

A′ ,

. -

C′

x A ′ ≠ x A ; y A ′ ≠ y A ; z A′ = z A .

B′

A ′x1′ y1′ z1′ Ax1 y1z1 ( . 102 ). ψ ′ = ψ ; θ′ = θ ; φ′ = φ ,

A′ -

. ,

, . , . -

( ), .

, .

. ,

, ,

. .

43. r rB

B r rA .

A ,

B r r r rB = rA + ρ . (2),r r r V B = V A + V BA ,

114

ρ.

. 103

, (2)

, (3)

r V BA -

B r r r r V BA = ω × ρ = ω × AB , -

ω ,

r a BA -

A, (4)

, -

,r . r r r VB = V A + ω × ρ , r r r r rε rω , a B = a A + a BA = a A + a BA + a BA

(5)

rε A; a BA -

rω a BA

: r r r r r r r r r r rε rω a BA = ε × ρ = ε × AB ; a BA = ω × ω × ρ = ω × ω × AB = ω × V BA . r (6) ε ,

(6) , .

B, .

, -

. 103 . §18.

, ,

. -

, , . .

, , , (

Oxyz

- Ax1 y1z1

. 104). . . 104

115

. r ( lativus -

re-

r - Vr .

, .

. (

)

, -

. e(

entraner -

r - Ve .

ωe ,

, . 44.

, . M

r r -

r r r r , . . ρ = x1i1 + y1 j1 + z1k1 . r r r r = rA + ρ .

40).

r rA ,

ρ,

. 105

( . -

M , ,

: ρ ≠ const .

r r ,

. 105 : r r r r r dr drA dρ r dρ V= = + = VA + , dt dt dt dt

r VA -

ρ, r

ρ:

116

(1)

; dρ / dt ,

ρ.

r r r r r r r r r r di1 dj1 dk1 dρ d . = ( x1i1 + y1 j1 + y1k1 ) = x&1i1 + y&1 j1 + z&1k1 + x1 + y1 + z1 (2) dt dt dt dt dt , , , r r r di1 / dt = ω × i1 ; , r r r r r r r dj1 / dt = ω × j1 ; dk1 / dt = ω × k1 , ω . (2) r r r r r r r r r r r r (3) x1 (ω × i1 ) + y1 (ω × j1 ) + z1 (ω × k1 ) = ω × ( x1i1 + y1 j1 + z1k1 ) = ω × ρ . r (2) ρ -

~r r r r dρ = x&1i1 + y&1 j1 + z&1k1 . dt (3) (4) (2), r ~r dρ d ρ r r = +ω × ρ . dt dt ρ = const , (4) r ρ, (5)

(4)

(5) . ,

, . (5)

r b , r , : ~r db db r r = +ω ×b , dt dt

(6) , . (

) .

. (5) ,

, r r r dρ / dt = Vr + ω e × ρ . (1), r r r r r V = V A + Vr + ω e × ρ . , r

(7)

(7) (8)

r Vr = 0 ,

117

r r V = Ve , (

)

. r r r r Ve = V A + ω e × ρ . (9) (2)

(8) (9)

. 35

(4)

. 40 ,

, )

(

. (9) :

(8) r r r V = Vr + Ve . . ,

(10) , .

45.

(

(8), r r r r r r r& r& a = V = V A + V&r + ω& e × ρ + ω e × ρ& . r r (11) V& = a A

(11) -

r r ; ω& e = ε e

r Vr

, , , ~r r r r ~r r r r r r r V&r = V&r + ω e × Vr ; ρ& = ρ& + ω e × ρ = Vr + ω e × ρ . (12) ,

(12) -

~r r V&r = ar . , (11) r r r r r r r r r r r a = a A + a r + ω e × Vr + ε e × ρ + ω e × [Vr + (ω e × ρ )] = r r r r r r r r = a A + ε e × ρ + ω e × (ω e × ρ ) + a r + 2(ω e × Vr ) .

r (13) a r = 0

(13) ,

r Vr = 0 .

(

r r a = ae . r r r r r r r a e = a A + ε e × ρ + ω e × (ω e × ρ ) ,

(13) (14) ,

, .

118

Ox1 y1z1

, ,

. 105

M. (14) (13), r r r r r a = ar + ae + 2(ω e × Vr ) . (15)

r ac .

(15) :

r r r r a = ar + ae + ac , :

(16) ,

, . r , r r ac = 2(ω e × Vr ) .

(17)

r r a c = 2ω eVr sin(ω e ∧Vr ) ,

(18)

r Vr (

ωe

. 106, a). ,

. . -

90o (

. 106, b). (18) r : 1) ω e = 0 ; , -

. 106 ,

r r 2) ω e ||Vr ; 3) Vr = 0 . , r

. , . M

V1 .

, ,

OB OM=R,

ω.

O, ,

. Oxy,

Ox1 y1 (

r r V1 = Vr ,

. 107, a). -

119

Ox1 y1 107

Oz (

ω = ωe .

re = OM . , -

. 107 . ,

Ve = ω e re = ωre , .

r Ve ⊥R 107, b.

. 107, b

r r r V = Vr + Ve . r r Vr = V1 . r Ve ⊥re , re = OM = R , Ve = ωR .

r V,

. OM=R.

r r V = Vr2 + Ve2 + 2VrVe cos(Vr ∧Ve ) = V12 + ω 2 R 2 ,

r r (Vr ∧Ve ) = π / 2 (

. 107, b).

. r r r ar = arτ + arn .

, ar = 0 .

ε e = dω e / dt = 0 .

r r r ae = aeτ + aen .

r a eτ = 0 ,

aen = ω e2 re = ω 2 ⋅ OM .

120

r a rn = 0 ,

ω e = ω = const

Vr = V1 = const ,

a rτ = dVr / dt = 0 .

r r r r a = ar + ae + ac . ,

a en = ω 2 R .

OM=R,

. 107, c.

a c = 2ω eVr = 2ωV1 ,

, OM=R, a c = 2ωV1 . .

Oz, r r (ω ∧Vr ) = π / 2 .

r Vr ,

. 107, c,

r a

. 107, c,

OM=R. r ∧r a = (a en ) 2 + a c2 + 2a en a c cos(a en a c ) = ω 4 R 2 + 4ω 2V12 = ω ω 2 R 2 + 4V12 , r r r ∧r aen ac π / 2 ( . 107, cos(a en a c ) = 0 ,

c).

OM=R: V = V12 + ω 2 R 2 ; a = ω ω 2 R 2 + 4V12 . , . Δϕ , M

Δt .

ωe

Vr (

r ΔVe ,

M′ . 108).

r ΔVr

. .

r ΔVr

r ΔVe

. ,

, -

. 108 -

. §19.

. .

121

. , , Oxyz

(

. 109, a). ΔO2 A2 B2 . O1 x1 y1z1

. . (

)

. .

. 109

O2 x 2 y 2 z2

, . r, e.

O2 x2∗ y2∗ z2∗ , O2 x2 y2 z2 (

O1 x1∗ y1∗ z1∗

. 109, -

b), O1

. . , ,

n n

. .

46.

ρ2 ;

r VO1

O1 x1 y1z1

Oxyz (

122

: -

. 110)

ω1

r VO2

O1 ;

ω2

O2 . O1 x1∗ y1∗ z1∗ , O2 x2∗ y2∗ z2∗ .

110 O2 x2 y2 z2 , ,

, . , -

. 110

M ,

. ,

r r r V = Vr + Ve .

r VO1 = 0

, :

(1)

ω1 = 0 ,

r r r r Vr = VO2 + ω 2 × ρ2 .

r VO2 = 0

(2)

( ω2 = 0 ,

r r r r r r r Ve = VO2 + ω 2 × r1 = VO2 + ω 2 × ( O1O2 + ρ2 ) , , M

r r1 -

; O1O2 , ( . 110). (2) r (3)r (1), r r r r r V = VO1 + VO2 + VO2 O1 + (ω1 + ω 2 ) × ρ 2 ,

r VO2O1 -

: (3) ,

(4)

O1 , 123

(7)

r r VO2 O1 = ω1 × O1O2 .

(5)

(4) r r r r r r r V2 = VO1 + VO2 + VO2 O1 ; Ω 2 = ω1 + ω 2 ,

(6)

r r r r V = V2 + Ω 2 × ρ2 .

(7)

, ,

r V2

r Ω2 ,

. (4) (7) . (5) r r r r r r VO2 O1 = ω1 × O1O2 = − (O1O2 × ω1 ) = O2 O1 × ω1 = mO2 (ω 1 ) . (8) (4), r r r r r r r r r r V = VO1 + VO2 + mO2 (ω 1 ) + mO2 (ω 2 ) + (ω 1 + ω 2 ) × ρ 2 , r r mO2 (ω 2 ) -

ω2

(8) (9) ,

O2 .

- On x n y n zn

, , r r r r r r r r r r r r r V = VO1 + VO2 + ⋅ ⋅ +VOn + mOn (ω1 ) + mOn (ω 2 ) + ⋅ + mOn (ω n ) + (ω1 + ω 2 + ⋅ + ω n ) × ρn (10)

On On

, , n r r r r r r r r r Vω = mOn (ω 1 ) + mOn (ω 2 ) + ⋅ ⋅ + mOn (ω n ) = ∑ mOn (ω i ) ,

(11)

n r n r r r r r r r Vn = VO1 + VO2 + ⋅ ⋅ +VOn + Vω = ∑ VOi + ∑ mOn (ω i ) .

(12)

i =1

n r r r r r Ω n = ω1 + ω 2 + ⋅⋅ +ω n = ∑ ω i .

(13)

i =1

124

(10) r r r r V = Vn + Ω n × ρn ;

: (14)

n r r n ⎛ n r ⎞ r r V = ∑ VOi + ∑ mOn (ω i ) + ⎜⎜ ∑ ω i ⎟⎟ × ρn . ⎝ i =1 ⎠ i =1 i =1 , n ,

(14) r Vn

On On xn yn zn

r Ωn (

(15)

. 111).

, .

(14)

(15) .

, , ,

On .

r Ωn ,

r Vω ,

. 111

, .

. .

. . ,

. .

, . . ω1 = ω 2 = ⋅⋅ = ω n = 0 (12), (14)

r Ωn = 0 . -

, : r r r r r V = Vn = VO1 + VO2 + ⋅⋅ +VOn . . O1 , O2 ,.., On

r r r VO1 = VO2 = ⋅⋅ = VOn = 0 . (12) , , r r r r r r r r Vn = Vω = mOn (ω1 ) + mOn (ω 2 ) + ⋅ ⋅ + mOn (ω n ) , (14)

r r r r V = Vω + Ω n × ρn .

(16) , -

(17) (18)

125

, . . O1 , O2 ,.., On

r Vω = 0 .

(13),

, .

(14) , On , , r Ωn , r r r r Ω n = ω1 + ω 2 + ⋅⋅ +ω n .

(19)

r r r r r r r V = Ω n × ρn = (ω1 + ω 2 + ⋅⋅ +ω n ) × ρn .

(20)

, , ,

, . ,

. . -

. .

r , VOi = 0 (i = 1, n) ,

, r

, , Oz , Oz1 , Oz2 ,.., Ozn

Oz1 , Oz2 ,.., Ozn

ω1 , ω 2 ,.., ω n , . , , :

Ω n = ±ω1 ± ω 2 ± ⋅⋅ ±ω n .

(21) -

P, . P

, ,

. V = Ω ⋅ MP ,

MP -

(22) .

. .

126

( . 37),

Ox 3 y 3z3 , Oxyz

(19),

ω = ω1 + ω 2 + ω 3 . r r r (23) ω1 , ω 2 , ω 3 -

(23)

Ox1 y1z1 , Ox2 y2 z2 , Ox3 y3 z3

, . 112 r r r r r r r r ω1 = ω1k1 = ω1k ; ω 2 = ω 2 i2 = ω 2 i1 ; ω 3 = ω 3 k 3 = ω 3 k 2 , r

(24)

ω1 = ψ& ; ω 2 = θ& ; ω 3 = φ& .

(25)

(23) , ω x3 = ω1x3 + ω 2 x3 + ω 3x3 ; ω y3 = ω1y3 + ω 2 y3 + ω 3 y3 ; ω z 3 = ω1z 3 + ω 2 z 3 + ω 3z 3 ;

(26)

ω x = ω1x + ω 2 x + ω 3x ; ω y = ω1y + ω 2 y + ω 3 y ; ω z = ω1z + ω 2 z + ω 3z . r r r ω1 , ω 2 , ω 3

(27)

. 112 (26)

(27),

, . , -

. (26)

(27)

: ⎡ω x ⎤ ⎡ω1x ⎤ ⎡ω 2 x ⎤ ⎡ω 3x ⎤ 3 3 3 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 3⎥ ⎢ + + ω ω ω ω = ; y 1 y 2 y 3 y ⎢ 3⎥ ⎢ 3⎥ ⎢ 3⎥ ⎢ 3⎥ ⎢ω z ⎥ ⎢ω1y ⎥ ⎢ω 2 z ⎥ ⎢ω 3z ⎥ ⎣ 3⎦ ⎣ 3⎦ ⎣ 3 ⎦ ⎣ 3 ⎦ (28) ,

. 112 ⎡ω x ⎤ ⎡ω1x ⎤ ⎡ω 2 x ⎤ ⎡ω 3x ⎤ ⎢ω ⎥ = ⎢ω ⎥ + ⎢ω ⎥ + ⎢ω ⎥ . ⎢ y ⎥ ⎢ 1y ⎥ ⎢ 2 y ⎥ ⎢ 3 y ⎥ ⎢⎣ω z ⎥⎦ ⎢⎣ω1z ⎥⎦ ⎢⎣ω 2 z ⎥⎦ ⎢⎣ω 3z ⎥⎦ r r r ω1 , ω 2 , ω 3 ,

(28)

127

Ox2 y2 z2 , Oxyz. r ω 3 = ω 3x 3 r ω 2 = ω 2 x2

[ [

[

ω 1 = ω 1x 1

ω3

. 112, r ω2 -

ω 3y3

ω 2 y2 ω1y1

] [ T ω 2 z ] = [ω 2 x T ω 1z ] = [ω1x T

ω 3z 3

= ω 3x 2

ω 3y2

ω 2 y1

ω1 -

Ox1 y1z1 ,

Ox2 y2 z2

ω1y

Ox3 y3 z3 Ox1 y1z1

] = [0 T ω 2 z ] = [θ& T ω 1z ] = [ 0 0 ω 3z 2

] T 0] ;

T 0 φ& ;

(29)

ψ& ] T .

: ⎡ω x ⎤ ⎡ω1x ⎤ ⎡ω 2 x ⎤ ⎡ω 3x ⎤ ⎡θ& ⎤ ⎡ 0 ⎤ ⎡0⎤ 3 1 2 3 ⎢ ⎥ ⎥ ⎥ ⎢ ⎥ T T ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ T ⎢ T T ⎢ ⎢ω y 3 ⎥ = Az (φ ) Ax (θ ) ⎢ω 1y1 ⎥ + Az (φ ) ⎢ω 2 y 2 ⎥ + ⎢ω 3 y 3 ⎥ = Az (φ ) Ax (θ ) ⎢ 0 ⎥ + Az (φ ) ⎢ 0 ⎥ + ⎢ 0 ⎥ ⎢ω z ⎥ ⎢ω1z ⎥ ⎢ω 2 z ⎥ ⎢ω 3z ⎥ ⎢ 0 ⎥ ⎢⎣φ& ⎥⎦ ⎢⎣ψ& ⎥⎦ ⎣ ⎦ ⎣ 3⎦ ⎣ 1⎦ 2 ⎦ ⎣ 3 ⎦ ⎣ (30) ⎡ω 3x ⎤ ⎡ω 2 x ⎤ ⎡ω1x ⎤ ⎡θ& ⎤ ⎡ 0 ⎤ ⎡ω x ⎤ ⎡ 0⎤ 2 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ω ⎥ = A ⎥ ⎢ ⎥ z (ψ ) Ax (θ ) ⎢ω 3 y 2 ⎥ + Az (ψ ) ⎢ω 2 y1 ⎥ + ⎢ω 1y ⎥ = Az (ψ ) Ax (θ ) ⎢ 0 ⎥ + Az (ψ ) ⎢ 0 ⎥ + ⎢ 0 ⎥ ⎢ y⎥ ⎢ ω 3z ⎥ ⎢ω 2 z ⎥ ⎢⎣ω 1z ⎥⎦ ⎢ 0⎥ ⎢⎣ψ& ⎥⎦ ⎢⎣ω z ⎥⎦ ⎢⎣φ& ⎥⎦ ⎣ ⎦ ⎣ ⎣ 2 ⎦ 1⎦ (31) , (7-10) . 37, 0 0 ⎤ ⎡1 ⎡cosψ − sin ψ 0⎤ ⎡cos φ − sin φ 0⎤ ⎢ ⎥ ⎢ ⎥ Ax(θ ) = 0 cosθ − sin θ ; Az(ψ ) = sin ψ cosψ 0 ; Az(φ ) = ⎢ sin φ cos φ 0⎥ . ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ 0 1⎦⎥ 0 1⎦⎥ ⎣⎢0 sin θ cosθ ⎦⎥ ⎣⎢ 0 ⎣⎢ 0 (26) (27), , ⎡ω x ⎤ ⎡ψ& sin θ sin φ ⎤ ⎡ θ& cos φ ⎤ ⎡ 0 ⎤ ⎡ψ& sin θ sin φ + θ& cos φ ⎤ ⎢ 3⎥ ⎢ (32) ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ & & ⎢ω y 3 ⎥ = ⎢ψ& sin θ cos φ ⎥ + ⎢− θ sin φ ⎥ + ⎢ 0 ⎥ = ⎢ψ& sin θ cos φ − θ sin φ ⎥ ; ⎢ω z ⎥ ⎢⎣ ψ& cosθ ⎥⎦ ⎢ 0 ⎥ ⎢⎣φ& ⎥⎦ ⎢ ⎥ ψ& cosθ + φ& ⎦ ⎣ ⎦ ⎣ ⎣ 3⎦

⎡ω x ⎤ ⎡ φ& sin ψ sin θ ⎤ ⎡θ& cos φ ⎤ ⎡ 0 ⎤ ⎡θ& cos φ + φ& sin ψ sin φ ⎤ ⎢ω ⎥ = ⎢− φ& cosψ sin θ ⎥ + ⎢θ& sin φ ⎥ + ⎢ 0 ⎥ = ⎢θ& sin φ − φ& cosψ sin φ ⎥ . ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ y⎥ ⎢ & cosθ ⎥ ⎥⎦ ⎢ 0 ⎥ ⎢⎣ψ& ⎥⎦ ⎢ & ⎢⎣ω z ⎥⎦ ⎢⎣ φ& cosθ ψ φ + ⎣ ⎦ ⎣ ⎦ , (26) (27) : ω x = θ& cos φ + φ& sin ψ sin φ , ω x 3 = ψ& sin θ sin φ + θ& cos φ , ω = θ& sin φ − φ& cosψ sin φ , ω = ψ& sin θ cos φ − θ& sin φ , y

ω z3

128

= ψ& cosθ + φ& ,

ω z = ψ& + φ& cosθ .

(33)

(34)

. (

. 113)

ω0 .

OB 3

r3 . . ,

, ,

, , -

. .

Oxy,

ω1 ( ω1 = 0 ), ω 2 , ω 3 . xOy

− ω0 ,

. -

(21), . 113 ω1′ = 0 − ω 0 ; ω 2′ = ω 2 − ω 0 ; ω 3′ = ω 3 − ω 0 ; ω 0′ = ω 0 − ω 0 = 0 . , , , O, C, B, ω1′ , ω 2′ , ω 3′ . , . 113, 1 2 , 1 3 . : V12 = ω1′ r1 = ω 2′ r2 ; V23 = ω 2′ r2 = ω 3′ r3 ; V12 2 3. 1 2, V23 , ω1′ r1 = ω 2′ r2 = ω 3′ r3 , r ω1′ r ω′ r ω′ =− 2 ; 1 = 3; 2 =− 3. r2 ω 2′ r1 ω 3′ r1 ω 3′ “-“ , . ω1′ , ω 2′ , ω 3′ , : r − ω0 − ω0 r2 r3 ω 2 − ω 0 = ; =− 3. =− ; ω2 − ω0 r2 r1 ω 3 − ω 0 r2 ω 3 − ω 0

129

ω 3 = (1 − r1 / r3 )ω 0 . 2.

ω1 (

114). ,

, ,

. 114

, . 2

ω1

ω0

“-“,

1 .

ω2

ω3 , ,

xOy ,

. ,

(21)

ω 3′ = ω 3 − ω 0 ; ω 0′ = ω 0 − ω 0 = 0 .

ω1′ = −ω1 − ω 0 ; ω 2′ = ω 2 − ω 0 ; ,

r − ω1 − ω 0 r3 ω 2 − ω 0 − ω1 − ω 0 r =− 2 ; = ; =− 3. ω2 − ω0 r1 ω 3 − ω 0 r2 r1 ω 3 − ω 0 ω 3 = ω 0 + (ω 0 + ω1 )r1 / r3 . , , ,

, , .

130

.......................................................................................................... , (3). (3). (4). (5). (6). (7). 1. § 1.

, 1.

................................. (8). ................................. (14).

(7). 2.

§ 2.

. 3.

(13). 4. (14).

7 13

(15). § 3.

...................................................................................... ( ) (19). (19). 6. ( ) (20). 7. (21). 8.

(20).

(21). 9. § 4.

(23). ................................................................... (24). (24). 11. (27).

, 10. (27). (28).

§ 5.

......................................................................... (28). 13. (

12. ) (29). 14.

(30).

15.

, (30). 16. ) (31).

(

§ 6. 32

................................................................................................... 17. 18.

(32). (33). (33). (35).

. (33). (34). (35). 19. (35).

§ 7.

. 20. (51).

. (46). 21. (49). (52).

......

(47). 22. (49). ,

§ 8. 24.

25.

(53). (53). 23. (54). (55) .................................................................................................... (59). (59). (59). (60). (62). (62). (62). (63).

2. § 9. § 10.

......................................................................... ...................................................................... (66). (66). (66). (66). (67). ..................................................................................... (68). (69). 28. (70). 29. (71). ................................................................................... (72). 31.

26.

§ 11. 27.

§ 12. 30.

65 65

(72). 32. (73).

(

)(75). § 13.

.. 33. (78). (78).

, , (79). (79). 34.

(79). .................................................. (80). 36.

§ 14. 35.

( ) (81). (81).

, (82). (83).

(84). (85). .........................................................

§ 15. 37.

, (88). (88).

, ,

(88).

(89). 38. (89). (91). 39. (96). (101).

(90). (96).

ยง 16. 40.

ยง 17. 42. ยง 18.

ยง 19. 46.

............................................................................................... (103). (103). 41. (109). ................................................... (112). 43. (113). ............................................. (114). 44. (115). 45. ( ) (117). ........................................................................ (121). (124). (125).

102

112

114

120

. . . .

60 84/16. . . . . . 7,67. .- . . 6,95. .3 . . . 1 18.07.94. . 634034, , . , 30.