Механика by Valdemar Dushinin

. .

2000

531 . .

. :

.–

, 2000. – 176 . .

, . . .

, .

. ,

. .

, . .

: . .

– ,

; . .

– ,

2000 ©

, 2000

1. §1.

, , . , , .

, , , .

, ,

. ,

, ,

. .

(

)

(

)

. . , . .

, . , ,

. «

» . ( -

). ,

. . , . , . 3

, .

, ,

. , . , ,

, , , . , . , . , . , (

)

. (

. ,

r r ma = F .

(1)

, .

, r r a = F / m.

, ,

, , .

, . , ,

, . , F = Îł (mm1 / r 2 ) ,

Îł m m1 .

; r -

, r = R, g = γ ( M / R 2 ) ≈ 9.8

/c2,

m1 = M ,

M,R -

. ( mg = P

). ,

m= P/g.

. . , 10-8.

. -

: ( );

( ); .

( ).

, . -

( ),

1 / .

1 . , .

. , , . ,

. ,

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

(1),

r r r r ma r + ma e + ma c = F .

(2) r

r ac = 0 ,

ωe = 0 , r r Ve = V .

r ae = 0 .

r r ma r = F .

(2) ,

, . , . , . . (

). . , .

, . .

, ,

. , , ,

. ,

, ,

, .

( ). . ,

, , ,

r r .r ( F1 , F2 ,.., Fn ), r r r r F = F1 + F2 + ⋅ ⋅ + Fn ,

ยง2.

, .

. .

, .

(

. .

r r ma = F , r , F

(1)

. (1) . ,

, . ,

r r r r a = dV / dt = d 2 r / dt 2 = && r,

r (r )

r r mr&& = F .

r F

r (r& )

(1)

(2) (t ) ,

, ,

, (

. .

r r r r F = F (t , r , r& ) .

(3)

, -

. ., , , .

(1)

a x = x&& ; a y = && y ; a z = && z,

: mx&& = Fx ; my&& = Fy ; mz&& = Fz .

(4)

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

(3)

Fx = Fx (t , x , y , z , x&, y&, z&) ; Fy = Fy (t , z , y , z , x&, y&, z&) ; Fz = Fz (t , x , y , z , x&, y&, z&) .

(1)

(5)

(

). , . . ab = 0 . aτ = &&s ,

a n = V 2 / ρ = s& 2 / ρ ,

: ms&& = Fτ ; m

= Fn , (m

s& 2

(6)

= Fn ) ; 0 = Fb .

, s, Fτ = Fτ (t , s, s&) ; Fn = Fn (t , s, s&) .

(

(7)

. .),

(1),

. ,

. , ,

, r , F

(1), .

, Fx , Fy , Fz

, (4)

Fτ , Fn , Fb

(6)

, . , . ,

. . . , . . . . 1. 2.

. ,

, .

, (

, .

(

. .),

. , ,

, .

, ,

, , , , . 5.

, .

6. .

, ,

, , ,

, .

. ,

f. , . 1).

( r mg . r N

r FT ,

fN , , (

. 1). ,

, . , .

( ,

), -

(

. 1). ,

r Oy|| N .

r Ox⊥N

Ox r Ox|| FT ,

r Oy⊥FT

: Fx = − mg sin α − FT ; Fy = N − mg cosα .

(4).

, :

mx&& = − mg sin α − FT ; my&& = N − mg cosα .

y = y& = && y = 0.

, ,

N = mg cosα , FT = mgf cos α .

, , : x&& = − (sin α + f cosα )g .

, . m

2. ,

R. ,

. , OO1

(

. 2).

r mg ,

r N

, ( ), . .2

, . , -

O1 ,

. 2.

r n

, r

, (

. 2).

, .

r N

, ( (

). :

Fτ = mg cosϕ ; Fn = N − mg sin ϕ ,

ϕ -

, (

. 2).

(6),

ms&& = mg cosϕ ; m

s& 2

ϕ s = Rϕ .

. 2,

= N − mg sin ϕ .

. ,

, s& = Rϕ& ; &&s = Rϕ&& ,

ρ = R,

: ϕ&& =

g cosϕ ; mRϕ& 2 = N − mg sin ϕ . R

, , . 2. . . . ,

. . , . . , .

, .

, . . ,

. ,

, .

, . . .

1. 2.

. ,

n r ri ri ri ri R = F1 + F2 + ⋅ ⋅ ⋅ + Fn = ∑ Fii = 0 ; i =1

(8)

n ri r ri r ri ri ri r r M O = mO (F1 ) + mO (F2 ) + ⋅ ⋅ + mO (Fn ) = ∑ mO (Fii ) = 0 . i =1

. ,

(8)

, ,

. , e,

r , Fe-

. ,

, .

, , , .

, . , -

, .

. m1 , m2 ,.., mn

, : r r r r r r r r r m1&& r1 = F1e + F1i ; m2 && r2 = F2e + F2i ,..., mn && rn = Fne + Fni .

(9)

n .

i 13

(9)

: r r r mi && ri = Fie + Fii

(i = 1, n ) .

(10)

(10) ,

3n :

zi = Fize + Fizi yi = Fiye + Fiyi ; mi && mi x&&i = Fixe + Fixi ; mi &&

(i = 1, n) .

(11)

3n , , 6n.

, . ,

. ,

, Fie ,

(10), r Fii -

, .

(11) ,

. .

, . ยง3.

, , .

. . ,

. . . : x = x (t ) ; y = y (t ) ; z = z (t ) . (4) . 1, : a x = x&& ; : Fx = mx&& ; Fx = my&& ;

, a y = && y ; a z = && z, Fz = mz&& .

: F = Fx2 + Fy2 + Fz2 ,

(1)

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

s = s( t ) ,

(6)

(2)

. 1, a n = s& 2 / ρ ,

aτ = &&s

, : Fτ = ms&& ; Fn = ms& 2 / ρ .

: F = Fτ2 + Fn2 ; tgα = Fτ / Fn = aτ / a n , r F

α -

(3) Fn .

, .

. ,

(1) - (3)

. .

. . 1.

. 1. 2. , . 3. ,

4. ,

, ,

, (1) - (3), . 5.

. m

1. x = a cosωt , y = b sin ωt . I

Oxy, ( . 3).

, . (4) . 1, mx&& = Fx ;

my&& = Fy .

V x = − aω sin ωt ; V y = bω cos ωt ; a x = x&& = − aω 2 cos ωt ; a y = && y = −bω 2 sin ωt .

, ,

Fx = mx&& = − mω 2 a cosωt = − mω 2 x ; Fy = my&& = − mω 2 b sin ωt = − mω 2 y .

(1)

(2):

r r r r F F x y F = mω 2 x 2 + y 2 ; cos(i ∧ F ) = x = − ; cos( j ∧ F ) = x = − . 2 2 2 2 F F x +y x +y

, . 3

. , OM = x 2 + y 2 .

r F

, OM,

, (

. 3).

, O. .

; ;

. ,

, ,

. x 2 / a 2 + y 2 / b 2 = 1,

(

. 3).

, ,

, .

, ,

, . ,

2. 1

4 / 2.

, . ,, ,

r mg r N(

4, a). , (

. 4,

b). , Q= N,

, (

. 4, c). .

, Oxy,

. 4, b. (

, .

), ,

Oy , (4)

, . 1,

: mx&& = mg − N ; my&& = 0 .

y = 0,

0 ≡ 0,

. a x = x&& = a .

Ox,

, , :

ma = mg − N .

, N = Q = mg − ma = m( g − a ) .

g = 9,8 Q = 1(9 ,8 − 4) = 5,8

, (

) . .

mg. , ,

, . ,

, .

. a = g,

N =Q=0

, ,

. ,

, , .

, ,

, . (a > g ) ,

, ,

. .

, , 18

, ,

0,3

. , ,

, 9 .

, .

, . 5a. , s r mg r T.

, , (6)

. 1.

Fτ = mg cosϕ ; Fn = T − mg sin ϕ ,

. . 1,

ϕ -

(6)

l , :

ms&& = mg cosϕ ; m

V2 = T − mg sin ϕ . l V = ωl ,

, :

T = mω 2 l + mg sin ϕ .

Tmax = mω l + mg 2

ϕ = π / 2(

. 5, a). ω = ω min ,

: ,

Tmax = Fl = 9

: ω min =

Fl − mg . ml

: ω min = 4,49 ( R = T′, ,

/ . . 5, b),

, . ,

, .

. ,

(4)

x = x (t ) ;

. 1, , y = y (t ) ; z = z (t ) , . , :

x = x (t , C1 , C2 ,.., C6 ) ; y = y (t , C1 , C2 ,.., C6 ) ; z = z (t , C1 , C2 ,.., C6 ) .

(4)

, . , ,

t0 = 0 .

: x (0) = x 0 ; y (0) = y 0 ; z (0) = z0 ; x&(0) = V0 x ; y&(0) = V0 y ; z&(0) = V0 z .

(4), :

(5)

x& = x& (t , C1 , C2 ,.., C6 ) ; y& = y& (t , C1 , C2 ,.., C6 ) ; z& = z&(t , C1 , C2 ,.., C6 ) .

(4)

(6)

t = t0 = 0 ,

. (4) ,

. (6) . 1. s = s(t , C1 , C2 ) . 20

(6)

: s(0) = s0 ; s&(0) = V0 . , , : s = s( t ) . ρ = ρ (t ) ,

. , . ,

. 1.

2. 3.

.1. .

, , .

4. . 1

.1 V0 = 15

/ ,

f = 0,1. x&& = − (sin α + f cosα )g .

: x&& = − kg . t0 = 0

(

r V0 ,

. 1

) -

V0 x = V0 .

t = t0 = 0 x(0) = 0 ; x& (0) = V0 .

Ox,

x& = V , x&& = V&

dV = − kg ; dV = − kgdt ; ∫ dV = − kg ∫ dt ; V = − kgt + C1 . dt

, t = 0 , V (0) = x& (0) = V0

V0 = 0 + C1 ,

C1 .

C1 = V0 .

V = V0 − kgt .

(a)

(a) x = V0 t −

kg 2 t + C2 . 2

(t = 0, x (0) = 0)

C2 = 0

: x = V0 t −

kg 2 t . 2

(b)

, .

V = 0,

t1 ,

(a)

S1 = x (t1 ) ,

(b)

: t1 = V0 / kg ; S1 = V02 / 2kg . k = 0,587 ,

, t1

S1 ,

2,61 2

19,57 .

, 60o .

ϕ&& =

g cosϕ ; mRϕ& 2 = N − mg sin ϕ . R

, N,

ϕ,

ϕ&& =

ϕ& = ω .

V = ωR ,

, , . . ω = ω (ϕ (t )) .

dω dω dϕ dω = =ω . dt dϕ dt dt

dω g = cosϕ ; N = mRω 2 + mg sin ϕ . dt R

(a)

ωdω =

(a)

ω (0) = 0 ; ϕ (0) = 0 . (a),

t = t0 = 0

ω2 g g g = sin ϕ + C1 . cosϕdϕ ; ∫ ωdω = ∫ cosϕdϕ ; 2 R R R

, 0 = 0 + C1 ,

C1 = 0

ω 2 = 2g sin ϕ / R

V 2 = 2 gR sin ϕ ; N = 3mg sin ϕ .

ϕ = ϕ1 = π / 3

(b)

V (ϕ1 ) = 4 3 gR ; N (ϕ1 ) = 3 3mg / 2 .

V. ,

L, ? , , , .

r mg ,

, . 6, Oy

Ox ,

xOy

. : Fx = 0 ; Fy = − mg .

, ,

. 13

m, : x&& = 0 ; && y = −g .

. t = t0 = 0

x (0) = 0 ; y (0) = h ; x& (0) = V0x = V ; y&(0) = V0 y = 0 .

, : 23

dx& = 0 ; ∫ dx& = ∫ 0 ; x& = C1 ;

(a)

dx = C1dt ; ∫ dx = C1 ∫ dt ; x = C1t + C2 .

(b)

: dy& = − gdt ; ∫ dy& = − g ∫ dt ; y& = − gt + C3 :

( )

dy = − gtdt + C3 dt ; ∫ dy = − g ∫ tdt + C3 ∫ dt ; y = − gt 2 / 2 + C3 t + C4 .

(d)

t=0

(a) - (d)

V = C1 ; 0 = 0 + C2 ; 0 = 0 + C3 ; h = 0 + 0 + C4 . C1 = V ;

C2 = 0 ;

C3 = 0 ;

C4 = h .

(a) - (d), : x& = Vx = V ; x = Vt ; y& = Vy = − gt ; y = h − gt 2 / 2 .

(e)

, . y = 0.

t1 ,

t1 = 2h / g .

(e)

. (e), L = x (t1 ) = V 2 h / g .

, , ,

, . )

(

Ox y = h − (g / 2V )x 2

(0, L).

(e), . , (

7. h, , 24

. . , m, S.

. . )

( r mg ,

(

)

r Fc .

, . 7, a.

Oxy -

- O1x1 y1 . O1x1 , ,

Ox Fx1 = mg − Fc .

Fx = mg kFc = kSV1 , V1 Fx = mg ;

.7 Fx1 = mg − kSV1 .

, : x&& = g ; mx&&1 = mg − kSV1 .

(a)

: x(0) = 0 ; x&(0) = V0 x = 0 .

(b)

x1 (0) = 0 ; x&1 (0) = V01x = V01 = V (h ) ,

( )

V (h ) -

O1x1

h. (a), ,

V, V = V (x (t )) .

x&& =

dV dV dx dV = =V dt dx dt dx

dV = g. dx

(d)

VdV = gdx .

(d), ,

, . , 0

(b), x

V (h ) V (h)

V2 VdV g dx = ; ∫ ∫ 2 0 0

V (h) h

= gx 0 ; 0

V 2 (h) = gh , 2

V (h) = 2 gh .

, V01 = V (h) = 2gh . O1x1 , x&1 = V1

x&&1 = dV1 / dt .

, dV1 / (mg − kSV1 ) = dt .

(a)

∫ du / u ,

u = mg − kSV1

du = − kSdV1 :

−

− kSdV1 1 − kSdV1 = dt ; = − kSdt . mg − kSV1 kS mg − kSV1

. V01 V1

∫

V01

V1 ,

(c), t V1 :

t − kSdV1 t = − kS ∫ dt ; ln(mg − kSV1 )|VV1 = − kSt 0 . 01 mg − kSV1 0

ln(mg − kSV1 ) − ln(mg − kSV ) = − kSt

, 0

t ,

, . ,

( ),

(

, . . Fx1 = Fx1 (V1x ) = Fx1 ( x&1 ) . 1

( (

)

). (

) Vl .

t→∞

(e).

Vl = mg / kS . V01 = 0 ,

, Vl (

. 7, b), ,

. , , .

V01 > Vl ,

, V01 < Vl ,

Vl ,

(

7, ).

V10 = 2gh > Vl = mg / ks ,

. 7, d. .

, . (

) , ,

. .

. . . , .

, , .

, 27

. , . 1.

Fx = Fx (t )

Fx (t ) = const . Fx = Fx (x&) .

Fx = Fx (x& 2 ) .

3. , 3 - 7. 4.

Fx = Fx (t , x , x&)

Fx = Fx (x , x&) ,

, Fx = Fx (x ) .

, .

, Fx = H (t ) − μx& − cx ,

c -

(

, ).

mx&& = − μx& − cx + H (t )

x&& +

μ m

x& +

c 1 x = H (t ) , m m

(7)

. ,

, x&& +

μ m

x& +

c x=0 m

(8)

, . . 4.

. , .

. ,

(

)

. ,

(

. 8, a). , ,

, . ,

F ,

. ,

. (

), l0 .

, ,

Fx = − F

= − cx ,

. 8, b, ,

. :

mx&& = − cx

mx&& + cx = 0 .

k 2 = c / m,

(9)

x&& + k 2 x = 0 .

(10)

(8).

x = e λt

, (10),

. (λ + k ) = 0 , 2

λ + k = 0. : λ1 = +ik ; λ2 = −ik . 2

λ ⋅t

(10) 29

x = C1∗ e ikt + C2∗ e − ikt .

, :

e ikt = cos kt + i sin kt ; e − ikt = cos kt − i sin kt .

(10) x = C1 cos kt + C2 sin kt , C1

(11)

, C1 = C1∗ + C2∗ ; C2 = i (C1∗ − C2∗ ) .

, (11) x& = − kC1 sin kt + kC2 cos kt .

(12)

. x0

x& 0 .

Ox ( x (0) = x 0 ; x&(0) = x& 0 )

(11)

, (12)

t = 0,

: x 0 = C1 ; x& 0 = kC2 . (11),

C1 = x 0 ; C2 = x& 0 / k .

: x = x 0 cos kt +

(11)

x& 0 sin kt . k

(13)

. (10),

. A

β,

C1 = A sin β ; C2 = A cos β .

(14)

x = A sin kt sin β + A cos kt cos β x = A sin(kt + β ) .

(15)

(14)

C2 :

(16)

A = C12 + C22 ; tgβ = C1 / C2 ; sin β = C1 / A ,

. :

A = x 02 + x& 02 / k 2 ; tgβ = x 0 k / x& 0 ; sin β = x 0 / A .

(17)

(16), (17) (14)

. 2π ,

. ,

, ,

(15),

, β

(17). . ,

(15)

(

). ,

(10) . , (

(

)

, (10), O (

. . 8, b),

, .

sin(kt + β ) = 1,

(15) x max = A .

(18)

, . (15) α = kt + β

(19)

β -

, , 2π ,

(15) . , (

, α

)

(15), α + 2π ,

(

b).

. 8,

T, .

A sin(kt + kT + β ) = A sin(kt + β ) .

f (t + T ) = f (t )

kT = 2π .

2π . k

(20)

. 9, α

O, A,

Oα .

β.

, .9

: ν=

1 k = . T 2π

(21)

(21) k = 2πν .

(22)

(

2π

)

, (k

(

(1/c). / . ,

, (

). ,

. , , ( (

. 10),

, ) )

(

. 10

. 2

. 3, 32

mlϕ&& = − mg sin ϕ

ϕ&& + (g / l ) sin ϕ = 0 .

. ,

sin ϕ = ϕ .

k2 = g / l,

: ϕ&& + k 2ϕ = 0 . (15), (20): T=

l 2π = 2π . k g

(23)

(

) , . , ,

, ,

. . .

, .

, Δ

. :

= mg ; cΔ

= mg ; c = mg / Δ

(24)

. Δ=Δ

+x,

Oxy. = Δ = (Δ

+ x) ,

: mx&& = mg − F ; mx&& = mg − c(Δ

+ x ) ; mx&& = mg − mg − cx ; mx&& + cx = 0 .

(25)

x&& + k 2 x = 0 , k = c/ m = g/ Δ

, ,

(

(26)

) , 33

. ,

, . Δ

, (26) .

(20) 8.

c2 .

. (26) T = 2π Δ

(20),

/g,

P. , P. Δ1

= Δ1

+ Δ2

1 2

/ (c1 + c2 )

c =

= P / c1

Δ2

= P / c2 ,

c + c2 p P P = . + =P 1 c1c2 c c1 c2

, c1

, .

. T = 2π

P c1 + c2 P = 2π . g c1c2 gc

, = Δ2

Δ1

=Δ

P = c1Δ 1 Δ

2Δ 2

= P / (c1 + c2 ) = P / c . = 1+

2 )Δ

, T = 2π

P P = 2π . g (c1 + c2 ) gc

. , . ,

. .

. . , , ,

, (

. 11),

. 11

r r Fc = − μV .

, ( ,

.12

)

r Fc

, ,

. (

. 8, b), (

, . 12).

. ,

, Fx = − F

− Fc ,

: mx&& = − F

− Fc ; mx&& = − cx − μVx ; mx&& + μx& + cx = 0; x&& + 2hx& + k 2 x = 0 .

(27)

; 2h = μ / m ,

(27)

: k = c/ m h -

. : λ2 + 2hλ + k 2 = 0 ; λ1,2 = − h ± h 2 − k 2 .

(27) ,

. 1.

h<k.

λ1,2 = − h ± ik1 ; i = − 1 ; k1 = k 2 − h 2 > 0 .

, (27) 35

x = e − ht (C1 cos k1t + C2 sin k1t ) ,

(28)

x& = − he − ht (C1 cos k1t + C2 sin k1t ) + e − ht (− C1k1 sin k1t + C2 k1 cos k1t ) .

( t = t 0 = 0 ; x (0) = x 0 ; x& (0) = x& 0 ) , C1 = x 0 ; C2 = (x 0 h + x& 0 ) / k1 . (28), ( )

t=0

(29)

(28), (29)

(27)

, x = e − ht (x 0 cos k1t +

x 0 h + x& 0 sin k1t ) . k1

β,

C1 = A sin β ; C2 = A cos β .

: (28)

(27) x = e − ht ( A sin β cos k1t + A cos β sin k1t ) = Ae − ht sin(k1t + β ) .

C1 , C2

A = C12 + C22 = x 02 +

(x 0 h + x& 0 ) 2 k12

(31)

; tgβ =

C1 x 0 k1 C x ; sin β = 1 = 0 . = C2 x 0 h + x& 0 A A

, (

(30) x

e − ht

. . . 13. e − ht

, x (t + T ) = x (t )

(30)

. T1

. 13

(30)

(31)

0 2π . (31),

. 2π ,

sin(k1t + k1T1 + β ) = sin(k1t + β )

(30). k1T1 = 2π ,

(32)

T1 = 2π / k1 = 2π / k 2 − h 2 .

T1 sin(k1t + β )

, 2π

(

. 14)

(

)

. , T1 =

(32)

2π k −h 2

2π k

1 1 − (h / k )

1 − (h / k )

T1 > T ,

h << k

(h / k ) 2 → 0 ,

T1 ≈ T ,

, .

. (

. 13)

, t1 ,

t1 + T1 / 2 .

, − A(t1 ) | x (t1 )| = =e A(t1 + T1 / 2) | x (t1 + T1 / 2)|

hT1 2

, D = e − hT1/ 2 ,

. : d = ln D = hT1 / 2 .

(33)

. ,

, 37

(33)

μ.

h 2.

h=k.

λ1,2 = −h ,

(27) x = e − ht (C1 + C2 t ) .

(34)

h>k.

λ1,2 = − h ± h 2 − k 2

(27)

x = C1e λ1t + C2 e λ2 t .

(35)

(

)

, λ1

λ2

, (

)

. 14

. , . 14. .

(

r Q( t ) ,

. ,

, . 15, . 15

, .

r Q

, x,

. 16

r Q

, Qx = H 0 sin ωt ,

ω -

H0 -

. ,

μx&

mx&& = − cx − μx& + H 0 sin ωt .

cx m,

. 13)

. 16

x&& + 2hx& + k 2 x = H sin ωt . c/ m = k h ); μ / m = 2h ,

(

(36)

; H0 / m = H .

. x = x1 + x 2 ,

x1 -

x2 -

(27), (36).

. 1.h < k ; x1 = A1e − ht sin(k1t + β ) ; k 1 = k 2 − h 2 . 2.h = k ; x1 = e − ht (C1 + C2 t ) . 3.h > k ; x1 = C1e λ1t + C2 e λ2 t ; λ1,2 = − h ± h 2 − k 2 . x2

(36) . x 2 = A2 sin(ωt + ϕ ) ,

(36)

ϕ -

(36) H sin ωt = H sin[(ωt + ϕ ) − ϕ ] = H sin(ωt + ϕ ) cos ϕ − H cos(ωt + ϕ ) sin ϕ x& 2

x&&2 : x& 2 = A2ω cos(ωt + ϕ ) ; x&&2 = − A2ω 2 sin(ωt + ϕ ) .

, x2

(36),

− A2ω 2 sin(ωt + ϕ ) + 2hA2ω cos(ωt + ϕ ) + A2 k 2 sin(ωt + ϕ ) = = H sin(ωt + ϕ ) cos ϕ − H cos(ωt + ϕ ) sin ϕ .

, ( A2 k 2 − A2ω 2 ) sin(ωt + ϕ ) + 2hA2ω cos(ωt + ϕ ) =

= H cosϕ sin(ωt + ϕ ) − H sin ϕ cos(ωt + ϕ ) . sin(ωt + ϕ )

cos(ωt + ϕ )

: 39

A2 (k 2 − ω 2 ) = H cosϕ ; 2 hωA2 = H sin ϕ .

ϕ:

A2 H

A2 =

( k 2 − ω 2 ) 2 + 4h 2 ω 2

; tgϕ = −

2 hω

k −ω2 2

(37)

−π

, x2 =

(36): H ( k − ω ) + 4h ω 2

2 2

sin(ωt + ϕ ) .

(38)

h<k,

x = A1e − ht sin( k1t + β ) +

( k − ω ) + 4h ω 2

2 2

sin(ωt + ϕ ) ,

(39)

(39) . ,

, .

(39))

(

− ht

. ,

(38). A2 =

H ( k 2 − ω 2 ) 2 + 4 h 2ω 2

H /ω2 [(1 − ω 2 / k 2 ) 2 + 4 (h 2 / k 2 )(ω 2 / k 2 )

z =ω / k, ξ = h/k; :

(

)

. A0 = H / k = (H 0 / m)(m / c) = H 0 / c . 2

. (

)

H0 ,

A0 ,

O ω (37)

: η=

A2 = A0

1 (1 − z ) 2 + 4ξ 2 z 2

; tgϕ = −

2 hω k −ω 2

= 2ξ

z 1− z2

, (40)

. .

(

)

ξ z

. 17. η

ω,

. (40)

, ,

ω << k ( z << 1 ),

. 17

( η ≈ 1 , ϕ ≈ 0 ). ω = 0,2 ÷ 0,3k ( z = 0,2 ÷ 0,3 ),

( η > 1 ), ,

. (z (40).

4ξ 2 z 2

1),

, . ,

ξ < 0,1,

( ω≈k

ξ = 0,05 , η ≈ 10 ), z ≈ 1.

, ,

A0 , −π / 2

, .

, ξ.

ω=k

k = 1,

h=ξ =0

, ω≠k x 2 = H sin ωt / (ω 2 − k 2 )

(41)

(36)

x 2 = Bt cosωt ,

x2 = −

Ht Ht cosωt = sin(ωt − π / 2), 2ω 2ω

(42)

. ,

(ω > k

ω→∞

z → ∞ , η → 0,

(37), (38)

z > 1) ,

ϕ → −π .

(40)

. .

1. 2. 3. ,

4. . 5. ,

, ,

. 6. , . 5. . , (11)

, .

. 2.

. ,

, .

. . , ( )

(

) . , , . , . ,

, ,

. , . ,

, . -

, , . . , ,

, . 43

. . §4. . . 6.

. ,

, : n

M = m1 + m2 + ⋅ ⋅ ⋅ + mn = ∑ mi .

(1)

i =1

, ,

(1)

∫ dm .

(2)

( m)

mi = pi / g ,

, ,

, . C,

r r r m1r1 + m2 r2 + ⋅ ⋅ ⋅ + mn rn r 1 = rC = m1 + m2 + ⋅ ⋅ ⋅ + mn M r ri -

∑ mi ri ,

(3)

i =1

. , :

1 xC = M

x i , y i , zi -

1 ∑ mi xi ; yC = M i =1

1 ∑ mi yi ; zC = M i =1

∑ mi zi ,

. ,

(4)

i =1

r rC = 0 .

, (4)

. :

∑ mi xi

i =1

= Mx C ;

i =1

∑ mi yi = MyC ; ∑ mi zi

= Mz C .

(5)

(3)

(4) :

1 r rC = M

∫ rdm ; x C ( m)

1 M

∫ xdm ;

1 M

yC =

( m)

∫ ydm ; zC

( m)

1 M

∫ zdm .

(6)

( m)

, -

. , ,

, .

mi g

. ,

(3) – (5) , .

, ,

. .

. . Jx , J y , Jz ;

: J xz = J zx ,

J xy = J yx ,

JO ;

J yz = J zy .

Oxyz. , . ,

, . ,

. 45

( Jx =

mhx2

= m( y + z ); J y = 2

. 18):

= m(x + z ) ; J z = mhz2 = m(x 2 + y 2 ) .

mhy2

(7)

: J O = mr = m(x + y + z ) . 2

(7)

(8)

: Jx + J y + Jz = 2JO

(9)

, (7),

. 18

(8). :

J xy = J yx = mxy ; J xz = J zx = mxz ; J yz = J zy = myz .

(10)

, .

i =1

i =1 n

i =1

J x = ∑ mi hix2 ; J y = ∑ mi hiy2 ; J z = ∑ mi hiz2 ;

(11)

J O = ∑ mi ri2 ;

(12)

i =1

J xy = J yx = ∑ mi xi yi ; J xz = J zx = ∑ mi xi zi ; J yz = J zy = ∑ yi zi .

(13)

: Jx =

∫

hx2 dm ; ( m)

∫ hy dm ;

Jy =

Jz =

( m)

JO =

∫r

∫ hz dm ; 2

(14)

( m) 2

dm ;

(15)

( m)

J xy = J yx =

∫ xydm ;

J xz = J zx =

∫ zxdm ;

∫ yzdm .

(16)

( m)

J yz = J zy =

, (9)

. (

Oz, ,

J z = Mρ z2 .

ρz

(17)

, ,

Oz .

(6) – (16)

, ⋅ . 2

. 19 Ou ,

(

J u = J x λ 2 + J y μ 2 + J zν 2 − 2 J xy λμ − 2 J yz μν − 2 J xz λν ,

(

λ , μ ,ν . 19),

(18)

Oxyz

Ou λ = cosα ; μ = cos β ; ν = cosγ .

(

)

, ⎡ Jx ⎢ θ = ⎢− J yx ⎢ − J zx ⎢

− J xy Jy − J zy

− J xz ⎤ ⎥ − J yz ⎥ , J z ⎥⎥

(19)

. θ A( x1y1z1 )

θ A(1) ,

Ax1 y1z1 . r ri , xi , yi , zi

. (

. 21)

, . . .6

(

, ; ,

. . 47

, , . , , ,

, .

J xy = J xz = 0 ,

Ox-

. ,

. , , , : ⎡Jx θ = ⎢⎢ 0 ⎢⎢ 0

0⎤ 0 ⎥. ⎥ J z ⎥⎥

Jy 0

(20)

, , . , (

. . 20)

, , . ,

. Oxyz,

, xOy

(

. 20

, 21).

Ox , zi = 0

J xz = J zx = 0 ; J yz = J zy = 0 .

Oz, .

(xi , yi ) (xi ,− yi ) ,

. 21.

J xy = J yx = ∑ mi xi yi − ∑ mi xi yi = 0 .

, Oxyz

, . -

21),

(

Cx c y c z c

. 21

. , . ; 2)

: 1)

, .

( ,

. 21)

- xOy, xOz,

Ox .

Oy ,

Oz ,

. . 22

. . ,

AB Cx (

. 22). ,

B .

. , , .

( .

)

, ,

, (

. )

, (

. 23, 24).

. 23

zc ,

, (

(14) (

. 23, a). m, l. γ dm = γdy , m / l ; dy . .),

hz = y

, Jz =

∫

hz2 dm =

( m)

m y3 γ ∫ y dy = ⋅ l 3 0

z. l

ml 2 . 3

(21) zc

: J zc

m y3 = ⋅ l 3

l/2

= −l/2

m l3 l3 ml 2 ( + )= . l 24 24 12

(22)

J zc .

, . . Oxyz,

, dm = γds = γdxdy ,

), dx

γ -

(14)

ds -

. 23, b: hx = y ; hy = x ;

dy. ,

∫ ( m)

(

m / ab ;

hz2 = x 2 + y 2 . Jx =

. 23, b. m,

hx2 dm = γ

2 ∫ hx ds = (S )

m m mb 2 2 2 ; y dxdy dx y dy = = ab ( S∫ ) ab ∫0 ∫0 3

(23)

∫

Jy =

∫

Jz =

( m) 2 hz dm =

( m)

hy2 dm =

∫

hy2 ds

(S ) 2

∫

( x + y )dm = 2

m m 2 ma 2 2 ; x dxdy = = x dx ∫ dy = 3 ab ( ∫S ) ab ∫0 0

( m)

(h y2 + hx2 )dm =

( m)

∫

hx2 dm +

( m)

(24)

∫ h y dm = J x + J y (25) 2

( m)

: ,

, ,

. (23)

(24) (25),

J z = mb 2 / 3 + ma 2 / = m(a 2 + b 2 ) / 3 .

(

(26)

). Cx c y c z c

(

. 23, c). m, J zc =

∫

hz2 dm = c

( m)

∫R

hzc = R ,

dm = R 2

∫ dm = mR

( m)

(27)

( m)

J zc = J xc + J yc . J xc = J yc ,

(28)

J xc = J yc = J zc / 2 = mR 2 / 2 .

(28)

, . (8),

zc 2J zc = J xc + J yc + J zc .

(28).

R. . 23, d.

, ρdϕ

dρ ,

, 23,d).

, ρ dϕ -

dρ -

(

dm = γdϕρdρ . J zc =

πR 2

, 2π

∫

γ = m / πR 2 , R

hz2 = ρ 2 ,

mR 2 . dϕ ∫ ρ dρ = ρ dρ = 2 ∫ 2 R 0 0 3

(29)

J xc = J yc = J zc / 2 = mR 2 / 4 .

(30)

J zc

dρ

ρ,

dm = γ ⋅ 2πρdρ = (2m / R 2 ) ρdρ . zc

dJ zc = ρ 2 dm = 2( m / R 2 ) ρ 3dρ .

. R

∫

dJ zc = (2m / R 2 ) ∫ ρ 3dρ = mR 2 / 2 .

( m)

J zc =

, .

dm = γdv .

m, ρ, dv = 2πρdρH ,

γ = m / V = m / πR 2 H .

dρ (

. H. . 24, a)

dm = (2 m / R 2 )ρdρ , dJ zc = ρ 2 dm = 2(m / R 2 )ρ 3dρ .

, J zc =

∫

dJ zc = (2m / R 2 ) ∫ ρ 3 dρ = mR 2 / 2 .

( m)

(31)

. 24

, dz,

(29)

dJ zc ,

H. zc

Cx c y c zc (

. 24, b)

, (14).

, : J xc = J yc

mR 2 m⎛ H2 2⎞ . = ⎜⎜ + R ⎟⎟ ; J zc = 2 4⎝ 3 ⎠

m, (

(32)

R. . 25, c). : J xc = J yc = J zc = J ,

J -

(9) 3J = 2 J C , JC ( C).

J = (2 / 3) J C .

. ,

dρ .

ρ,

γ = m/V .

dv = 4πρdρ , V = (4 / 3)πR 3 ,

dJ C = ρ 2 dm = (3mρ 4 dρ ) / R 3 . JC =

∫

dm = γdv .

(15) r 2 = ρ 2 , : ,

dJ C =

∫ρ R3

dρ =

( m)

3 mR 2 . 5

(33)

(34)

J xc = J yc = J zc = (2 / 5)mR 2 .

, (

. 24, c),

, ,

10.

( -

). z

zc ,

( Oxyz

25).

Cx c y c z c ,

, Oz

Cz c

. 25

(10), Jz = ∑

(6):

mi (xi2

yi2 ) ;

2 2 + y ci ). J zc = ∑ mi (x ci

. 26 xi = x ci ; yi = y ci + d ; zi = z ci . Jz , 2 2 2 2 J z = ∑ mi (x ci + y ci + 2 y ci d + d 2 ) = ∑ mi (x ci + y ci ) + 2d ∑ mi y ci + d 2 ∑ mi .

J zc ;

∑ mi y ci

= My C ,

yC = 0 ;

Md 2 .

, (35)

J z = J zc + Md 2 .

. ,

Cx c y c z c

(18), . ,

. 9. .

, .

1. (

. 26), Ox

J x = mb 2 / 3 ; J y = ma 2 / 3 ,

, m -

. . 26

, . J y = J yc + md 22

J xc = J x − md 12 ; J yc = J y − md 22 .

J x = J xc + md 12 ;

d1 = b / 2 ,

Cx c

Cyc - d 2 = a / 2 (

. 26).

, -

, xc

J xc =

yc :

ma 2 ma 2 ma 2 mb 2 mb 2 mb 2 . − = − = ; J yc = 3 4 12 3 4 12

J zc = J xc + J yc ,

J zc = mb 2 / 12 + ma 2 / 12 = m(a 2 + b 2 ) / 12 .

2. (

27). 2 l1 ,

m1 ,

. .

z. ,

Jz =

J zC

J zK1

J zK 2 ,

, J zC

J zK2

J zK1

: -

. J zK1 = J zK 2 = J zK .

z, J z = J zC + 2 J zK .

(22), m = m1 ,

l = 2 l1 ,

J zC = ml 2 / 12 = m1 (2l1 ) 2 / 12 = m1l12 / 3 .

z -

. 27

. Cz c ,

Czc

(28), m = m2 , R = r , M = m2 ,

. 27).

z, d = l1 + r .

J zc = mR / 2 = m2 r / 2 . 2

zc ,

J zK = J zc + Md 2 = m2 r 2 / 2 + m2 (l1 + r ) 2

(35), 2 J zK = m2 r 2 + 2 m2 (l1 + r ) 2 .

J z = J zC + 2 J zK = m1l12 / 3 + m2 (3r 2 + 2 l12 + 4 l1r ) .

§5. . (

. . 5).

11. . (

r F -

r r mdV / dt = F .

) ,

- m

, : r mV

r Fdt -

r r d (mV ) = Fdt .

(1)

, (

), :

. .

r q

r V.

⋅ / . -

, , r V0

. ,

r V,

(1). : t

r r r mV − mV0 = ∫ Fdt .

(2)

(2) (

) : , ⋅ .

(1)

(2)

. ,r

r r F = F (t )

(2)

F = const .

r F=0

. (2)

r r mV = const = mV0 ,

(3)

, ,

, . , mV x = const = mV0 x .

Fx = 0 .

Ox, 12.

(10)

. .2

r r r mi dVi / dt = Fie + Fii , i = 1, n .

, n

(

), r re ri d m V R = +R . ∑ i i dt i =1

n r n r r d e m V F ( ) = + ∑ dt i i ∑ i ∑ Fii i =1 i =1 i =1 n

r Q

, . . n r r Q = ∑ miVi .

(4)

i =1

, :

r dQ r e =R , dt

(5)

r Q0

: dQ = R e dt .

(5), t0

r Q,

, : t r r r Q − Q0 = ∫ R e dt .

(6)

. (5)

(6)

, :

dQ y

dQx dQz = R ye ; = R xe ; = Rze ; dt dt dt Q x − Q0 x =

∫

Rxe

; Q y − Q0 y =

∫

R xe

(7) t

; Qz − Q0 z = ∫ Rze . t0

(8)

Qx , Qy , Qz

Q0x , Q0 y , Q0z -

. .

r Re = 0

(6)

r r Q = const = Q0 ,

(9)

. R xe = 0 . Qx = const = Q0x .

Ox,

(8) (10)

. (4),

. (

)

. . ,

(4)

: n

i =1

Qx = ∑ miVix ; Qy = ∑ miViy ; Qz = ∑ miViz .

(11)

r , Vi

(4), ,

Vix , Viy , Viz

(11) -

. ,

r r MrC = ∑ mi ri .

(4) M-

r VC -

r r Q = MVC ,

(12)

M-

r r MVC = ∑ miVi ,

r r MdrC / dt = ∑ mi dri / dt

r VC -

, (12)

, :

Qx = MVCx ; Q y = MVCy ; Qz = MVCz .

(13)

, k, , r Q=

∑ Q j = ∑ M j VC j , j =1

(14)

j =1

r , VC j -

Mj -

, . , .

, ,

r r r Vi = Vie + Vir , n n r r r = + m V m V ∑ i i ∑ i ie ∑ miVir n

1=1

i =1

r r Qe = MVC .

i =1

r r r Q = Qe + Qr . r

ωe = 0 ,

r r Qr = MVCr ,

(12)

r VCr -

. (

r VCr = 0 , r r r Q = Qe = MVC .

, ( -

)

, . .

. , ,

. ;

: );

(

. .

, .

, ,

. ,

. , (

. . , .

, . 13.

. (5), (12). ,

r r dVC = Re M dt

, : r r Ma C = R e .

(15)

. : , . (15) : Mx&&C =

R xe ;

My&&C =

R ye ;

(16)

Mz&&C = Rze .

, . (16) (4) .6

, :

i =1

yi ; Mz&& = ∑ mi && Mx&&C = ∑ mi x&&i ; My&&C = ∑ mi && zi .

(17)

r Re = 0 ,

(15),

r r VC = const = V0C , r V0C = 0 ,

, .

, Ox, Rxe = 0 .

(16), ,

Ox VCx = const = V0Cx , V0Cx = 0 ,

, x C = const = x 0C ,

x 0C -

. 14.

. .

1. . . .

, , , .

, , ,

, m

. 61

, .

, . 2.

, (

, . ,

, .

, . , . 15. . , .

r aC

. , .

r r -

r r Mr&& = R e , r , a -

r r r Ma = Ma C = R e

(18)

, :

Mx&& = Rxe ; My&& = R ye ; Mz&& = Rze ,

x, y, z -

(19)

. .

, ,

16. . . 62

. ,

, .

. .

, .

( t > 0 ).

2. 3.

, .

, (

6. ,

7. . .

8. . ,

. . . : -

, . , (17) .

. , : ) (

)

; 63

) ( ) . 1

m1 ,

V01 m2 ,

(

. 28).

, ω = ω (t ) . ,

, .

. ,

. 28 r m1g ,

r m3 g .

r m2 g

r N1

r N2.

, .

. Oxy, .

. 28,

, , Rxe

= 0.

, Ox: Qx = const = Q0x . 64

, V01 (

. 28, a). ,

Q0 x = Q01x + Q02 x

r V01 . + Q03 x = m1V01 + m2V01 + m3V03 = (m1 + m2 + m3 )V01 . ( ) Q01x , Q02 x , Q03x -

( )

. , .

V 3 = ωr .

, V1 ,

28, b).

( ,

r V1 .

r V3

, Qx = Q1x + Q2 x + Q3x

r V1 (

. 28, b).

r r V3 + V1 . = m1V1 + m2V1 + m3 (ωr + V1 ) = (m1 + m2 + m3 )V1 + m3ωr .

(a)

(b)

(b),

V1 = V01 − m3 rω (t ) / (m1 + m2 + m3 ) .

, , Ox l

, . m1

m2 .

. ,

, .

. .

r m1g

r m2 g ,

r N,

. ,

. , .

. 29,

Ox , . ( Ox

), , :

. 29

x C = x 0C = const .

( )

x 01

x 02 ,

x2 .

. m1x1 + m2 x 2 m1x 01 + m2 x 02 = . m1 + m2 m1 + m2

M = m1 + m2 , m1 ( x1 − x 01 ) + m2 ( x 2 − x 02 ) = 0 .

, Δx1

Ox Δx 2 ,

m1Δx1 + m2 Δx 2 = 0 . Δx1 .

Ox Δx1 . Δx 2 = l + Δx1

Δx1 = −

, m1Δx1 + m2 (l + Δx1 ) = 0 .

m2 l. m1 + m2

l. (

. 29),

, .

. ,

, , ,

. ,

(

), x,

, ,

m1Δx1 + m2 Δx 2 + ⋅ ⋅ ⋅ + mn Δx n = 0 , , Δx1 , Δx 2 ,..., Δx n -

m1 , m2 ,..., mn -

(20)

x. , .

1 .

3. m2 ,

3 .

2 ,

3 .

. 1-7,

, . ,

8, ,

Ox .

(20). Δx1

2 , (

. 30, , h). 3

3 . 30

Δx 3 = Δx1 .

, 2 , ,

h cosα . Ox Δx1 . Δx 2 = h cosα + Δx1 . (20), m1Δx1 + m2 (h cosα + Δx1 ) + m3Δx1 = 0 ,

Δx1 = −

m2 h cosα . m1 + m2 + m3

, , Ox. AB

Ox.

2 , 2l

c. ϕ = ωt .

l0 ,

, , m2 .

. ,

r m1g

. 31

r m2 g ,

Oxy

r r N1 N2, r F ,

. 31.

. ,

ϕ. R xe = − F ;

R ye = N 1 + N 2 − (m1 + m2 ) g .

N ,

= cx ,

Rxe = − cx ; R ye = N − (m1 + m2 ) g .

, My&&C = N − (m1 + m2 )g .

(16)

My&&C = m1 && y C1 + m2 && y C2 .

x C1 , y C1

x C2 , y C2 C2

a 68

Mx&&C = − cx ; Mx&&C = m1x&&C1 + mx&&C2 ,

A b

(

. 31

x C2 = x + a ; y C2 = y + b ,

x C1 = x + a + l cosϕ ; y C1 = y + b − l sin ϕ .

ϕ = ωt ,

, ,

(a) y + lω sin ωt ) + m2 && y = N − (m1 + m2 )g . m1 ( && x − lω cos ωt ) + m2 && x = − cx ; m1 ( && , y = const y& = && y = 0. (a) (b) m ( && x − lω 2 cos ωt ) + m && x = − cx ; m lω 2 sin ωt = N − (m + m ) g . 2

(b)

N, N′, N ′ = N = (m1 + m2 )g + m1lω 2 sin ωt .

( ) sin ϕ = −1 ,

ϕ = π / 2,

, -

ϕ = 0,

: N min ′ = (m1 + m2 )g + m1lω 2 . ′ = (m1 + m2 )g − m1lω 2 ; N max ( ) . . . . ,

. , .

N min ′ > 0,

ω = (m1 + m2 ) g / (lm1 ) . Ox ,

(b). x&& +

m1 c lω 2 cosωt , x= m1 + m2 m1 + m2

k 2 = c / (m1 + m2 ) ,

x&& + k 2 x = H cosωt ,

H = m1lω 2 / (m1 + m2 ) .

(d)

. ,

ω ≠k,

ω < (m1 + m2 ) g / (lm1 ) ω, k (

. 69

. 4). ,

, ( ).

, .

, ,

(d). ,

, m1 ,

l = 0 ).

(

(a) - (d), ,

. , . . §6.

, , 17.

. .

r r d (mV ) / dt = F . r r,

, : r r d r r r × (mV ) = r × F , dt r r r r r × F = mO ( F )

(1):

(1)

r r r × mV ,

r r r r d r r r r d r r d r dr r d (r × mV ) = × mV + r × (mV ) = V × mV + r × (mV ) = r × (mV ) , dt dt dt dt dt

r r V × mV = 0 ,

r V

r mV

(1), : r d r r r (r × mV ) = mO (F ). dt

(2)

(2) O

r kO ,

r r r r r r × mV = mO (mV ) = k O .

(3)

O, . 32):

(

. ,

, ;

, ,

h = k O / mV .

. 32

, (2) : ,

. , .

, .

, ,

, .

. ,

(2)

Oxyz :

r r d r r r d r r r d r (r × mV ) x = mx ( F ) ; (r × mV ) y = mx (F ) ; (r × mV ) z = mz (F ) , dt dt dt

(4)

r r r r r r r r r (r × mV ) x = mx (mV ) = k x ; (r × mV ) y = m y (mV ) = k y ; (r × mV ) z = mz (mV ) = k z -

. , , r r mx ( F ) , mz ( F ) .

- mx ( F ) ,

, . (2) x, y, z

r r

(4),

(

), . r r mO ( F ) = 0 ,

(2)

r r r k O = r × mV = const . r mz (F ) = 0 ,

Oz,

r r k z = (r × mV ) z = const .

(4)

, .

. ,

18.

. ,

O: r r d r r r (ri × miVi ) = mO (Fie ) + mO (Fii ) ; (i = 1, n ) . dt

, , r d r r r r r (ri × miVi ) = ∑ mO (Fie ) + ∑ mO (Fii ) ∑ dt i =1 i =1 i =1 n

r r d n r (ri × miVi ) = M Oe , ∑ dt i =1

O 72

(5)

r KO ,

O n r r r K O = ∑ (ri × miVi ) .

(6)

i =1

(5)

r r dK O = M Oe , dt

(7)

: (

) .

Oxyz, : dK y dK x dK z = M xe ; = M ze , = M ye ; dt dt dt

(8)

Kx , K y , Kz -

M xe , M xe , M ye -

. , . .

, , . .

(7) M ze

r K O = const .

= 0.

(7)

(8) (8)

r M Oe = 0 .

Oz, K z = const .

, , ,

. .

, , , .

. , , ,

, ,

. ,

. , ,

. 15.

r rC -

O, M

r r Q = MVC ,

r r r r r K O = rC × MVC = mO ( MVC ) ; r r r K x = mx ( MVC ) ; K y = m y ( MVC ) ; K z = mz ( MVC ) ,

r VC -

(9) (10)

Oz, hzi ,

, , ω -

k zi

Vi = ωhzi ,

r = mz (miVi ) = ± miVi hzi = ± mi hzi2 ω .

(11)

(11) , ,

k zi

Oz, . .

Oz, n ⎞ ⎛ r K z = ∑ mz (miVi ) = ± ∑ mi hzi2 ω = ±⎜⎜ ∑ mi hzi2 ⎟⎟ ω = ± J zω . ⎠ ⎝ i =1 i =1 i =1 n

(12)

. , .

(11)

(12)

ω,

Kω = Jω ω ,

(13)

Kω -

Jω -

, .

Jω Jz

. 19. ,

. . 16.

. , , . ,

(

) . 1. m ,

= bt ,

b -

m2 .

. . , . . 35,

. r r m2 g , m1g

m , r Y.

r X

Oz: dK z / dt = M ze .

r r m2 g , X

r Y

, . (

r V

M ze = m

. 33

− m1gr .

)

(

. 33).

r K z = K1z + K 2 z = mz (m1V ) ± J 2 z ω = m1Vr + J 2 z ω = m1Vr + (m2 r 2 / 2)ω , K1z K2z

, J 2z -

, m2 r 2 / 2 (

. . 9).

K z = (m1 + m2 / 2)r 2ω .

V = ωr , m d (m1 + 2 )r 2ω = m dt 2

− m1gr ;

2 (bt − m1gr ) dω = , dt (2 m1 + m2 )r 2

, .

dω =

2b (2m1 + m2 )r

tdt −

2m1 gr (2m1 + m2 )r

dt ; ω =

bt 2

(2m1 + m2 )r 2

−

2m1grt (2m1 + m2 )r 2

(

ω (0) = ω 0 = 0 ),

0 = 0 + C1 ,

+ C1 .

t = 0, C1 = 0 .

, : ω=

bt − 2 m1gr

(2m1 + m2 )r 2

2. 500

ρ = 1,5 . ,

240

. 10 .

. . 34,

r mg

mc ,

. 34

, , (

r YB .

r r r X A , YA , Z A

. 36).

r XB ,

Oz. ,

M ze

= − mc . K z = J z ω = mρ 2 ω .

mρ 2 = const ,

: mρ 2ω& = − mc .

, dω = −

ω (0) = ω 0 ),

mc mρ

dt ; ω = −

ω 0 = 0 + C1 ,

mc mρ 2

t + C1 .

t = 0;

(

C1 = ω 0 .

, : t1 ,

ω = ω 0 − (mc / mρ 2 )t . 0 = ω 0 − (mc / mρ 2 )t1 .

ω = 0,

mc = ω 0 mρ 2 / t1 .

ω 0 = 240 ⋅ π / 30 = 8π

/ ,

t1 = 10 ⋅ 60 = 600 .

mc ,

: mc = (8π ⋅ 500 ⋅ 1,5 ⋅ 1,5) / 600 = 15π = 47,12 · . 3. EL AB. DE = b D. ω0 . . J, l. , m. , Axyz, . 35, .

. Az

r Mg

r mg

r r r X A , YA , Z A

r r X B , YB

, ,

. M ze

= 0,

Az ,

K z = K z 0 = const .

. 35

ω0 ,

K 0 z = K 01z + K 02 z .

K 01z = Jω 0 .

r Ve 0 .

(

Ve0 = ω 0 ⋅ DE = ω 0 b .

. 35),

r K 02 z = mz (mVe0 ) = mb 2ω 0 .

K z 0 = ( J + mb 2 )ω 0 .

, K z = K1z + K 2 z .

, K1z = Jω .

ω.

r Ve .

EL Ve = ω ⋅ EL = ωl .

(

r Vr

.37), (

35),

r. V.

, r r r K 2 z = mz (mV ) = mz (mVr ) + mz (mVe ) . r mz (mVr ) = 0 , Az,

r mVr

r K 2 z = mz (mVe ) = ml 2ω .

K z = ( J + ml 2 )ω . K 0z Kz ,

: ω = ω 0 ( J + mb 2 ) / ( J + ml 2 ) .

ω, ω

, ,

20.

, .

(8),

(12), K z = J zω . J z = const ,

J zω& = M ze .

ω& = ε = ϕ&& ,

, J zϕ&& =

(8) : (14)

M ze .

. ,

, (14) 78

mz&& = Fz .

(14)

, . § 7.

. 21. .

r r mdV / dt = F . r r dr / dt = V r, r r r r dV r r r m ⋅ dr = F ⋅ dr ; mVdV = F ⋅ dr . dt r F ,

δA

r dr

: r

(1)

δA = F ⋅ dr mV 2 / 2 ,

(2)

r r r r r r ⎛ mV 2 ⎞ ⎛ mV 2 ⎞ r r ⎛ mV ⋅ V ⎞ mVdV mVdV ⎟⎟ . ⎜⎜ ⎟⎟ = d ⎜ = d ⎜⎜ ; mVdV d = + ⎟ 2 2 ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠

(2)

(1), :

⎛ mV 2 ⎞ ⎟⎟ = δA . d ⎜⎜ ⎝ 2 ⎠

(3)

(3) .

: , . ,

, -

. ,1

·( 2/ 2) = (

· /c2) ·

· .

(

)

(

) (3). 2 2 mV / 2 − mV0 / 2 = ∫ δA .

M0 M

A , : A=

∫ δA .

(4)

M0M

, : mV 2 / 2 − mV02 / 2 = A ,

(5)

: ( ,

)

. , . . . ,

, .

. ,

r F

(2) ,

(2)

(6)

. ,

r r r r r r r r r r r r r r δA = (F1 + F2 + ⋅ ⋅ + Fn ) ⋅ dr = F1dr + F2 dr + ⋅ ⋅ + Fn dr = δA(F1 ) + δA(F2 ) + ⋅ ⋅ +δA(Fn ) (6)

, ,

(6)

r r r r r r r δA( F ) = F ⋅ dr = Fdr cos( F ^ dr ) = Fdr cos( F ^V ) , r r , dr = Vdt .

(7)

(

), ( ).

( ),

( , (

r r dr = Vdt ,

r r r r r δA(F ) = F ⋅ Vdt = FV cos(F ^V )dt . | ds / dt | = V , | ds| = Vdt ,

(8)

r r

δA(F ) = F cos(F ^V )ds . r F

(9)

r dr

(7)

: r r r r r r r δA(F ) = (Fx i + Fy j + Fz k )(dxi + dyj + dzk ) = Fx dx + Fy dy + Fz dz . ,

(10) .

r δA ( F ) ,

(10)

r dA(F ) .

. , :r

r N (F ) = δA(F ) / dt .

(11)

(8),

: r r r r r F ⋅ Vdt r r N (F ) = = F ⋅ V = FV cos(F ^V ) , dt , ,

(12)

, . .

(11) :

(10),

r dx dy dz N (F ) = Fx + Fy + Fz = Fx x& + Fy y& + Fz z& . dt dt dt

, (13)

r r r N = N (F1 ) + N (F2 ) + ⋅ ⋅ ⋅ + N (Fn ) ,

(14)

. , 1

· /c = 1

/ . .

∫

r δA( F1 ) +

M0 M

r r r r A F = A F + A F + ⋅ ⋅ ⋅ + A F δ ( ) ( ) ( ) ( n n). 1 2 ∫

r δA( F1 ) + ⋅ ⋅ ⋅ +

∫ M0 M

(4) (6)

M0 M

(15)

, .

, , . , :

r A(F ) =

∫ δA ( F ) ,

(16)

M0M

, ( )

. , ,

, (8), ,

(9).

, .

, . .

, ,

. ( - M0 ,

. 36). - M, . 36

- M 1. (9), s0 ,

s1 .

r r

δA(F ) = F cos(F ^V )ds ,

r A(F ) =

r δA ( F ) =

∫ M 0 M1

r^ r V )ds .

∫ F cos(F

F = const ,

r r cos(F ^V ) = const ,

. , : s1

r r r r r r r r r A( F ) = F cos( F ^V ) ∫ ds = F cos( F ^V ) s|ss1 = F cos( F ^V )( s1 − s0 ) = Fs cos( F ^V ) , (17) 0

s -

. ,

, .

, 1

( ), =1

( ), .

(17)

m M0 M1 (

. 7),

Oz ,

(10). Oxyz Fx = 0 ; Fy = 0 ; Fz = − mg ,

. 37

δA(mg ) = − mgdz ,

(10)

(16)

: r A(mg ) =

∫

1 r δA(mg ) = − mg ∫ dz = − mg (z1 − z 0 ) = mgh ,

M 0 M1

h = z0 − z1

z0 < z ,

. .

r A(mg ) = ± mgh ,

h -

, (18)

. ,

. (18)

, , .

. , , .

r F M 1,

M (r . r38) r δA( F ) = F ⋅ dr .

. 38

(8), ,

r r

r r / r,

r δA( F ) = F

r r rdr = rdr .

r r F = F re , r r r r r rdr ⋅ dr = F r . r r r r r ⋅ r = r2 ,

r e-

r r r r rdr + rdr = 2rdr ,

: r r δA( F ) = F r dr ; A( Fc ) =

r δA( F ) =

∫ M 0 M1

r1 -

∫ F r dr ,

(19)

(

(19)

. 38).

, , (

)

, .

. , .

, .

, l0 .

r F

M2 (

39),

(19) .

F = −F ; dr = dl ; r δA(F ) = − c(l − l 0 )dl ; r0 = l1 ; r1 = l 2 . F

= Δ = c( l − l0 ) ;

. 39

l1 , l , l 2 -

Δ -

, .

(19) l2 r c A( F ) = − c ∫ (l − l 0 ) dl = − (l − l 0 ) 2 2 l 1

l2 l1

c c = − (l2 − l 0 ) 2 + (l1 − l 0 ) 2 . 2 2

Δ1 , Δ 2 -

r c c c A( F ) = − Δ22 + Δ21 = − ( Δ22 − Δ21 ) , 2 2 2

(20)

, ,

(

)

r FT ,

(19) ,

, FT = km / r , 2

. , (20),

r A(FT ) = km(1 / r1 − 1 / r0 ) ,

r0 -

(21)

( ,

r1 -

)

22. .

, : miVi / 2 − 2

miVi20

/2=

Aie

−

Aii ,

i = 1, n ,

Aie -

, Aii -

, 2

n mV2 i i0

mV ∑ i2 i − ∑ i =1 i =1

i =1

= ∑ Aie + ∑ Aii .

, T0 Ae

, , A

. , : T − T0 = A + A , e

(22)

: (

) ,

(22)

. ,

, -

dT / dt = dA e / dt + dA i / dt .

, dT / dt = N e + N i .

(23)

(23) : . (22)

(23),

, . , ,

. . .

. ,

1 n miVi2 , ∑ 2 i =1

(24)

, . , (

. 40)

. 40

(

r r r Vi = Vie + Vir .

r VC

r r r Vi = VC + Vir .

r Vi2 = Vi2 ,

r r Vie = VC ,

(24),

r r 2 1⎛ n ⎞ 2 ⎛ n r ⎞ r 1 n 1 n T = ∑ mi (VC + Vir ) = ⎜ ∑ mi ⎟ VC + ⎜ ∑ miVir ⎟ ⋅ VC + ∑ miVir2 , 2 i =1 2 ⎝ i =1 ⎠ 2 i =1 ⎠ ⎝ i =1 r r Qr = MVCr ,

. ,

r VCr = 0 . T=

1 1 n 1 MVC2 + ∑ miVir2 = MVC2 + TCr , 2 2 i =1 2

(25)

TCr -

, . (25) : (

)

, . 87

. . . 1.

r V.

(24) T=

1 2 n 1 V ∑ mi = MV 2 . 2 i =1 2

(26) Vi = ωhzi , z, hzi -

2. .

ω -

(24)

1 n 1⎛ n 1 2 2 2⎞ 2 2 m h ω m h = ⎜ ∑ ∑ i zi i zi ⎟ ω = J zω , 2 i =1 2 ⎝ i =1 2 ⎠

T= Jz -

(27)

3. (

)

Vi = ωhω ,

ω (

hω -

, . 41).

, .

. 41

(27) : T=

1 2

Jω ω 2 ,

(28)

Jω -

. Ox1 y1z1 (

. 43),

r r

λ = cos(i1^ ω ) = ω x1 / ω ;

r r

μ = cos( j1^ ω ) = ω y1 / ω ; ν = cos(k1^ ω ) = ω z1 / ω .

(18) Jω =

ω x2 J x1 1 2 ω

ω y2 J y1 1 2 ω

ω z2 J z1 1 2 ω

− 2 J x1y1

ω x1ω y1 ω2

− 2 J y1z1

ω y1ω z1 ω2

− 2 J x1z1

ω x1ω z1 ω2

(28), T=

1 (J ω 2 x1 x1 2

J y1ω y2 1

Ox1 y1z1

: 88

J z1ω z2 1

− 2 J x1y1 ω x1ω y1 − 2 J y1z1ω y1ω z1 − 2 J x1z1ω x1ω z1 ) .(29)

(29)

1 2

2 x1 ω x1

)

(30)

+ J y1 ω 2y + J z1ω z2 , 1

, . (29) ,

(29) ω x1 = ω y1 = 0 ,

Oz1

ω z1 = ω .

, 4. ,

Cx1 y1z1 ,

r r Vir = ω × ρi ,

(

. 42).

ω -

ρi -

, ,

T= TCω -

(25)

r r 1 1 1 MVC2 + ∑ mi (ω × ρ i ) = MVC2 + TCω , 2 2 i =1 2

(31)

, :

(

) . 42

)

, ,

, -

(28) TCω = 21 Jω ω 2 .

C, TCω

, (29)

(30), . r

, Cz ∗

xOy,

Cz1 .

, Cz1 ,

(27) TCω = 21 J Cz1ω 2 , , . 2 T = 21 MVC + 21 J Cz1ω 2 .

J Cz1 -

(31) (32)

Cz1

, . .

. 1. ,

, ,

2. ,

r F ( . 43, a), r^ r r^ r δA( F ) = FV cos( F V )dt = FωR cos( F V ) .

V = ωR .

(8) ,

r r | F cos(F ^V )| = FΠ , FΠ -

Oz,

. ,

r | mz (F )| = FΠ R , r r r FR cos( F ^V ) = ± mz ( F ) .

. 43

r r r δA(F ) = ± mz (F )ωdt = ± mz (F )dϕ ,

dϕ = ωdt

(33)

. , ,

. ,

, ,

r mz (F ) ,

, . .

. 43, a

. r A(F ) =

∫

ϕ r r δA(F ) = ∫ ± mz (F )dϕ .

(34)

ϕ0

M0M

r r A(F ) = ± mz (F )ϕ ,

ϕ -

(

(35)

). .

r r r N (F ) = δA(F ) / dt = ± mz (F )ω .

(36)

, . 3.

r V

r r r V =ω×r,

r r-

(

r F

. 43, b) , r , ω -

(8)

r r r r r r r r r δA(F ) = F ⋅ Vdt = F ⋅ (ω × r )dt = ω ⋅ (r × F ) , r r r r r × F = m0 ( F ) -

r r r r r r F ⋅ (ω × r ) = ω ⋅ (r × F ) .

r r r r r r r r r δA( F ) = ω ⋅ mO ( F )dt = mO ( F ) ⋅ ω ⋅ cos(mO ( F ) ^ ω )dt = mω ( F )ω ⋅ dt = mω ( F )dϕ , (37) , dϕ r r r r ^r r r mO (F ) mω ( F ) = mO ( F ) cos(mO ( F ) ω ) -

, . r r r r r r N (F ) = δA(F ) / dt = mO (F ) ⋅ ω = mω (F ) ⋅ ω .

(

. 43, c),

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

A, ρ i -

Fi ,

r Vi

: (38)

r r r r Vi = V A + ω × ρi ,

r VA -

. (6)

(8)

n r r r ⎛ n r⎞ r r r r δA = ∑ Fi (V A + ω × ρi ) = ⎜ ∑ Fi ⎟ ⋅ V A dt + ∑ Fi ⋅ (ω × ρi )dt . ⎝ i =1 ⎠ i =1 i =1 n

, r⎞ r ⎛ n r r ⎞ r ⎛ n r⎞ r ⎛ n r ⎛ n r⎞ r δA = ⎜ ∑ Fi ⎟ ⋅ V A dt + ⎜ ∑ ρi × Fi ⎟ ⋅ ωdt = ⎜ ∑ Fi ⎟ ⋅ V A dt + ⎜ ∑ mA (Fi )⎟ ⋅ ωdt . ⎠ ⎝ i =1 ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠

(39)

, ,

r r r Fi = Fie + Fii .

, (39),

⎛ n r ri ⎞ r ⎛ n ri⎞ r ⎛ n r re ⎞ r ⎛ n r e⎞ r δA = ⎜ ∑ Fi ⎟ ⋅ V A dt + ⎜ ∑ mA (Fi )⎟ ⋅ ωdt + ⎜ ∑ Fi ⎟ ⋅ V A dt + ⎜ ∑ mA (Fi )⎟ ⋅ ω ⋅ dt . ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠

r r ∑ Fii = R i = 0 ;

r r r ∑ mA (Fii ) = M Ai = 0 .

δA

Ai ,

: i

. ,

, n

δA = δA e = ∑ Fie ⋅ V A dt + ∑ mA (Fi ) ⋅ ω ⋅ dt = R e ⋅ V A dt + M Ae ⋅ ω ⋅ dt . i =1

i =1

(40)

r M Ae

, , ,

, . 5.

, .

r r r r r r m F M F F M ( ) = ( , ) = , ′ ∑ A i

(40) ,

r r r r r r r δA(F , F ′ ) = δA( M ) = M ⋅ ω ⋅ dt = M cos( M ^ ω ) ⋅ ω ⋅ dt = M ω ⋅ ω ⋅ dt , Mω r

, (41)

ω.

. (40)

r r r r r N (F , F ′ ) = N ( M ) = M ⋅ ω = M ω ⋅ ω .

, ,

Mz ,

: 92

(42)

r r

δA( F , F ′) = δA( M z ) = ± M z ωdt = ± M z dϕ ; N ( F , F ′) = N ( M z ) = ± M z ω , dϕ z. , , z1

(43)

( ).

. ϕ

∫ δA ( M z ) =

A( M z ) =

∫ ± M z dϕ .

(44)

ϕ0

M0M

A ( M z ) = ± M zϕ ,

ϕ -

(45)

. . 6.

, Az1 ,

xOy, A.

(39) r

δA = ∑ Fie ⋅ V A dt + ∑ m Az1 ( Fie ) ⋅ ωdt = R e ⋅ V A dt + M eAz ωdt = R e ⋅ drA + M eAz dϕ , (46) i =1

i =1

ω -

Az1 , dϕ e , M Az 1

. . . B r V2

A ( r . 44, a), V1 , B

A A

V2 > V1 .

B, r FT

r FT′

(

. 46, b)

, . . 46

δATi = δA(FT ) + δA(FT′ ) = FT ⋅ V2 dt + FT′ ⋅ V1dt = − FT V2 dt + FT′ V1dt = − FT (V2 − V1 ) . , V2 − V1

δATi = − FT Vr dt .

Vr ,

. ,

, . . , (

( .

)

, . ( (

)

. .

, . 2.

(

3. ,

4. .

(

, .

. ,

6. ,

, ,

. ,

. 3.

C m3 (

. 47, a).

, ,

, , . ,

dT / dt = N + N . e

: ,

N = 0. dT / dt = N e . i

. . 45, a.r

x, ,

x. T = T1 + T2 + T3 ,

T1 , T2

T3 -

. T1 = T2 =

1mV2. 2 1 1 1m V2 2 2 C2

(26) (32)

, + 21 J C2 ω 22 ,

VC2 C2

J C2

, , (

ω2 -

), .

. 45

, T=

1 2

J 3ω 32 ,

(27)

J3 -

, ,

ω3 -

(31) J C2 =

1 m r2; 2 2 2

J3 =

1 m r2, 2 3 3

r3 -

(29) .

. .9

, T = 21 m1V12 + 21 m2VC2 + 41 m2 r22ω 22 + 41 m3r32ω 32 . 2

, ,

ω 3 = V1 / r3 , E L

. ,

(

V1 .

. 47, a)

P , ω 2 = VE / EP = V1 / (2r2 ) , VC2 = ω 2 ⋅ CC2 P = ω 2 r2 = V1 / 2 . T=

1mV2 2 1 1

1 m V2 + 1m V2 = + 81 m2V12 + 16 2 1 8 3 1

1 (8m 1 16

+ 3m2 + 2m3 )V12 .

. 45, a,

r r r r r r N e = N (m1 g ) + N (m2 g ) + N (m3 g ) + N ( R ) + N ( FT ) + N ( N 2 ) , r r r r m1g , m2 g , m3 g , R r r , FT N2 r FT

r m3 g r N2

r R

. P (12)

VP = 0 . , r r r r r r r r N e = m1g ⋅ V1 + m2 g ⋅ VC2 = m1gV1 cos(m1g ^V1 ) + m2 gVC2 cos(m2 g ^VC2 ) = m1gV1 , r r r r (m1g ^V ) = 0 , (m2 g ^VC2 ) = π / 2 .

, dV1 ⎛ 8m1 + 3m2 + 2m3 ⎞ d ⎡⎛ 8m1 + 3m2 + 2m3 ⎞ 2 ⎤ ⎜ ⎟ V1 ⎥ = m1gV1 ; 2V1 ⎜ ⎟ = m1gV1 , ⎢ ⎠ ⎠ ⎦ dt ⎝ 16 16 dt ⎣⎝

, ,

V1 -

V1 , , dV1 / dt = x&& = a1 , m m1 g = 1 g, a1 (m1 + 83 m2 + 41 m3 ) = m1g ; a1 = 3 1 m m1 + 8 m2 + 4 m3 a1

, ,

δ,

. r2 .

, .

, ,

. 47, b

, M TK .

M TK = δ ⋅ N 2 .

, N2

M TK

( y C2 = const ,

. 45, b). y

N 2 − m2 g = 0 .

m2 && y C2 = N 2 − m2 g .

M TK = δ ⋅ m2 g .

N 2 = m2 g ,

r r r N = N (m1g ) + N ( M TK ) = m1g ⋅ V1 ± M TK ω 2 = m1gV1 − δ ⋅ m2 gω 2 , e

(43). V1 :

ω 2 = V1 / 2r2 , N e = m1gV1 −

δ ⋅ m2 gV1 2r2

⎞ ⎛ δ = ⎜ m1 − m2 ⎟ gV1 . 2r2 ⎠ ⎝

, a1 = m

⎛ m m1 δ m2 δ m2 ⎞ ⎟g, g− g = ⎜⎜ 1 − 2r2 m 2r2 m ⎟⎠ m ⎝m

= m1 + 83 m2 + 41 m3 .

B . i,

C D

, E

(

. 46,

a).

, -

, , J J , = M − sω ,

M , s, ω

, .

. 46

: dT / dt = N e + N i . N i = 0,

, dT / dt = N . e

(

. 46, b). -

. T =T +T +T ,

, T = mV 2 / 2 ,

. T = Jω 2 / 2 ,

T = J ω2 / 2.

, T=

1 mV 2 2

+ 21 Jω 2 + 21 J ω 2 .

ϕ (

, L

, ω

. 46, b).

VL = ω ⋅ r , V = VL = ω ⋅ r ,

, ,

(

ω = ω ⋅i .

. 46, a).

, T = 21 mr 2ω 2 + 21 Jω 2 + 21 J i 2ω 2 = 21 ( J + J i 2 + mr 2 )ω 2 = = J + J i 2 + mr 2

1 2

J ω2,

, . ,

, .

r r r N e = N (mg ) + N (FT ) + N ( N ) + N ( M

r mg -

r FT -

, N ,

(

)

, ,

, . ,

M , ω r r r N (N ) = 0 , N⊥V , r r r r N e = mg ⋅ V + FT ⋅ V ± M ω = − mgV sin α − FT V + M ω .

FT = fN ,

, (

y = const , , FT = f cosα ⋅ mg .

N = mg cosα .

, 46,

c),

y: my&& = N − mg cosα . N − mg cosα = 0 ,

N e = − mg sin αV − mgf cosαV + M ω = M ω − (sin α + f cosα )mgV .

ω = ω ⋅i ,

V =ω ⋅r,

ω: N e = M i ⋅ ω − (sin α + f cosα )mgr ⋅ ω .

= M − sω = M − sω ⋅ i ,

N e = ( M i − si 2ω )ω − (sin α + f cos α )mgrω = M i ⋅ ω − si 2ω ⋅ ω − kmgr ⋅ ω , k = sin α + f cosα .

, ω,

: J = si 2

dω = M i − si 2ω − kmgr ; J dt M =M i -

dω +s ω = M dt

− kmgr ,

, , ω = ϕ& ,

J ϕ&& + s ϕ& = M

dω / dt = ϕ&& , − kmgr .

. 1.

. ,

, ,

. 99

2. , 3.

(

)

. ,

(

)

. 4.

. (

)

. 5. ,

. A

4. ,

. 47, a.

OA m,

l ,

AB C, , . , .

: T − T0 = A e + A i . , T0 = 0 . , T = Ae , T -

, A = 0. i

, , . (

. 47, b)

. T = T1 + T2 + T3 ,

, :

1 J ω2 2 1 1

+ 21 m2VC2 + 21 J C2 ω 22 + 21 m3VC2 + 21 J C3 ω 32 , 2

J1 -

, A; J C2 , J C3 , ; VC2 ,VC3 -

(22)

m2 , m3 -

. (29)

. 9,

J1 = 13 m1l 2 = 13 ml 2 ; J C2 =

100

; ω1 ,ω 2 ,ω 3 (21),

1 12

m2 l 2 =

1 12

ml 2 ; J C3 =

1 m r2 2 3 3

1 mr 2 , 3 2

m1 -

r3 -

, m.

T = 16 ml 2ω 12 + 21 mVC2 + 2

, 1 ml 2 ω 2 2 24

+ 21 mVC2 + 41 mr32ω 32 .

r VA ,

ω1 ,

A V A = ω1 ⋅ l ,

. ,

B, .

VC3

ω 2 = V A / APAB = ω 1l / l = ω1

PAB = VB = VPAB

B, ( = 0,

. 47, b).

, .

VC2 = ω 2 ⋅ C2 PAB = 21 ω 1l

P ,

ω 3 = VB / BP = 0 / r3 = 0 .

T = 16 ml 2ω12 + 81 ml 2ω12 +

, r FT

1 24

(

ml 2ω12 = 13 ml 2ω 12 .

r R

r N3

r r r m1g , m2 g , m3 g r r FT N3

r R

. 47).

, ,

r r r A e = A(m1 g ) + A( m2 g ) + A(m3 g ) = ± m1 gh1 ± m2 gh2 ± m3 gh3 ,

(18)

h1 , h2 , h3 -

( 47, b).

. 47, a)

(

, h3 = 0 .

r m1g ,

l / 2,

h1 = 21 l sin 30 o .

, ,

h2 = 21 l sin 30o .

, A e = 21 m1 gl sin 30o + 21 m2 g sin 30o = mgl sin 30 o = 21 mgl .

101

1 2 2 1 ml ω 1 = mgl ; ω1 = 3 2

3g . 2l

, , . 5. , A ,

. A . T = A e + Ai .

. 47

A( M T ) = − M T ϕ r ,

ϕr -

. (

ϕ 0r = 120o ,

. 47, a) (

ϕ1r = 180o ,

. 49, b)

ϕ r = ϕ 1r − ϕ 0r = 60o = π / 3 .

, A e + A i = 21 mgl − 13 π ⋅ M T ,

: ω1 =

3g π ⋅ M T − . 2l ml 2

m1 = m ,

m2 = 3m .

2 3

m3 = m .

ρ=r 2.

R = 2r .

, ,

h1 ,

, 1

. :

T − T0 = A e + A i .

, ,

102

T = Ae .

, 1

, . 48).

(

T = T1 + T2 + T3 ,

1 3 2

, , T=

1mV2 2 1 1

1 2

J 2ω 22

1 m V 2, 2 3 3

A .

(17)

J 2 = m2 ρ . 2

, . 48

1mV2 2 1 1

+ 21 m2 ρ 2ω 22 + 21 m3V32 .

1 V1 ,

ω 2 = V1 / R ,

V3 = ω 2 r = V1 ⋅ r / R ,

3 . 48. T = 21 m1V12 + 21 m2 ρ 2V12 / R 2 + 21 m3r 2V12 / R 2 = mV12 ,

, . . 48.

: h1 -

r r r r A = A(m1 g ) + A(m2 g ) + A(m3 g ) + A( R A ) = ± m1 gh1 ± m3 gh3 = mgh1 − mgh3 , 1, h3 3, e

, . h1 , V3 = V1 ⋅ r / R , dh3 = dh1 ⋅ r / R . h3 0 h1 ,

A e = mgh1 − mgh1

V1 = dh1 / dt ,

V3 = dh3 / dt .

h3 = h1 ⋅ r / R .

r 1 = mgh1 . R 2

: V1 = gh1 / 2 .

103

ยง 8. 23. . . . , ,

. ,

( , ),

. ,

.. ,

, . . , ,

. , ,

. .

, . ,

(16) .

, U (x , y , z ) , r

ฮดA(F ) = dU , Fx dx + Fy dy + Fz dz =

(1)

U U U dz . dy + dx + z y x

. ,

Fx = U / x ; Fy = U / y ; Fz = U / z .

, ,

(48),

. ,

, . .

, , M0

104

(2)

. 21. ,

r F

(

)

r A(F ) =

r δA ( F ) =

∫ M0M

∫ dU = U (x , y , z) − U (x0 , y 0 , z0 ) = U − U 0 .

Π = −U + const ,

, : r r r A( F ) = ( − Π + const ) − ( − Π 0 + const ) = Π 0 ( F ) − Π( F ) .

(3)

, , . M0 ,

(49)

, :

r r Π( F ) = − A( F ) ,

, (

(4)

)

M0 .

. , l0 ,

, (20) :

r Π ( F ) = cΔ2 / 2 ,

Δ -

(5)

. ,

(18)

. 21

r Π(mg ) = mgh ,

(6)

, , . .

U = ∑ U i ( x i , y i , z i ) ; Π = ∑ Π i ( x i , y i , zi ) .

, ,

(3), . .

r r r A = ∑ A(Fi ) = ∑ Π 0 (Fi ) − ∑ Π (Fi ) = Π 0 − Π .

(7)

. ,

, A = Π0 − Π .

, (5)

, 105

mV 2 / 2 − mV02 / 2 = Π 0 − Π ,

mV 2 / 2 + Π = mV02 / 2 + Π 0 .

E = mV 2 / 2 + Π = h , h = mV02 / 2 + Π 0

(8)

(54) .

, , , . , ,

, -

. ,

. , .

. A e + Ai

(22)

Π0 − Π , T + Π = h,

(7) T − T0 = Π 0 − Π ,

h -

(9)

, T0 + Π 0 .

(9)

: , ,

. , ,

, , .

, , ,

106

. .

, (

. .),

. . . . .

, . . ,

, ,

A∗

A e + A i = (Π 0 − Π ) + A∗ ,

, : ∗

E − E 0 = A∗ .

∗

T − T0 = Π 0 − Π + A ; T + Π − (T0 + T ) = A ,

(10)

( A∗ > 0 ), (

, ( A∗ < 0 ),

. .). (

) ,

(

, . .).

, . ,

, . , ,

107

, , . § 9. ,

, ,

. ,

, ,

. -

(

), ,

. (

)

. ,

. .

24. Ax1 y1z1 (

. 49). i

r r r r a i = a ir + a ie + a ic ,

r aie -

r a ir

, ,

r a ic -

r r r a ic = 2ω e × Vir ,

ωe

. 49

Oxyz,

, -

r Vir -

. r Fie

108

r Fii -

r r r r r mi (air + aie + a ic ) = Fie + Fii ,

. : r r r r r mi air = Fie + Fii + Φ ie + Φ ic ( i = 1, n ), r r Φ ie = − mi a ie

(1) r r Φ ic = − mi a ic

. ,

, , ,

, . .

(1)

, §5 - §8 , -

. ,

r r r r dQr / dt = R e + Φ e + Φ c , r r r r r r R e = ∑ Fie ; Φ e = ∑ Φ ie ; Φ c = ∑ Φ ic -

(2)

r r Qr = ∑ miVir -

, . , A

, r r r r r r dK Ar / dt = M Ae + M A (Φ ie ) + M A (Φ ic ) . r r r K Ar = ∑ ρi × miVir -

ρi -

(3)

, (

. 51); M Ae r

A; M A (Φ ie )

r r M A (Φ ie ) -

r r dTr δA e δA i δA(Φ ie ) = + + = N e + N i + N (Φ ie ) , dt dt dt dt

(4)

109

Tr =

1 2

∑ miVir2

(4)

, ,

dρi

r Vir

, .

, r

( aic ⊥Vir ). 25.

. , .

. , ,

. -

, , . , §5 Oxyz,

r r r K O = ∑ ri × miVi

§6

r r r r dQ / dt = R e ; dK O / dt = M Oe , r r Q = ∑ miVi

(5)

(

r r r M Oe = ∑ ri × Fie

O. (5)

, . .

110

r r . K O = ∑ ri × miVi ,

r ri r Oxyz, Vi r r r r . 51) ri = rA + ρi , rA -

(

Ax1 y1z1 ,

ρi -

: n n n r r r r r r r r r r r K O = ∑ (rA + ρ i ) × miVi = rA × ∑ miVi + ∑ ρ i × miVi = rA × Q + K A , i =1

i =1

r Q = ∑ miVi -

i =1

r r K A = ∑ ρi × mi Vi -

A. A

r Vi -

(6)

∑ mi ρi

r = Mρ C ,

r r Q = MVC ,

(7)

ω -

r r r r r r K A = ∑ ρi × mi (V A + ω × ρi + Vir ) .

r KA ,

r r r r K O = r A × MVC + K A .

r r r r r r r Vi = Vie + Vir = V A + ω × ρi + Vir , r , Vir -

(6)

ρC -

r VA

r KA

, r r r r r K A = ρ C × MV A + K ωA + K rA . r K ωA -

(8)

n r rω r r K A = ∑ ρi × mi (ω × ρi ) ,

(9)

i =1

“ ω “.

r K Ar ,

111

n r r r K Ar = ∑ ρi × miVir ,

(10)

i =1

. ,

r K Ar = 0 ,

r Vir = 0

(8)

ρC = 0 ,

r r r r K A = ρ C × MV A + K ωA .

(11)

r r K C = K Cω ,

(12)

(11)

. O

r r , V A = VO = 0 , r rω KO = KO ,

(11) (13)

. .

(6),

r r r r r r dK O / dt = V A × Q + rA × (dQ / dt ) + dK A / dt , r r r r r r (5), dK O / dt = M Oe = ∑ r × Fie , dQ / dt = R e = ∑ Fie . r r r r r r r r rA dK A / dt = Q × V A + ∑ ri × Fie − ∑ rA × Fie ,

, , , ,

r r Q = MVC ,

r r r r r r ∑ (ri − rA ) × Fie = ∑ ρi × Fie = M Ae

A, :

r r r r dK A = Q × V A + M Ae dt

112

r r r r dK A = MVC × V A + M Ae , dt

(14)

r KA ,

(6) K A = ∑ ρi × miVi , r Vi -

A, .

. -

. :

r r r r r dQ r e dK A = Q × V A + M eA . =R ; dt dt

(15)

, Ax1 y1z1 (

. 49).

r Q

r KA ,

( Oxyz,

)

(

)

r r r r Q = Qx1 i1 + Q y1 j1 + Qz1 k1 ,

Ax1 y1z1 ,

r r r r K A = K Ax1 i1 + K Ay1 j1 + K Az1 k1 .

r r r ~r dQ / dt = dQ / dt + ω × Q , r r r r (15) dK A / dt = Q × V A + M eA ,

r r r ~r dK A / dt = dK A / dt + ω × K A . r r dQ / dt = R e , ~r ~r r e dK r e dQ r r r r r r A R = + ω × Q ; Q ×VA + M A = +ω × KA. dt dt

Ax1 y1z1 : r r r r r dQ r r r e dK A r r + ω × K A + V A × Q = M Ae . +ω × Q = R ; dt dt r r Qi = ∑ miVi , r r r K A = ∑ ρi × miVi

(8) -

(16)

A, .

113

( (12).

(16) . r r r MVC = Q||VC , Cx1 y1z1

, C

(16)

r r r e dK C r r r r dQ r + ω × MVC = R ; + ω × K C = M Ce . dt dt

(17)

(

(12). ,

, r

r V A = VO = 0 ,

Ox1 y1z1

r r r dQ r r r e dK O r r +ω × Q = R ; + ω × K O = M Oe , dt dt

r KO

(18)

(13),

. Ax ∗ y ∗ z ∗ , r

ω = 0, r r r r re r dQ / dt = R ; dK A / dt + Q × V A = M Ae .

(15)

(19)

, (5). ∗ ∗ ∗

, (19)

r r r r MVC = Q||V A = VC ,

r r r r Q × V A = MVC × VC = 0 , r r r r dQ / dt = R e ; dK C / dt = M Ce .

114

Cx y z . r r V A = VC

C -

(20)

(5)

. 26. . .

, ,

, , .

(8)

( ω = 0 ),

(8)

Ax ∗ y ∗ z ∗ , r r r r K A = ρ C × MV A + K Ar .

r Kω A = 0,

C ,

r KA

∗ ∗ ∗

Cx y z ,

r r r K A = K C = K Cr .

ρC = 0 , (21)

, ,

r KC

r K Cr .

(21)

(20), r r dK Cr / dt = M Ce .

(22)

, . (10),

ρi

r Vir -

, , -

, .

115

r r aie = a C ,

r a ic = 0 ,

: dTCr / dt = N e + N i ,

(23)

. -

, , . § 10. .

. ,

. , . ,

, ,

. ,

. 25

Ax1 y1z1

(16),

(11) (9)

r r r r r dQ r r r e dK A r r +ω × Q = R ; + ω × K A + V A × Q = M Ae ; dt dt n r r r rω rω r r r K A = ρ C × MV A + K A ; K A = ∑ ρi × mi (ω × ρi ) , i =1

r Kω A -

: (1)

(2)

116

, .

27. .

(2)

A Ox1 y1z1 .

r VA = 0 ,

r r K A = Kω A.

(2)

r VA ,

(2) K ωA

ρ C × MV A ,

r ρ i = ri -

. -

(2) O

r r r r r r k O = ri × mi (ω × ri ) = mi ri × (ω × ri ) . r r r r r r r r r a × (b × c ) = b ⋅ ( a ⋅ c ) − c ⋅ ( a ⋅ b ) r , kO r r r r r r r r r r r k Oi = mi ω (ri ⋅ ri ) − mi ri (ω ⋅ ri ) = mi ri2ω − mi (ri ⋅ ω )ri .

r k Oi

Ox1 y1z1 : r r = mi (x12i + y12i + z12i )ω − mi (x1i ω x1 + y1i ω y1 + z1i ω z1 )ri ,

ω x1 , ω y1 , ω z1 -

x1i , y1i , z1i -

. Ox1 ,

k x1 = mi ( x12i + y12i + z12i )ω x1 − mi ( x1i ω x1 + y1i ω y1 + z1i ω z1 ) x1i = = mi ( y12i + z12i )ω x1 − mi x1i y1i ω y1 − mi x1i z1i ω z1 .

117

n ⎛ n ⎞ ⎛ n ⎞ ⎛ n ⎞ K x1 = ∑ k x1 = ⎜ ∑ mi ( y12i + z12i )⎟ ω x1 − ⎜ ∑ mi x1i y1i ⎟ ω y1 − ⎜ ∑ mi x1i z1i ⎟ ω z1 . ⎝ i =1 ⎠ ⎝ i =1 ⎠ ⎝ i =1 ⎠ i =1

K y1

K z1

. 7,

K x1 = J x1ω x1 − J x1y1ω y1 − J x1z1ω z1 ; K y1 = − J y1x1ω x1 + J y1ω y1 − J y1z1ω z1 ;

(3)

K z1 = − J z1x1ω x1 − J z1y1ω y1 + J z1ω z1 .

: ⎡ Kx ⎤ r ⎢ 1⎥ K O = ⎢ K y1 ⎥ = ⎢ Kz ⎥ ⎣ 1⎦

⎡ Jx 1 ⎢ ⎢− J y1x1 ⎢− J z x ⎣ 11

− J x1y1 J y1 − J z1y1

− J x1z1 ⎤ ⎡ω x1 ⎤ ⎥ ⎥⎢ − J y1z1 ⎥ ⎢ω y1 ⎥ , J z1 ⎥⎦ ⎢⎣ω z1 ⎥⎦

(4)

: r r K O = θ O(1) ω ,

θ O(1) -

(5)

ω-

Ox1 y1z1 ,

, .

, O, , , K x1 = J x1ω x1 ; K y1 = J y1ω y1 ; K z1 = J z1ω z1 . Oz1 ,

(6)

ω x1 = ω y1 = 0

(3)

K x1 = − J x1z1ω z1 ; K y1 = − J y1z1ω z1 ; K z1 = J z1ω z1 .

(7)

, . . Oxyz, 118

§ 3,

. Ox1 y1z1

. ,

Ax1 y1 z1

(2),

(5), r r r r K A = ρ C × MV A + θ A(1) ω ,

θ A(1) -

(8)

ρC = 0 ,

. (8)

r r K C = θ C(1)ω ,

θ C(1) -

(9)

Cx1 y1 z1

r r Q = ∑ miVi

, .

r r r r Vi = V A + ω × ρ i , r

, ρi .

r VA -

, -

, ω -

r r r r r r Q = ∑ mi (V A + ω × ρi ) = V A ∑ mi + ω × ∑ mi ρi . r r r ρC ∑ mi = M , ∑ mi ρi = MρC ,

, r r r r Q = MV A + M (ω × ρ C ) .

(10)

. .

28. .

(18) 119

Ox1 y1z1 .

. 25

(5), , θ O(1)

r dω r r + ω × K O = M Oe . dt

(11)

(11) :

r k

r i1

r j1

ω x1

ω y1

ω z1 = (ω y1 K z1 − ω z1 K y1 )i1 + (ω z1 K x1 − ω x1 K z1 ) j1 + (ω x1 K y1 − ω y1 K x1 ) k1

K x1

K y1

K z1

⎡ 0 r ⎢ ω × KO = ⎢ ω z1 ⎢− ω y 1 ⎣ r

− ω z1 0

ω x1

ω y1 ⎤ ⎡ K x1 ⎤ r ⎥⎢ ⎥ − ω x1 ⎥ ⎢ K y1 ⎥ = ΩK O

(12)

0 ⎥⎦ ⎢⎣ K z1 ⎥⎦

Ω-

, .

(12)

(11),

(5), θ O(1)

(10)

r r r dω + Ωθ O(1)ω = M Oe . dt

(13)

(4), (12), : ⎡ Jx 1 ⎢ J − ⎢ y1x1 ⎢− Jz x 1 1 ⎣ ⎡ 0 ⎢ + ⎢ ω z1 ⎢− ω y ⎣ 1

− ω z1 0

ω x1

− J x1y1 J y1 − J z1y1

ω y1 ⎤ ⎡ J x1 ⎥⎢ − ω x1 ⎥ ⎢− J y1x1 0 ⎥⎦ ⎢⎣ − J z1x1

− J x1z1 ⎤ ⎡ω& x1 ⎤ ⎥⎢ ⎥ − J y1z1 ⎥ ⎢ω& y1 ⎥ + J z1 ⎥⎦ ⎢⎣ω& z1 ⎥⎦ − J x1y1 J y1 − J z1y1

− J x1z1 ⎤ ⎡ω x1 ⎤ ⎥ ⎥⎢ − J y1z1 ⎥ ⎢ω y1 ⎥ = J z1 ⎥⎦ ⎢⎣ω z1 ⎥⎦

(14)

⎡Me ⎤ ⎢ x1 ⎥ ⎢ M ye ⎥ . ⎢ e1 ⎥ ⎢⎣ M z1 ⎥⎦

(14)

120

. (14)

Ox1 y1z1

. (14)

⎡ J x ω& x ⎤ ⎡ 0 ⎢ 1 1⎥ ⎢ ⎢ J y1ω& y1 ⎥ + ⎢ ω z1 ⎢ J z ω& z ⎥ ⎢− ω y ⎣ 1 1⎦ ⎣ 1

− ω z1 0

ω x1

ω y1 ⎤ ⎡ J x1ω x1 ⎤ ⎡⎢ M x1 ⎤⎥ ⎥ ⎥⎢ − ω x1 ⎥ ⎢ J y1ω y1 ⎥ = ⎢ M ye ⎥ . ⎢ 1⎥ 0 ⎥⎦ ⎢ J z ω z1 ⎥ ⎢ M e ⎥ e

⎣

⎦

⎣

z1 ⎦

, ⎡ J x ω& x ⎤ ⎡ − J y ω y ω z + J z ω y ω z ⎤ 1 1 1 1 1 1 ⎥ ⎢ 1 1⎥ ⎢ & ⎢ J y1ω y1 ⎥ + ⎢ J x1ω x1 ω z1 − J z1ω x1ω z1 ⎥ = ⎢ J z ω& z ⎥ ⎢− J x ω x ω y + J y ω x ω y ⎥ ⎣ 1 1⎦ ⎣ 1 1 1 1 1 1⎦

⎡M e ⎤ ⎢ x1 ⎥ ⎢ M ye ⎥ . ⎢ e1 ⎥ ⎢ Mz ⎥ ⎣ 1⎦

, ⎡ J x1ω& x + ( J z1 − J y1 )ω y1ω z1 ⎤ 1 ⎥ ⎢ & ω J y y ⎢ 1 1 + ( J x1 − J z1 )ω x1ω z1 ⎥ = ⎢ J z ω& z + ( J y − J x )ω x ω y ⎥ 1 1 1 1⎦ ⎣ 1 1

⎡M e ⎤ ⎢ x1 ⎥ ⎢ M ye ⎥ . ⎢ e1 ⎥ ⎢ Mz ⎥ ⎣ 1⎦

: J x1ω& x1 + ( J z1 − J y1 )ω y1ω z1 = M xe ; 1

J y1ω& y1 + ( J x1 − J z1 )ω x1ω z1 = M ye ;

(15)

J z1ω z1 + ( J y1 − J x1 )ω x1ω z1 = M ze . 1

(14)

(15)

, , . . ,

121

ω x1 = ψ& sin θ + θ& cosφ ; ω y1 = ψ& sin θ cosφ − θ& sin φ ; ω z1 = ψ& cosθ + φ& .

(16)

, . , ,

. .

, ,

, J x1 = J y1 ,

Oz1 ,

. .

, A,

(1)

r Q

r r r r r dQ r r r e dK A r r +ω × Q = R ; + ω × K A + V A × Q = M Ae , dt dt

r KA

(17)

: r r r r r r r r Q = MV A + M (ω × ρ C ) ; K A = ρ C × MV A + θ A(1) ω ,

(10)

(18)

(8).

, (17)

(18)

r r r dQ r r r e dK C r r + ω × K C = M Ce ; +ω × Q = R ; dt dt r r r r Q = MVC ; K C = θ C(1)ω , r r r r ρ C = 0 , Q = MVC ||VC .

(19) (20)

(20) θ C(1) -

, r

, VC -

Cx1 y1z1

. (17)

(19)

122

, ,

(16)

(

). , .

r r ω × Q = ΩQ , r

Ω -

(19)

r r r r r Q& + ΩQ = R e ; θ C(1)ω& + Ωθ C(1)ω = M Ce .

(21)

(21) ⎡ Q& x ⎤ ⎡ 0 ⎢ & 1⎥ ⎢ ⎢ Qy1 ⎥ + ⎢ ω z1 ⎢ Q& z ⎥ ⎢− ω y 1 ⎣ 1⎦ ⎣

(12) -

− ω z1 0

: ω y1 ⎤ ⎡ Qx1 ⎤ ⎡⎢ R x1 ⎤⎥ ⎥⎢ ⎥ − ω x1 ⎥ ⎢ Qy1 ⎥ = ⎢ R ye ⎥ , e

⎢ e ⎢ Rz ⎥ ⎣ 1⎦

0 ⎥⎦ ⎢⎣ Qz1 ⎥⎦

ω x1

(22)

1⎥

(20), ,

⎡ a Cx ⎤ ⎡ 0 ⎢ 1⎥ ⎢ M ⎢ a Cy1 ⎥ + M ⎢ ω z1 ⎢ a Cz ⎥ ⎢− ω y 1 ⎣ 1⎦ ⎣

− ω z1 0

ω x1

ω y1 ⎤ ⎡VCx1 ⎤ ⎡⎢ R x1 ⎤⎥ ⎥ ⎥⎢ − ω x1 ⎥ ⎢VCy ⎥ = ⎢ R ye ⎥ . e

0 ⎥⎦ ⎢⎢VCz ⎥⎥ ⎣ 1⎦ 1

(23)

1⎥

⎢ e ⎢ Rz ⎥ ⎣ 1⎦

(23)

Cx1 y1z1 (

, ,

(20)

(14), -

, . , (23)

(14),

(16). .

, . ,

123

. * * *

Cx y z ,

. ,

, ( . 26).

Cx1 y1z1 (

50), -

. 50

, : r r r r r r Ma C = R e ; dK C / dt + ω × K C = M Ce . r (24) K C r ; Re r ; M Ce r

(24)

; ω -

, ,

, . (24)

(16), , . (12) -

(24)

: r r r r Ma C = R e ; θ C(1)ω& + Ωθ C(1)ω = M Ce ,

124

(20),

(25)

r a = [ x&&C

&& yC

&& zC ] r , R e = R xe R ye T

[

]

Rze

; θ C(1) -

Ω -

. (25)

⎡ Jx 1 ⎢ J − ⎢ y1x1 ⎢− Jz x 1 1 ⎣ ⎡ 0 ⎢ + ⎢ ω z1 ⎢− ω y 1 ⎣

⎡ x&&C ⎤ M ⎢ && yC ⎥ = ⎢ ⎥ ⎢⎣ && z C ⎥⎦

⎡ Rxe ⎤ ⎢ e⎥ ⎢R y ⎥ ; ⎢R e ⎥ ⎣ z⎦

− J x1y1

− J x1z1 ⎤ ⎡ω& x1 ⎤ ⎥⎢ ⎥ − J y1z1 ⎥ ⎢ω& y1 ⎥ + J z1 ⎥⎦ ⎢⎣ω& z1 ⎥⎦

J y1

− J z1y1

ω y1 ⎤ ⎡ J x1 ⎥⎢ − ω x1 ⎥ ⎢− J y1x1

− ω z1 0

0 ⎥⎦ ⎢⎣ − J z1x1

ω x1

− J x1y1 J y1

− J z1y1

(26)

(27)

− J x1z1 ⎤ ⎡ω x1 ⎤ ⎥⎢ ⎥ − J y1z1 ⎥ ⎢ω y1 ⎥ = J z1 ⎥⎦ ⎢⎣ω z1 ⎥⎦

⎡Me ⎤ ⎢ x1 ⎥ ⎢ M ye ⎥ . ⎢ e1 ⎥ ⎢⎣ M z1 ⎥⎦

(26) ,

(27)

. , ,

, ,

. xOy,

. .

(26)

(27),

&& z C = 0 , ω x1 = ω y1 = 0 , ω& x1 = ω& y1 = 0 ,

⎡ Jx 1 ⎢ J − ⎢ y1x1 ⎢− Jz x ⎣ 1 1

⎡ x&&C ⎤ M ⎢ && y ⎥= ⎢ C⎥ ⎢⎣ 0 ⎥⎦

⎡ R xe ⎤ ⎢ e⎥ ⎢Ry ⎥ ; ⎢Re ⎥ ⎣ z⎦

− J x1y1

− J x1z1 ⎤ ⎡ 0 ⎤ ⎥⎢ ⎥ − J y1z1 ⎥ ⎢ 0 ⎥ + J z1 ⎥⎦ ⎢⎣ω& z1 ⎥⎦

J y1

− J z1y1

125

− ω z1 0

⎡ 0 + ⎢ω z1 ⎢ ⎢⎣ 0

0⎤ 0⎥ ⎥ 0⎥⎦

⎡ Jx 1 ⎢ ⎢− J y1x1 ⎢− Jz x 1 1 ⎣

⎡M e ⎤ ⎢ x1 ⎥ ⎢ M ye ⎥ . ⎢ e1 ⎥ ⎢ Mz ⎥ ⎣ 1⎦

− J x1z1 ⎤ ⎡ 0 ⎤ ⎥⎢ ⎥ − J y1z1 ⎥ ⎢ 0 ⎥ = J z1 ⎥⎦ ⎢⎣ω z1 ⎥⎦

− J x1y1

J y1

− J z1y1

, 2 ⎡ − J x z ω& z ⎤ ⎡ J y1z1ω z1 ⎤ 11 1 ⎥ ⎢ ⎥ ⎢ 2 & ⎢ + − − ω ω J J x1z1 z1 ⎥ = ⎢ y1z1 z1 ⎥ ⎥ ⎢ ⎢ J z ω& z ⎥ 0 ⎥⎦ 1 1 ⎦ ⎢ ⎣ ⎣

⎡M e ⎤ ⎢ x1 ⎥ ⎢ M ye ⎥ . ⎢ e1 ⎥ ⎢Mz ⎥ ⎣ 1⎦

: Mx&&C = Rxe ; My&&C = R ye ; 0 = Rze ;

(28)

− J x1z1ω& z1 + J y1z1ω z2 = M xe ; − J y1ω& z1 − J x1z1ω z2 = M ye ; J z1ω& z1 = M ze . 1

(28)

0 = Rze ; − J x1z1ω& z1 + J y1z1ω z2 = M xe ; − J y1ω& z1 − J x1z1ω z2 = M ye 1

(29)

, .

, Rze

= 0;

M xe 1

= 0;

M ye 1

: -

= 0,

xOy, (29)

x1Cy1 J x1z1 ≠ 0

M xe

M ye

. J y1z1 ≠ 0 ,

, Cx1

Cy1

J x1z1 = J y1z1 , x1Cy1 ω& z

, ,

Cz1

ω z1 .

, x1Cy1 ,

, ,

, .

126

Mx&&C = Rxe ; My&&C = R ye ; J z1ω& z1 = M ze .

(30)

(30) ω& z1 ϕ

Cz ∗ ,

Cz1

ω& z1 = ϕ&& ,

, . , , ,

. -

. (

α ,

. 51).

f .

, r mg ;

: r N

r FT .

. Oxy

. 51.

Cx1 y1 .

. 51

(

(30) -

Rxe

= mg sin α − FT ;

. 51

R ye

. 51). = N − mg cos α ; M ze = FT r ,

(30),

mx&&C = mg sin α − FT ; my&&C = N − mg cosα ; J z1ϕ&& = FT r .

(a)

, . P

CP = r = const ,

127

ω z1 = VC / CP ; ω& z1 = ϕ&& =

1 ( dV C CP

/ dt ) = a Cτ / r = a C / r = x&&C / r , τ aC

, x&&C .

(b)

FT = J z1 a C / r 2 .

(a),

a C = x&&C ,

(b)

(a)

: a C = x&&C =

mg sin α m + J z1 / r 2

2 g sin α , 3

( )

J z1 = mr 2 / 2 .

FT FTp = fN .

, && yC = 0 ,

N = mg cosα ,

(a)

FT ≤ fmg cosα .

( ),

1 mr 2 2 g sin α 2 ≤ fmg cosα 2 3 r

tgα ≤ 3 f .

§ 11. ,

. , .

29. .

r r r ma = F + R ,

r F

r R -

r ma r r Φ = −ma ,

(1)

, , 128

r r r F + R + Φ = 0.

r r r F , R, Φ

(2)

: ,

. . ,

r r r Φ = Φτ + Φ n , r r r r : Φ τ = −maτ ; Φ n = − ma n .

(1), . -

. .

, .

, -

, .

. ,

(2)

: Fx + Rx + Φ x = 0 ; Fy + R y + Φ y = 0 ; Fz + Rz + Φ z = 0 ,

(3)

, , .

, , ,

. 30. . Fii -

r r r r r mi ai = Fie + Fii + Rie + Rii

(i = 1, n ) ,

r Fie

129

r Rie

r Rii -

r r Φ i = − mi a i ,

r r r r r Fie + Fii + Rie + Rii + Φ i = 0 (i = 1, n ) .

(4)

(

)

, . -

. , O(

)

r r r r r r r r r r mO (Fie ) + mO (Fii ) + mO (Rie ) + mO (Rii ) + mO (Φ i ) = 0 (i = 1, n ) .

(4)

(5)

(5).

. r r r r r r r r r F e + R e + Φ = 0 ; M O (Fie ) + M O (Rie ) + M O (Φ i ) = 0 ,

(6)

: ,

. (6)

, :

∑ Fix

Fxe

+ R ye + Φ y = 0 ; ∑ Fiz = Fze + Rze + Φ z = 0 ; + Φ x = 0 ; ∑ Fiy = r r ∑ mx (Fi ) = M x (Fie ) + M y (Rie ) + M z (Φ i ) = 0 ; r r r r (7) ∑ my (Fi ) = M y (Fie ) + M y (Rie ) + M y (Φ i ) = 0 ;

R xe

Fye

r r re re = + + Φ m ( F ) M ( F ) M ( R ) M ( ∑ z i z i z i z i ) = 0.

130

(7),

, -

, ,

(6)

r r r Φ = − (F e + R e ) .

r r (F e + Re )

r Ma C ,

r r Φ = − Ma C ,

(8)

r aC -

, ,

. -

, ,

. (6)

, O

r r r r r r M O (Φ i ) = − ( M O (Fie ) + M O (Rie )) .

r r r r r M O (Fie ) + M (Rie ) = dK O / dt . r r r M O (Φ i ) = − dK O / dt ,

(9)

O . ,

r M z (Φ i ) = − dK z / dt .

(10)

r r r r ⎞ ⎛ dK A r M A (Φ i ) = − ⎜ + V A × MVC ⎟ , ⎠ ⎝ dt r KA -

(11)

131

r r V A = VC ,

(11)

r r r M C (Φ ) = − dK C / dt ,

r KC -

(12)

. , , r r r M C (Φ i ) = − dK Cr / dt ,

(12) (13)

r K Cr -

. ,

(12)

(13)

, -

, . , -

. , , ,

, . ( ), . (6) O (

r Φ

, (8)

r r M O (Φ i )

, .

. 52, a

. 52

52, b

, C.

, 132

r r r r M O (Φ i ) ≠ M C ( Φ i ) .

. -

. .

. ,

(8)

r r r Φ ∗ = Φ = − Ma C .

(14)

, . , ,

, ,

, . 53),

. O, z,

. Φ = Ma C ,

, O

r aC

(

. 53). -

, i,

mi .

r r r ai = a iτ + a in , r , Φ in -

. 53

r r r r r Φ i = − mi a iτ − mi a in = Φ τi + Φ in ,

r Φ τi -

. z, .

r Φ in

r r mz ( Φ i ) = mz ( Φ τi ) .

r r r mz (Φ i ) = mz (Φ τi ) + mz (Φ in ) . rn z ( . 55), mz (Φ i ) = 0 ,

aiτ = εhiz ,

ε -

133

hzi -

z, r r M z ( Φ i ) = ∑ mz ( Φ τi ) = − ∑ ε (mi hiz )hiz = −ε ∑ mi hiz2 = − J z ε

(15)

. 53).

(

r r Φ = − Ma C ; M z (Φ i ) = − J z ε .

(16)

r aC = 0,

, r r Φ = 0 ; M zC ( Φ i ) = − J zC ε .

C J z = J zC ,

(17)

, z1 ,

xOy, . 54),

(

, . -

, , . , Φ = Ma C ,

, -

, (

. 54

. 54).

, Cz1 ,

, i,

mi . r r r r rτ rn + aiC ai = a C + aiC = a C + aiC , rτ rn , aiC aiC -

r a iC -

, r r r r r r rn r rτ r M C ( Φ i ) = ∑ ρi × ( − mi ai ) = ∑ ρi × ( − mi a C ) + ∑ ρi × ( − mi aiC ) + ∑ ρi × ( − mi aiC ), r

ρi -

(

∑ ρi × ( −mi a C ) = −(∑ ρi mi ) × a C 134

= −M∑

ρi mi M

. 54).

. r r r × a C = − Mρ C × a C = 0 ,

ρC = 0 . r

τ τ ∑ ρi × (− mi aiC ) = ∑ ρi × Φ iC .

n = 0, ∑ ρi × (− mi aiCn ) = ∑ ρi × Φ iC

n ρ i ||Φ iC (

, M C ( Φ i ) = M C ( Φ τiC ) ,

. 54).

r r rτ M Cz1 (Φ i ) = M Cz1 (Φ τiC ) . , , a iC = ερi , r r M Cz1 (Φ i ) = − ∑ mz1 (Φ τi ) = − ∑ mi ερ i ρ i = − (∑ mi ρ i2 ) ⋅ ε = − J Cz1 ε .

, Cz1

(

. 54). , ,

, ,

, (

r M Cz1 (Φ i )

, . 54

r r Φ = − Ma C ; M Cz1 = − J Cz1 ε .

(18)

, .

. . .

r r r r r r Φ = − Ma C ; M C ( Φ i ) = − dKC / dt = − dKC( r ) / dt . Cx1 y1z1

Cx ∗ y ∗ z ∗

, , Oxyz, (

. 55). ,

. 55

135

. ,

r ~r r r dK C dK C r r M C (Φ i ) = − =− − ω × KC , dt dt

r KC Cx1 y1z1 ,

ω -

, . , r i1

r k1

r j1

r r r r r M C ( Φ i ) = − K& Cx1 i1 − K& Cy1 j1 − K& Cz1 k1 − ω x1

ω y1

ω z1 ,

K Cx1

K Cy1

K Cz1

r r r i1 , j1 , k1 -

r M Cx1 (Φ i ) = − (K& Cx1 + ω y1 K Cz1 − ω z1 K Cy1 ) ; r M Cy1 (Φ i ) = − (K& Cy1 + ω z1 K Cx1 − ω x1 K Cz1 ) ;

(19)

r M Cz1 (Φ i ) = − (K& Cz1 + ω x1 K Cy1 − ω y1 K Cx1 ) .

(19) (3) . 27,

, .

, ,

K Cy1 = J y1ω y1 ; K Cz1 = J z1ω z1 . (19) , r M Cx1 (Φ i ) = −[ J x1ω& x1 + ( J z1 − J y1 )ω y1ω z1 ] ; r M Cy1 (Φ i ) = −[ J y1ω& y1 + ( J x1 − J z1 )ω x1ω z1 ] ; r M Cz1 (Φ i ) = −[ J z1ω& z1 + ( J y1 − J x1 )ω x1ω y1 ] .

(8),

136

(3)

. 27

(19), (20) .

K Cx1

Cx1 y1z1 = J x1ω x1 ;

(20)

ω = ω x1 i1 + ω y1 j1 + ω z1 k1 = 0 ,

r r M C (Φ i ) = 0 , r r Φ ∗ = − Ma C ,

, . , xOy, ω x1 = 0 ; ω y1 = 0 ; ω z1 = 0 .

x1Cy1 ,

J x1z1 = J y1z1 = 0 ,

K Cx1 = K Cy1 = 0 ; r (20) M Cx1 (Φ i ) = M Cy1 = 0 ,

(3)

K Cz1 = J z1ω z1 = J Cz1ω . r M Cz1 (Φ i ) = − J Cz1ω& = − J Cz1 ε .

27:

r r Φ = − Ma C

(8). , . , (

. .),

, ,

ω x1 = ω y1

O z, = 0 , ω z1 = ω .

, z1 ,

(20),

. (3)

. 27, :

r r r M x1 (Φ i ) = J x1z1ω& − J y1z1ω 2 ; M y1 (Φ i ) = J y1z1ω& + J x1z1ω 2 ; M z1 (Φ i ) = − J z1ω& = − J z1 ε .

Φ = Ma C ,

r aC .

, J x1z1 = J y1z1 = 0 , r r r r M x1 (Φ i ) = 0 ; M y1 (Φ i ) = 0 ; M z (Φ i ) = M z1 (Φ i ) = − J z ε = − J z1 ε .

, ,

. ,

137

31. . ,

(

, .

, ,

, , . , ,

, ,

. .

, . ,

y1 (

. r Φ = 0,

r aC = 0

. , . (

, )

. ,

138

. , . . 1. DE

A B (

. 56). , DE

, ω.

. 56

, .

xOy

Oxy,

. , . D,

r mg

r XB

. r

r X A , YA

DE;

DE. . i r r r a i = a iτ + a in .

(

. 56, a),

ω = const , r r ai = a in .

mi .

r a iτ = ω& ri = 0 ,

ri -

ai = a in = ω 2 ri ,

Φ i = mi a i = mi ω 2 ri

, , .

. , (8)

Φ = Φ ∗ = ma C = mω 2 rC =

1 mω 2 l sin α 2

139

r Φ

, H/3

, (

. 56, a). . :

= 0 , X A + X B + Φ = 0 ; ∑ Fiy = 0 , YA − mg = 0 ; r ∑ mA (Fi ) = 0 , − 2 X B l − 21 mgl sin α − Φ (l − 23 l cosα ) = 0 .

∑ Fix

Φ,

: −

X A + X B + 21 mω 2 l sin α = 0 ; YA − mg = 4 X B − mg sin α − (1 − 23 cosα )mω 2 l sin α

(a)

= 0.

D( D,

r r X D , YD ,

. 56, b) -

DE: r m ( F ∑ D i ) = 0 , − 21 mgl sin α + 23 Φl cosα = 0 .

α = arctg

2 2 ω l 3g

2 2 ω l. 3g

. cosα ≈ 1; sin α ≈ tgα = 2ω 2 l / 3g . YA = mg ; X B = −

α ,

( ) α

3gω 2 l − 2ω 4 l 2 3 gω 2 l + ω 4 l 2 m. m; X A = 18g 18 g

, .

10o ).

( ( ),

r r X A, XB

140

tgα =

Φ,

r YA -

3( m,

. 57).

, . A . . , . :

r m3 g

r r m1g , m2 g ,

;

r r X A , YA ,

Oxy,

r N

. 57;

. 57

. ,

r a1 .

, 2

. a 2 = a1

ε 3 = a1 / r3 , , .

1 r3 -

Φ 1 = m1a1 ;

Φ 2 = m2 a 2 = m2 a1 .

( A,

. 57).

, .

xOy.

r M ( Φ i ) = J Az ε 3 = 21 m3r32 ⋅ a1 / r3 .

ε3,

. 2

. T′ = T .

. 2

∑ Fix = 0 , − Φ 2 + T = 0 ; 1

∑ Fix

= 0 , − T ′ + X A = 0 ; ∑ Fiy = 0 , YA − m3 g + Φ 1 − m1g = 0 ; r r ∑ mA (Fi ) = 0 , T ′r3 + Φ1r3 − m1 gr3 + M (Φ i ) = 0 .

: 141

T ′ = T ; − ma1 + T = 0 ; − T ′ + X A = 0 ; YA − 2 mg + ma1 = 0 ;

(a)

T ′ + 23 ma1 − mg = 0 .

(a)

T T = ma1 .

(b) 1: a1 =

(a), (b)

2g. 5

XA =

(a)

(b) T′ = T ,

, T=

2 mg . 5

2 mg , 5

YA = 85 mg .

. . 1. . 2. 3.

. , ,

. (

. .).

, .

(

), , ,

6. , . : ) .5

; b) ; )

, . -

3. ,

1 ,

142

m1 ,

c, (

m2 -

. 58).

l ,

. , -

. ,

, . -

r r m1g , m2 g r N

: ;

r F

; (

. 58

. 58, ).

, ,

. Oxy, ,

, C

Cx1 y1

. x,

, x)

r V1

(

r a1 ,

ϕ)

( ω

ε,

. ,

, r a1 (

Φ 1 = ma1 ,

. 58, a). ,

r r r Φ 2 = −m2 a 2 , a2 r r r r a 2 = a 2e + a 2r + a 2c . r , a 2c = 0 .

143

r r r , a 2 r = a 2τ r + a 2nr ,

a 2 r = εl -

a 2nr = ω 2 l -

, .

r r a 2 e = a1 .

(

r a1 ,

r r r r r r r r r r r a 2 = a 2 e + a 2τ r + a 2nr ; Φ 2 = − m2 a 2 e − m2 a 2τ r − m2 a 2nr = Φ 2 e + Φ τ2 r + Φ 2n r , r r r Φ 2 e , Φ τ2 r , Φ 2n r . 60, a). x, ϕ Φ1

: Φ 1 = m1a1 = m1x&& ; Φ 2 e = m2 a1 = m2 x&& ; Φ τ2 r = m2 εl = m2ϕ&&l ;

( )

Φ 2n r = m2ω 2 l = m2ϕ& 2 l .

∑ Fix

= 0, − F

− Φ 1 − Φ 2 e − Φ τ2 r cosϕ + Φ n2 r sin ϕ = 0 . r r C. X C , YC

, (

. 58,b). C: r m ( F ∑ C i ) = 0 , − Φ 2e l cosϕ − Φ τ2r l − m2 gl sin ϕ = 0 .

(a)

= cx ,

: (m1 + m2 )x&& + cx = m2 l (ϕ& 2 sin ϕ − ϕ&& cosϕ ) ; lϕ&& + g sin ϕ = − x&& cosϕ .

(b)

2. , . .

(1736 - 1813) 1778

”,

“

. -

144

, -

. , .

. ,

.( ,

§ 12. : -

; ;

. .

32. ,

. ,

. . . ,

. ,

. . ,

. ,

. 59,a -

y ≥ ax 2 .

, ,

y = ax 2

(

. 59, b),

145

. 59

, f (x , y , z , x& , y& , z&, t ) = 0 .

(1)

: 1) ; 2)

; 3) , ,

. y = ax , 2

. 59,b. , ,

, (

. 61, d).

x& O = rω ,

x& O − rϕ& = 0 .

. .

, .

r V (

. 61, c).

y = ax − Vt . 2

. ,

, . , (

)

, .

, : dx O dϕ ; dx O = rdϕ ; =r dt dt

∫ dx O = r ∫ dϕ

; x O = rϕ + C ,

. , .

146

, , .

, ,

f (x , y , z , t ) = 0.

33.

(

(2)

) ,

. i.

r δ ri

. (2),

, r

δ ri = δ xi i + δ yi j + δ zi k

(3)

r grad f ⋅ δ ri = 0 ,

grad f -

(4)

(2)

t, ,

grad f =

(3) (4), δ x i , δ y i , δ zi

f r f r f r k. j+ i + zi yi xi

(5)

(5), r

δ ri ,

f f f δ xi + δ yi + δ zi = 0 . zi yi xi r

(6) r ri , x i , y i , zi .

δ ri , δ x i , δ yi , δ zi

, (4)

(6).

147

δ x i , δ yi , δ zi

δr

, (

): f f f δ xi + δ yi + δ zi = 0 . zi yi xi

δ f ( xi , yi , zi , t ) =

(7)

, ,

, .

, ,

. r r dri = Vi dt ,

r Vi ,

. -

: ) , ,

; b) , ; c)

, . , (

. 59, b

c). . 60, a

(

t − Δt , t, t + Δt ,

t ,

, .

Δt -

, t

r V,

δr,

t. ,

)

. 60, b

, -

148 . 60

r dr ,

; -

r dr

r δr.

r r r (δ r1 , δ r2 ,..., δ rn ) ,

(8)

∑ gradf k (ri , t ) ⋅ δ ri

(k = 1, l ) ,

(9)

i =1

(8)

r r

(9)

x1 , y1 , z1 ,..., x n , y n , z n . r r gradf k (r , t ) ⋅ δ r = 0 ( k = 1, l ) .

r δr

“

”

“ ,

”. virtualis

“

”. 34.

r δ ri

r Vi∗ ,

δ x i , δ y i , δ zi

(7) ,

δ ri

r Vi∗ .

x&i∗ , y& i∗ , z&i∗

r r δ ri = kVi∗ ,

r Vi

r Vi∗ .

35.

, . . l

. , n δ x1 , δ y1 , δ z1 ,..., δ x n , δ y n , δ zn

, 149

3n − l .

s = 3n − l .

(10)

, ,

. ,

, .

36.

. .

r δ A( F )

r r r δ A(F ) = F ⋅ δ r .

(11)

(9) . , (

. .22).

, ,

, -

r r r F1 , F2 ,..., Fn ,

r δr , r r r δ r1 , δ r2 ,..., δ rn ,

: n r r r r r r r r r r r δ A = δA(F1 ) + δA(F2 ) + ⋅ ⋅ ⋅ + δA(Fn ) = F1δ r1 + F2δ r2 + ⋅ ⋅ ⋅ + Fnδ rn = ∑ Fi δ ri .

(12)

i =1

. ,

r Vi∗ ,

(12) ,

r r δA = ∑ Fi ⋅ kVi∗ ,

. k,

: n r r r ∗ r ∗ r N = δA / k = N (F1 ) + N (F2 ) + ⋅ ⋅ ⋅ + N (Fn ) = ∑ Fi ⋅ Vi∗ . ∗

i =1

150

(13)

37.

. .

, :

r r r δA( Ri ) = Ri ⋅ δ ri = 0 .

(14)

, , n r r r δA( R ) = ∑ Ri ⋅ δ ri = 0.

(15)

i =1

. ,

, .

, ,

; )

; -

. , b)

: a) , ,

. .; d)

, ;

§ 13.

. .

, . , ,

. .

, .

38.

. -

, (

. . 30).

, , (

)

. 151

r r r Fi + Ri + Φ i = 0

: i = (1, n ) .

, -

n r r r r r n r r n r r r ∑ Fi ⋅ δ ri + ∑ Ri ⋅ δ ri + ∑ Φ i ⋅ δ ri = ∑ ( Fi + Ri + Φ i ) ⋅ δ ri = 0 , n

i =1

(1)

i =1

: ,

. ,

, -

. , (15)

. ,

, . 37,

n r n r r r r r r ∑ Fi ⋅ δ ri + ∑ Φ i ⋅ δ ri = ∑ (Fi + Φ i ) ⋅ δ ri = 0 ,

(1)

i =1

(2)

i =1

. ,

, :

1) 2)

; ,

; 3)

; 4) 5)

; , ,

;

152

6) ,

(1); -

7) (1); 8)

.4-7

, . 39.

. (

, -

)

. ,

, r r F ∑ i ⋅ δ ri = 0 . n

(3)

i =1

(2) ,

. (2),

, , . (3)

(2),

r Vi = 0 = const

r ai = 0 ,

, Φi = 0. (3) (3)

r r F ∑ i ⋅δ r ≠ 0,

, . -

, r

r r dri = δ ri

∑ Fi ⋅ dr ≠ 0 . T − T0 ≠ 0 ,

, .

, . r r δ ri = kVi∗ ,

(3), (3)

∑ Fi ⋅ kVi∗ = 0 . 153

, : n

∑ Fi ⋅ Vi∗ = 0 ,

(4)

i =1

, . , “

”,

, . .

, . , , ,

. 1.

, ,

4. 5.

. , . , .

a) 6a.

. , ,

(

)

, 7a.

. , , .

154

8a. ,

9a.

6a - 8a ,

, .

10a.

, ,

. b) 6b.

. , ,

, .

7b.

, , ,

8b. ,

. 9b.

6b - 8b , .

10b.

. 8b

: ,

, . , . ,

(

155

AB -

B , -

Q. P

. 61

. (

. 61, a), ϕ.

ϕ,

. ,

. .

δϕ

( Q

r δ rB .

. 61). r δ rA ,

P-

, r

∑ δA( Fi ) = δA = δA( M ) + δA(Q) + δA( P) = ± Mδϕ + Q ⋅ δ rA + P ⋅ δ rB

= 0.

, r δA(Q) = Qx δ x A + Q y δ y A ,

P rQ δA( P) = Px δ x B + Py δ y B . r P ,

, . 61, Oxy:

Qx = 0 ;

Qy = − Q ;

Px = P ;

Py = 0 .

δA = Mδϕ − Qδ y A + Pδ x B = 0 .

A x B = 2 l cosϕ ; y B = 0 .

156

B: x A = l cosϕ ; y A = l sin ϕ ; ( -

δ x A = − l sin ϕδϕ ; δ y A = l cosϕδϕ ; δ x B = −2l sin ϕδϕ ; δ y B = 0 . δ yA

δ xB

: Mδϕ − Ql cosϕδϕ − 2 Pl sin ϕδϕ = 0 ; M − Ql cos ϕ − 2 Pl sin ϕ = 0 .

: P=

M Q − ctgϕ . 2l sin ϕ 2

. ω∗ (

. 61, b).

r V A∗

r VB∗ ,

, -

. ,

r VB∗

r V A∗ ⊥OA

. , r r r r∗ r r∗ ∗ r ∗ ∗ ∗ ∗ ∗ ( ) = = ( ) + ( ) + ( ) = ± + ⋅V A + P ⋅VB = 0 , N F N N M N Q N P M Q ω ∑ i

N∗ -

. .

V A∗

∗

= ω ⋅ OA = ω l .

, PAB ,

. 61,b.

, A

r r r r N ∗ = Mω ∗ + QV A∗ cos(Q^ V A∗ ) + PVB∗ cos( P ^ VB∗ ) = Mω ∗ + QV A∗ cos(180o − ϕ ) + + PVB∗ cos180o = Mω ∗ − QV A∗ cosϕ − PVB∗ = 0 . OB = 2 l cosϕ ; BPAB

OBPAB = OBtgϕ = 2l sin ϕ ; OPAB = OB / cosϕ = 2l ; APAB = OPAB − l = l .

, B: V A∗

ω ∗AB VB∗

= V A∗

∗

/ APAB = ω ;

VB∗

ω ∗AB

∗

⋅ BPAB = 2lω sin ϕ .

157

Mω ∗ − Qω ∗ l cosϕ − 2 Pω ∗ l sin ϕ = 0 ; M − Ql cosϕ − 2 Pl sin ϕ = 0 .

P. 2. (

r P

. 62),

α. r P

r P

r Q

r Q.

. ,

, ,

, . -

. 62 r

δ r1 ,

. r P,

(

. 64),

δ r2

r r r r r r δA = δA( P ) + δA(Q) = P ⋅ δ r1 + Q ⋅ δ r2 = Pδ r1 − Qδ r2 = 0 .

( 2π

δϕ = δ r1 / a .

Pδ r1 − Q

158

, δ r2 = δ r3 tgα , δ r2 = tgα h / (2π a ) ⋅ δ r1 . , -

. 62). ,

, -

k = h / (2π ) , δ r3 = kδϕ = h / (2π ) ⋅ δϕ .

(

)

r δ r1

2π 1 htgα htgα δ r1 = 0 ; P − Q ⋅ ⋅a . = 0; Q = P h tgα 2π a 2π a

, .

3. m1 = m2 = m .

C, 2.

1 ,

m3 .

1 .

. 63 r r r m1g , m2 g , m3 g .

r FT ,

. , . , .

x ( 2,

. 63, a). .

r V1∗ ,

1 x.

. 63, a,

P1 -

B, ω B∗ = V1∗ / (2 R ) ,

R-

V3∗ = VB∗ = ω B∗ R = V1∗ / 2 .

3 ,

r r r r r r r r r N 1∗ = N 1∗ (m1g ) + N 1∗ (m3 g ) + N 1∗ (FT ) = m1g ⋅ V1∗ + m3 g ⋅ V3∗ + FT ⋅ V1∗ = r r r r r r = m1 g cos(m1 g ^ V1∗ ) + m3 gV3∗ cos(m3 g ^ V3∗ ) + FT V1∗ cos( FT ^ V1∗ ) = m3 gV3∗ − FT V1∗ = = m3 gV1∗ / 2 − FT V1∗ = 0 .

159

V1∗ ,

, : m3 g / 2 − FT = 0 .

(a) r V2∗ ,

2 . 65, b).

B, ω B∗ = V2∗ / (2 R) .

V3∗ = VB∗ = ω B∗ R = V2∗ / 2 .

, r r r r r r r N 2∗ = N 2∗ (m2 g ) + N 2∗ (m3 g ) = m2 g ⋅ V2∗ + m3 g ⋅ V3∗ = m2 gV2∗ cos(m2 g ^ V2∗ ) + r r + m3 gV3∗ cos(m3 g ^ V3∗ ) = m2 gV2∗ − m3 gV3∗ = m2 gV2∗ − m3 gV2∗ / 2 = 0 . V2∗ ,

, : m2 g − m3 g / 2 = 0 .

(b)

(

. Oy, m1 && y = N − m1 g . , N = m1g .

63, c). y = const ,

, FT = fN ,

: FT = fm1g

( )

(a) - ( )

: m3 g / 2 − FT = 0 ; mg − m3 g / 2 = 0 ; FT = fmg .

: m3 = 2m ; f = 1. , r r

. , ,

, ,

X ,Y

M ,

, , , , ,

160

r X,

, ,

r Y.

M .

, . ยง 14.

(

)

(

. . .

40.

. , . n

r r ri = ri (q1 , q 2 ,..., q s , t ) q j ( j = 1, s) -

(i = 1, n) ,

(1)

. , s

, . , -

ฮดq j ( j = 1, j ) .

. . . -

, ,

, , . q& j

161

q& j = dq j / dt ( j = 1, s) .

(2)

. -

41. .

δq1 -

, q1 ,

)

( -

. r (δ ri )1 =

r ri δq1 (i = 1, n) . q1

r r r r r rr1 r rr2 r rrn r r δA1 = F1 (δ r1 ) 1 + F2 (δ r2 ) 1 + ⋅ ⋅ + Fn (δ rn ) 1 = F1 δq1 + F2 δq1 + ⋅ ⋅ + Fn δq1 q1 q1 q1 . δA1 -

: r ⎛ n r ri ⎞ δA1 = ⎜ ∑ Fi ⋅ ⎟ δq 1 . q1 ⎠ ⎝ i =1

δq 2

. r (δ ri ) 2 =

r ri δq 2 (i = 1, n) . q2

δA1 , r ⎛ n r ri ⎞ δA2 = ⎜ ∑ Fi ⋅ ⎟ δq 2 . q2 ⎠ ⎝ i =1

, :

r ⎛ n r ri ⎞ δAs = ⎜ ∑ Fi ⋅ ⎟ δq s . qs ⎠ ⎝ i =1

, : 162

r r r ⎛ n r ⎛ n r ⎛ n r ri ⎞ ri ⎞ ri ⎞ δA = δA1 + δA2 + ⋅ ⋅ +δAs = ⎜ ∑ Fi ⋅ ⎟ δq s .(3) ⎟ δq1 + ⎜ ∑ Fi ⋅ ⎟ δq 2 + ·· + ⎜ ∑ Fi ⋅ qs ⎠ q1 ⎠ q2 ⎠ ⎝ i =1 ⎝ i =1 ⎝ i =1

Q1 , Q2 ,..., Qs ,

δA = Q1δq1 + Q2δq 2 + ⋅ ⋅ + Qsδq s =

∑ Q j δq j .

(4)

j =1

, q& ∗j ,

(3)

r r r ⎛ n r ⎛ n r ri ⎞ ∗ ⎛ n r ri ⎞ ∗ ri ⎞ ∗ N = δA / k = ⎜ ∑ Fi ⋅ ⎟ q& s , ⎟ q&1 + ⎜ ∑ Fi ⋅ ⎟ q& 2 + ··· + ⎜ ∑ Fi ⋅ q q q ⎝ i =1 ⎝ i =1 ⎝ i =1 1⎠ 2⎠ s⎠ ∗

N ∗ = N 1∗ + N 2∗ + ⋅ ⋅ + N s∗ = Q1q&1∗ + Q2 q& 2∗ + ⋅ ⋅ + Qs q& s∗ =

(5)

∑ Q j q& ∗j .

(6)

j =1

Q1 , Q2 ,..., Qs

(4)

. -

. [Q j ] = [ A] / [q j ] ,

[ A] -

, -

, ,

, . . , .

, . . .

163

. -

, , s

N ∗ = N 1∗ + N 2∗ + ⋅ ⋅ + N s∗ =

(7)

∑ N ∗j = j =1

∑ Q j q& ∗j = j =1

∑ Q j q& j .

(7)

j =1

, ,

(q& j ≠ 0) , (q& k = 0 ( k ≠ j )) .

(3) ⎛ n r rri ⎞ ⎟δq j , δA = ∑ ⎜⎜ ∑ Fi q j ⎟⎠ j =1⎝ i =1 s

r r ri Q j = ∑ Fi ⋅ qj i =1 n

(9)

( j = 1, s) .

(8)

(9)

, :

⎛ x y z ⎞ Q j = ∑ ⎜⎜ Fix i + Fiy i + Fiz i ⎟⎟ ( j = 1, s) , qj qj qj ⎠ i =1 ⎝ r Fi . Fix , Fiy , Fiz n

(10)

. ,

Π = Π (q1 , q 2 ,..., q s ) ,

Q j = − Π / q j ( j = 1, s) .

(11)

: (

) .

164

. OM M1

l2 ,

. ϕ1

MM 1

ϕ2 (

. 64). , -

(10). Oxy,

r m1g

. 64

. 66,

r m2 g

: P1x = m1g ; P1y = 0 ; P2 x = m2 g ; P2 y = 0 ;

(a)

x1 = l1 cosϕ 1 ; y1 = l1 sin ϕ1 ; x 2 = l1 cosϕ1 + l 2 cosϕ 2 ; y 2 = l1 sin ϕ1 + l 2 sin ϕ 2 .

(b)

(10) Q j = P1x

ϕj

+ P1y

ϕj

(a)

+ P2 x

ϕj

+ P2 y

ϕj

( j = 1,2 ) .

(b)

Q1 = − m1gl1 sin ϕ 1 − m2 gl1 sin ϕ 1 = − (m1 + m2 )gl1 sin ϕ 1 ; Q2 = − m2 gl 2 sin ϕ 2 .

, .

r m1g

r m2 g

, ,

Ox.

r m1g

h1 = l1 − l1 cosϕ1 , h2 = h1 + l 2 (1 − cosϕ 2 ) ,

r m2 g

Π = m1 gh1 + m2 gh2 = m1 gl1 (1 − cos ϕ1 ) + m2 g[l1 (1 − cos ϕ 1 ) + l2 (1 − cos ϕ 2 )] .

(11) Q1 = −

ϕ1

= − (m1 + m2 )gl1 sin ϕ1 ; Q2 = −

ϕ2

= − m2 gl 2 sin ϕ 2 .

165

. 42. (

(3)

. 39), :

∑ Q j δq j

= 0.

, δ j ( j = 1, s)

, Q j = 0 ( j = 1, s) .

(12)

, .

(11) Π / q j = 0 ( j = 1, s) .

43. (

(1)

(13)

. 38)

n r r r r r ∑ (Fi + Ri ) ⋅ δ ri + ∑ Φ i ⋅ δ ri = 0 , n

i =1

. ,

. , s

j =1

i =1

j =1

∑ Q j δq j + ∑ Q j δq j = ∑ (Q j + Q j )δq j Qj -

(14)

, r r ri Qj = ∑ Φi ⋅ qj i =1 n

δq j ( j = 1, s)

166

= 0,

( j = 1, s) .

(15)

(14)

Q j + Q j = 0 ( j = 1, s) ,

(16)

. ,

. . , -

§ 15. ,

, . . ,

( ,

. , . . 44.

. − Q j = Q j ( j = 1, s) .

(15) Φ i = −mi ai = −mi dVi / dt ,

r r dVi ri ∑ mi dt ⋅ q = Q j ( j = 1, s) . j i =1 n

(1)

167

r r dVi ri d ⎛ T⎞ T ⎟⎟ − ⋅ = ⎜⎜ Q j = ∑ mi dt q j dt ⎝ q& j ⎠ qj i =1 n

(2)

( j = 1, s) .

(2)

(1)

: d ⎛ T⎞ T ⎟⎟ − ⎜⎜ = Q j ( j = 1, s) , dt ⎝ q& j ⎠ qj

(3)

. .

(3) s

q j (t ) ,

. 2.

. ,

(3) ,

, .

45. ,

Qj = − Π / q j

( j = 1, s) , Π d ⎛ T⎞ T ⎟⎟ − ⎜⎜ + = 0 ( j = 1, s) . dt ⎝ q& j ⎠ qj qj

, (T − Π ) / q& j = T / q& j ( j = 1, s) .

L = T −Π,

(4)

. :

168

d ⎛ L⎞ L ⎟⎟ − ⎜⎜ = 0 ( j = 1, s) . dt ⎝ q& j ⎠ qj

(5)

, Qj = − Π / q j Qj ,

(3)

Π d ⎛ T⎞ T ⎟⎟ − ⎜⎜ =− + Q j ( j = 1, s) . dt ⎝ q& j ⎠ qj qj

, , ,

, -

. . ,

(3)

(5)

. ,

. 46. ,

(

)

(

)

. -

: 1) 2) 3)

; ; ,

; 4)

;

, ;

6) ; 7) 8)

; (

) ;

169

9) ,

. : (

)

. ,

( )

. .

. , . -

, , . m1 = 8m

. 2

m2 = 2 m ,

c. . . 65). ,

( . 65

. s, ,

l0 ,

q1 = s ,

, q2 = x .

d ⎛ T⎞ T d ⎛ T⎞ T = Q1 ; ⎜ ⎟ − = Q2 . ⎜ ⎟− dt ⎝ s& ⎠ s dt ⎝ x& ⎠ x

(a)

T = T1 + T2 ,

T2 -

. ,

170

T1 -

(

)

, ,

1mV2 2 1 1

+ 21 m2VC2 + 21 J Cω ω 22 ,

(b)

, ω2 -

VC -

, J Cω , J Cω = m2 R 2 / 2 .

, r V1

, -

r V2 (

. 67), , s& = V1 , x& = V2 . r r r , VC = V1 + V2 . , VC = V1 + V2 = s& + x& . , ω e = 0 , ω 2 = ω 2r , ω 2r , ω 2 r = V2 / CP = V2 / R = x& / R , P , R, ω 2 = x& / R . (b) , , J Cω ,

1 m s& 2 2 1

+ 21 m2 (s& + x& ) 2 + 41 m2 x& 2 .

, &&2 . T = 5ms& 2 + 23 mx& 2 + 2msx

(a): T T T T = 0; = 0. = 10ms& + 2mx& ; = 3mx& + 2ms& ; s& x& s x

( ) r m2 g

r m1g

r F

r F′ ,

, ,

= F′ ,

(

. 65).

P-

, ,

. , . 171

N 1∗ : V2 = x& = 0 , r r r r r r r r r r N 1∗ = N 1∗ (m1 g ) + N 1∗ (m2 g ) + N 1∗ ( F ′ ) + N 1∗ ( F ) = m1 g ⋅ V1 + m2 g ⋅ V1 + F ′ ⋅ V1 + r r + F ⋅ V1 = F ′ s& − F s& = (F ′ − F )s& = 0 . V1 = s& ≠ 0 ,

, Q1 = 0 . V2 = x& ≠ 0 , V1 = s& = 0 , N 2∗ : r r r r r r N 2∗ = N 2∗ (F ) + N 2∗ (m2 g ) = F ⋅ V2 + m2 g ⋅ V2 = − F x& = − cxx& .

, Q2 = − cx . ( )

(a),

d d (10ms& + 2mx&) = 0 ; (3mx& + 2 ms&) = − cx . dt dt

, : 10&&s + 2 && x = 0 ; 3mx&& + cx + 2ms&& = 0 .

(d)

, ,

&&s = −0,2 x&& .

(d)

&&s

(d), : 2,6mx&& + cx = 0 x&& + k 2 x = 0 ,

k 2 = c / 2,6m .

(e)

2. R

, k , τ = 2π / k . : k = c / 2,6m ; τ = 2π 2,6m / c . ( . 66) 2 l m2 , M1

1 -

M2.

, . ,

, .

172

. ,

Oxyz, ,

Oz . Cx1 y1z1 ,

66.

q1 = ϕ 1 -

q2 = ϕ2 -

(

. 66).

. 66

d ⎛ T ⎞ T d ⎛ T⎞ T = Q1 ; ⎜ = Q2 . ⎟− ⎜ ⎟− ϕ2 dt ⎝ ϕ& 2 ⎠ ϕ1 dt ⎝ ϕ&1 ⎠

(a)

T = T1 + T2 ,

T1 -

T2 -

Oz. T1 =

1mV2 2 1 C

+ TCr ,

VC -

, TCr . ,

Cx1 y1z1

ω x1 , ω y1 , ω z1 -

TCr = 21 ( J x1ω x2 + J y1 ω 2y + J z1 ω z2 ) , 1

(

)

. Oz

T2 =

Jz -

Oz, . T=

J z = J 1z + J 2 z ,

J zω z2 ,

ωz -

+ 21 ( J x1 ω x2 + J y1ω y2 + J z1ω z2 ) + 21 J z ω z2 . 1 1 1 J x1 = m1R 2 ; J y1 = J z1 = m1R 2 / 2 . J 2 z = m2 l 2 / 3 . J1z = J ∗ + m1l 2 = m1R 2 / 2 + m1l 2 = 21 m1 (R 2 + z1

1mV2 2 1 C

Oz,

1 2

ϕ1,

2l 2 ) .

Cz1∗ ,

z1∗

173

, T=

1mV2 2 1 C

+ 21 m1R 2ω x2 + 41 m1R 2ω y2 + 41 m1R 2ω z2 + 16 m2 l 2ω z2 + 21 m1R 2ω z2 + m1l 2ω z2 1

ω1 ω 2 r r ω1 ω 2 ( . 66), ω1 = ϕ&1 , ω 2 = ϕ&2 .

, ,

Cx1 y1z1 : ω x1 = ω 1 = ϕ&1 ; ω y1

ω z1 = ω 2 cosϕ1 = ϕ& 2 cosϕ1 .

, r r r ω = ω1 + ω 2 , = ω 2 sin ϕ1 = ϕ& 2 sin ϕ1 ; Oz

ω2 ,

ω z = ω 2 = ϕ& 2 ,

VC = ω 2 l = ϕ& 2 l .

1 m R 2ϕ& 2 1 2 1

1 2

m2 l 2 2 1 ϕ& 2 + 2 m1 (2R 2 + 3l 2 )ϕ& 22 , 3

sin 2 ϕ1 + cos2 ϕ1 = 1.

: ⎛ m l2 ⎞ T T T T = m1R 2ϕ&1 ; = 0; = 0. = ⎜⎜ 2 + 2m1 R 2 + 3m1l 2 ⎟⎟ ϕ& 2 ; ϕ&1 ϕ1 ϕ2 ϕ& 2 ⎝ 3 ⎠

(b) , -

ϕ& 1 ≠ 0

ϕ& 2 ≠ 0

N 1∗

ϕ& 2 = 0

, = ± M1ω1 = M1ϕ&1 ,

N 2∗ = ± M 2ω 2 = M 2ϕ& 2 ,

ϕ& 1 = 0

Q1 = M 1 .

Q2 = M 2 . (a),

(b)

: m1R 2ϕ&&1 = M 1 ; (m2 l 2 / 3 + 2 m1R 2 + 3m1l 2 )ϕ&&2 = M 2 .

ϕ&&1

ϕ&&2

, ,

. J y1 = J z1 .

174

Oz,

Cx1

, ,

1. § 1.

, ,

............................... (3).

(3).

§ 2. 7

....................................................................... (7). (7). (8).

(9). 2.

(12). (12).

(13).

(14). § 3. 14

.. 3.

(14).

(15). (15).

(20).

(21). (27). 4. (28).

(

)

(29). (35).

(38). 5.

(43). § 4. 6.

, (48).

....................................................................................…… (44). 7. (45). 8. (49). 9. (50). 10. ( ) (54).

§ 5.

… (56). 12.

11. (57).

(59). (61). 14.

(60). 13. (61). 15. (62). 16.

(63). § 6.

17.

..................................................... (71). 18. (73).

70 (74).

175

(74). 19. (75). 20. ,

(79).

(79). § 7. 79

......................................................... 21.

(79). , (82). 22. (86). .

(87).

(81). (88).

(90).

(93). -

(94). § 8.

104

............................... 23. (104). ,

(104).

(106).

(106). § 9.

...................... (108). 25.

24. (110). (110).

108

(111). (112). 26. (114).

§ 10.

.................... (116). (119). (119). -

27. 28.

116

(121). (123). (125). § 11. 29. 30. (129). (132). (133). (138).

176

......................................………… 128 (128). (131). (135). 31. (137). -

2. ยง 12. 32. ( (149). 35.

...................................... (144). 33. (146). 34. (149). 6. (150).

) (149). 37.

ยง 13.

151

.................. -

160

38.

144

(151).

(152). 39. (152).

(153).

ยง 14. 40.

(160). 41. (161). (162). 42. (165). 43.

(165). ยง 15.

166

......................................................... 44.

(166). 45. (167). 46.

(168).

. . . .

60 84/16. . . . . . 8,84. .- . . 8. .3 . . . 1 18.07.94. . 634034, , . , 30.

177