ISSN 1991-3087

:   77-24978 05.07.2006 .

ISSN 1991-3087

42457

- 1 .

: 305008, ., , .7.

.: 8-910-740-44-28

E-mail: jurnal@jurnal.org

@Mail.ru Rambler's Top100
.

 

,

.

 

 

, [3, .197-202], [9, .139-142], [10, .364], [11, .28-41]. [2, .359], [4, .189], [12, .105], [13, .218]. , , , [4, .172], [10, .280], [13, .218], [14, .95]. . : [15], [16, .63-68], [20, .218-219]. . .

 

1. .

 

. . , .

mw = F, (1.1)

F .

, , . . . . ( , )

w = w + w. (1.2)

(1.1)

mw = F - mw, (1.3)

mw - , - , - . [2,.359]. [3, .198] . [2,.359] .

, . : v, , , . w, r. , v . [2,.362]

w = w - 2[vw ] + (d v/dt) - w2r^ + [(dw/dt)r],. (1.4)

r^ - - r, . , , [2,.364]

mw = mw + 2m[vw ] - m(d v/dt) + mw2r^ m[(dw/dt)r]. (1.5)

mw = F + F + F + F + F, (1.6)

: F ; F ; F ; F ; F .

F . F , . F . F . F .

, , .

, . [2, .77]

dP/dt = ∑F, (1.7)

: P ; ∑F .

, , ,

P(t) = m(t)v(t), (1.8)

: m(t) ; v(t) .

- ,

v(t) = dr(t)/dt, (1.9)

r -.

: , -. (1.9) (1.8) (1.7)

d(m (dr/dt))/dt = ∑F. (1.10)

m [1,.295],

d[(d(mr)/dt) r(dm/dt)]/dt = ∑F.

d[(d(mr)/dt)]dt d[r(dm/dt)]/dt = F.

m(d2r/dt2) + (dm/dt)(dr/dt) + (dm/dt)(dr/dt) +

+ r(d2m/dt2) r(d2m/dt2) - (dm/dt)(dr/dt) = ∑F. (1.11)

(1.11)

m(d2r/dt2) = ∑F - (dm/dt)(dr/dt). (1.12)

(1.12) . ,

F = - (dm/dt)(dr/dt). (1.13)

, - . (1.13) - , - () . , ( ) [2, .120] () . (1.12) - [2, .120], , ( ). (1.12) , , , (1.11) [17, .102], , . (1.11), (1.13), ,

m(d2r/dt2) + r(d2m/dt2) +(dm/dt)(dr/dt) = ∑F + F + r(d2m/dt2) +(dm/dt)(dr/dt). (1.14)

(1.14) Fm, F ,

m(d2r/dt2) + r(d2m/dt2) + (dm/dt)(dr/dt) = ∑F + F + Fm + F. (1.15)

Fm ,

Fm = r(d2m/dt2). (1.16)

(1.16),

r = Fm /(d2m/dt2). (1.17)

, , . Fm .

F ,

F = (dm/dt)(dr/dt). (1.18)

F - , , . , , . , F F . , (1.15) . , (1.15), , , , , .

 

2. .

 

. , .

[18, .88-90], . , [4, .101] .

(.2.1) m R.

 

. 2.1.

 

F , , [2, 365]

F = m ω2 R. (2.1)

ω = v /R, v R, (2.1)

F = m v2 / R. (2.2)

R. , . , v, . , .

m (.2.2) R, w .

 

. 2.2.

 

( ), , . , , . w , , . , (.2.2), , . , S ,

S = w t2 /2, (2.3)

, S

S = w t2 /2. (2.4)

(2.3) (2.4)

S / S = w / w .

w S / S = 1/ cosΨ

w = w /cosΨ, (2.5)

0 £ Ψ £ π/2.

, (.2.2) . ,

(.2.3).

 

. 2.3.

 

, , . π/2 ³ Ψ ³ 0.

, , , .

ω v, [2, .365]

F = 2m [v ω]. (2.6)

.2.4.

 

. 2.4.

 

(2.3),(2.4),(2.5) , , , , , , , . . (.4) -π/2 £ Ψ £ π/2, π/2 ³ Ψ ³ 0. , (.2.5) , .

 

. 2.5.

 

0 £ Ψ £ π/2, π/2 ³ Ψ ³ -π/2.

[2,.365] F (.2.6),

F = -m w, (2.7)

w .

 

. 2.6.

 

, () . (.2.6) , , , , , . .

F ( , ) [2, .365]

F = -m [(dω/dt)R]. (2.8)

R ω, (2.8)

F = -m (dω/dt)R. (2.9)

(.1.7) , () .

 

. 2.7.

 

ω = v /R, v R, (2.9)

F = -m (dv/dt). (2.10)

dv/dt = w , w , (2.10)

F = -m w (2.11)

, (2.11) (2.7) , w α II α (.2.8) .

 

. 2.8.

 

, w , , . w II FII, , w II.

FII = -m w II. (2.12)

w II = w cosΨ,

FII = -m w cosΨ, (2.13)

Ψ -π/2 £ Ψ £ π/2.

, (2.13) . .

, : , , , . , [3, .198].

,

F = - (dm/dt)(dr/dt). (2.14)

()

u = dr/dt, (2.15)

(2.14)

F = -u (dm/dt). (2.16)

(2.16) ─ , u , u . , , , . [2, .116]. (1.12) ∑F = 0,

m(d2r/dt2) = - (dm/dt)(dr/dt). (2.17)

d2r/dt2 = dv/dt,

v , (2.17) (2.15)

m(dv/dt) = - (dm/dt)u. (2.18)

(2.17) dt

mdv = -udm, (2.19)

, u = uO , , mO m v

m

v = -uO ∫ dm /m = uOln(mO/m). (2.20)

mO

mO/m = v/uo. (2.21)

(2.21) - .

 

3. .

 

(. 3.1) R, ω . , ∆m,

F = ∆m ω2 R.

, . .

 

. 3.1.

 

[19, .76-82], (.3.2). , , , - . , , , , .

 

. 3.2.

 

, m ℓ:

F =m ω2 R. (3.1)

,

m = ρV. (3.2)

ℓ = π R,

π .

V = π2 Rr2 = πR π r2 = ℓ π r2 ,

r .

V = ℓ π r2 .

,

= RΨ,

V = π r2 RΨ. (3.3)

(3.3) (3.2) :

m = ρ π r2 RΨ. (3.4)

(3.4) (3.1),

F = ρ π r2 ω2 R2Ψ.

, (.2)

F = ∆Fcos((π/2)- Ψ).

, cos((π/2)- Ψ) = sin Ψ,

F = ∆F sin Ψ.

F

F = ρ π r2 ω2 R2 sin ΨΨ.

, 0 Ψ

Ψ

F = ∫ ρ π r2 ω2R2 sin ΨdΨ.

0

,

Ψ

F = - ρ π r2 ω2R2 cosΨ│. (3.5)

0

, w w,

w = 10 w .

, (2.5)

cos Ψ = 0,1.

Ψ ≈ 0,467 π,

84 .

,

0 £ Ψ £ 84 96£ Ψ £ 180 . 6,7% (, , 1% ). , , (3.5) , , 2

84

F = - 2ρ π r2 ω2R2 cosΨ│. (3.6)

0

F= 1,8 ρ π r2 ω2 R2.

,

ω = v/R,

F= 1,8 ρ π r2 v2 .

, , ,

v = w t, (3.7)

F= 1,8 ρ π r2 (w t)2. (3.8)

[1, .451] , ,

t

F = ((1,8ρ π r2w2)/t) ∫t2 dt.

0

F = 0,6ρ π r2w2t2. (3.9).

, , .

(.3.3): R. S R1 S1, . S < S1

R1 < R. . [5, .333] ,

v/v1 = S1/S = r12 /r2, (3.10)

r1 r .

,

v/v1 = w / w1. (3.11)

, (3.10) (3.11)

w1 = w r2 / r12. (3.12)

, (3.9), , (3.12)

F┴1 = 0,6 ρ π r12 w12 = 0,6ρ π r2w2t2 (r2 / r12 ) = F(r2 / r12) (3.13)

(3.9) (3.13) , (r2 / r12).

r < r1 , .

 

. 3.3.

 

, ( ) , . , F , 3.3, . F

F = 2 F - 2F┴1 = 1,2ρ π r2w2t2 (1- (r2 / r12)) (3.14)

, , . . , , .

, , , .

r = 0,025; r1 = 0,05; ρ = 1000 /3; w = 5/2, t = 1, F.44.

 

4. .

 

, m , ω v. F F = 2m[vw].

 

. 4.1.

 

.4.1 ,. .4.2 .

 

. 4.2.

 

.4.1 .4.2. . 4.3 .

 

. 4.3.

 

() . 180 . F| | , , . F^ , .

, . , . v - . v v , , (.3) v α

v = v cosα, v = v sinα.

(.4.3) , v , v v. , R, v v, v , ( ). R . α 90 0 . , , 1/4 R0. , . 1/4 R0 r

m = ρπ2 r2 R0 /2, (4.1)

ρ .

F^ = 2m v ω cos b, (4.2)

v ; ω ; b F (-90 £ b £ 90).

, . F (-90 £ b £90). α 90 0 ,

0

v = 1 / (0 - π/2) ∫ v cos α dα = 2 v / π. (4.3)

π/2

v π /2R

ω = (1/ ((v π /2R) - v R))) ∫ ω dω = (v /2R) ((π /2.) +1). (4.4)

v/R

(4.4) . , , v /R.

ℓim (v /R) = ℓim (v sinα /R), (4.5)

v 0 α 0

R 0 R 0

R .

[7, .410] : vsinα /R (R= 0, α = 0) R = kα , . , . , .

α = 0 R= 0, α = π /2 R= R (.3), = 2R/π , (5) ,

ℓim (v π sinα /2R α) = (v π/2R) ℓim sinα/α = v π/2R. (4.6)

α 0 α 0

(4.1), (4.3) (4.4) (4.2)

F^ = ρ π r2 v2((π /2.) +1) cos b.

(-90 £ b £ 90) .

90

F^ = ρ π r2 v2((π /2.) +1) ∫ cos b db = 2 ρ π r2 v2((π /2.) +1).

-90

∑F^ = 4ρ r2 v2((π /2.) +1). (4.7)

(3.7), (4.7)

∑F^ = 4ρ r2 (w t)2((π /2.) +1). (4.8)

,

t

F = ∑F^ = 4ρ r2 w2((π /2.) +1) / t) ∫t2 dt.

0

F 1,3ρ r2 w2((π /2.) +1)t2. (4.9)

r = 0,02; w = 5/2; ρ = 1000/3; t = 1c, F ≈ 33.

(.4.3), , , . , , .4.4 r, .

 

. 4.4.

 

(3.5) Ψ = 180, F,

F = 2 ρπ r2 v2. (4.10)

, R, (. (3.5)) ρ, r v . R , , (.3.2) (4.10). , , , .

(.4), , ( ) , , , . . .

 

5. . .

 

1. .

[6, .427]. , R ω (. 5.1): . R ³ ℓ/2. 0 £ α £ π/2. , , α = 45 , , 5.1.

 

. 5.1.

 

α ω t

α = ωt/2, (5.1.1)

. , , .

F1 = mω2 (R - (ℓ/2) cos α) sin 2α (5.1.2)

F2 = mω2 (R + (ℓ/2) cos α) sin 2α (5.1.3)

F3 = - mω2 (R + (ℓ/2) sin α) sin 2α (5.1.4)

F4 = - mω2 (R - (ℓ/2) sin α) sin 2α (5.1.5)

, .

F2-1 = mω2cosα sin2α. (5.1.6)

F3-4 = - mω2sinα sin2α. (5.1.7)

π /2

F 2-1 = (1/(π/2))∫2cosα sin2αdα = 4mω2ℓ/3π 0,4mω2ℓ, (5.1.8)

0

π /2

F 3-4 = (1/(π/2))∫2sinα sin2αdα = -4mω2ℓ/3π -0,4mω2ℓ. (5.1.9)

0

, . ( ), : , .

F 2-1 F 3-4

F = | F 2-1 | + | F 3-4 | = 0,8 mω2ℓ. (5.1.10)

(.5.2) , .

 

. 5.2.

 

, (.5.3).

 

. 5.3.

 

, (.5.3)

F = 4F = 3,2mω2. (5.1.11)

m = 0,1; ω =2πf, f = 10/; ℓ = 0,5, F ≈ 632.

 

2. .

, R ω (. 5.4): .

 

. 5.4.

 

m1 m2, m3 m4 . . (.5.5)

v1 = v2 = (ωℓ/4) sin (Ψ/2), (5.2.1)

Ψ = ωt.

r1 r2

v1R = v2R = (ωℓ/4) sin (Ψ/2) cos b, (5.2.2)

cos b = R /r1 = R /r2 =R/Ö(R2 +(ℓ2 /4) cos2 (Ψ/2)), (5.2.3)

R , r1, r2 , r1 = r2.

 

. 5.5.

 

vR1 = ω r1, (5.2.4)

vR2 = ω r2. (5.2.5)

, , ,

ω1 = (vR1 - v1R)/r1 = ω[1 (ℓRsin (Ψ/2))/4(R2 +(ℓ2/4)cos2(Ψ/2))], (5.2.6)

ω2 = (vR2 + v2R)/r2 = ω[1+ (ℓRsin(Ψ/2))/4(R2 +(ℓ2/4)cos2(Ψ/2))]. (5.2.7)

F1 = mω12 r1

F2 = mω22 r2

F1 = 2 [(1 (ℓRsin(Ψ/2))/4(R2 +(ℓ2/4)cos2(Ψ/2))]2Ö(R2 +(ℓ2/4)cos2(Ψ/2)), (5.2.8)

F2 =2 [(1+ (ℓRsin(Ψ/2))/4(R2 +(ℓ2/4)cos2(Ψ/2))]2Ö(R2 +(ℓ2/4)cos2(Ψ/2)). (5.2.9)

, ℓ= 4R. , Ψ=180 ω1 = 0 , ω2 = 2ω (.5.6).

 

. 5.6.

 

ℓ= 4R

F1 = 2R[(1+ 4cos2(Ψ/2) sin(Ψ/2))/(1+4cos2(Ψ/2))]2Ö(1 + 4cos2(Ψ/2)), (5.2.10)

F2 =2R[(1+ 4cos2(Ψ/2)+ sin(Ψ/2))/(1+4cos2(Ψ/2))]2Ö(1 + 4cos2(Ψ/2)). (5.2.11)

, Ψ 0 180 Ψ = b = 60 F2 .

,

0 £ Ψ £ 60, ,

π /3

F 1-2 = (1/(π /3))∫ (F1 sin(b + Ψ) - F2 sin(b - Ψ))dΨ ≈ 0,6mω2 R, (5.2.12)

0

b = arccos (1/Ö(1 +4 cos2 (Ψ/2))) (5.2.3).

F 1-2 (5.2.12) , . 60 £ Ψ £ 180

π

F 1+2 = (1/(π-(π/3)))∫(F1 sin(Ψ + b)+ F2 sin(Ψ- b))dΨ ≈ 1,8mω2 R, (5.2.13)

π /3

0 £ Ψ £ 180, ,

F = (F 1-2 + 2F 1+2)/3 ≈ 1,4 mω2 R. (5.2.14)

m3 m4 , .

, ,

F = 4 F = 5,6mω2 R. (5.2.15)

m = 0,1; ω =2πf, f = 10/; ℓ= 4R , R = 0,1, F ≈ 220.

 

3. , .

, , , R ω (. 5.7): .

 

. 5.7.

 

m1 m2, m3 m4 . ℓ = 2R , . , 1,5ω 0,5ω , ω. 2R 0, RÖ2.

 

. 5.8.

 

0 £ Ψ £ 36 (. 5.8) , 36 £ Ψ £ 72 (. 5.8, . 5.9) , 72 £ Ψ £ 90 (. 5.9) .

 

. 5.9.

 

.

ω 1 = (ω + 0,5ω + ω)/2 = 1,25ω. (5.3.1)

ω 2 = (ω - 0,5ω + ω)/2 = 0,75ω. (5.3.2)

R 1 = (2R + RÖ2)/2 = R(2 + Ö2)/2. (5.3.3)

R 2 = (0 + RÖ2)/2 = (RÖ2)/2. (5.3.4)

, ,

F 1 = mω2 1 R 1 cos(Ψ /2)sin2Ψ 2,67mω2 R cos(Ψ /2)sin2Ψ. (5.3.5)

, ,

F 2 = mω2 2 R 2 sin(Ψ /2)sin2Ψ 0,4mω2 R sin(Ψ /2)sin2Ψ. (5.3.6)

0 £ Ψ £ 36

0,2 π

F 1 + 2 = (1/0,2π) ∫ (F 1 + F 2 )dΨ 1,47mω2 R. (5.3.7)

0

36 £ Ψ £ 72

0,4π

F 1 - 2 = (1/0,2π) ∫(F 1 - F 2 ) 1,95mω2 R. (5.3.8)

0,2π

72 £ Ψ £ 90

0,5π

F - (1 + 2) = - (1/0,1π) ∫(F 1 + F 2 )dΨ -3,72mω2 R. (5.3.9)

0,4π

0 £ Ψ £ 90

F = (2F 1 + 2 + 2F 1 2 + F - (1 + 2) )/5 0,622 R. (5.3.10)

.

, ,

F = 4F = 2,48mω2R. (5.3.11)

m = 0,1; ω =2πf, f = 10/; R = 0,25, F ≈ 245.

 

6. .

 

, , ,

, (.6.1) , .

 

. 6.1.

 

(), m, , . . . , , . w , . , . . (. 6.2), , , .

 

. 6.2.

 

, . 6.3.

 

. 6.3.

 

¢¢ , () . , . , , , .

.

= = r, = R.

, Ψ R 0 (.6.4)

r + r = R, (6.1)

Ψ =180 (.6.5)

Ð = 90. (6.2)

, , ,

r = 2R/(2+Ö2), (6.3)

= (3 - 2Ö2)R. (6.4)

. , w.

 

. 6.4.

 

w , ,

w = (180/225)w. (6.5)

w ∆t = 225/w = 5π/4w

w = w - w = - 0,2w. (6.6)

,

dω/dt = ∆w/∆t = - 0,16w2/ π. (6.7)

(2.8)

F = -m [(dω/dt)R] = 0,16mw2 R/ π. (6.8)

 

. 6.5.

 

F = 0,16mw2 RsinΨ/π. (6.9)

π

F = 0,16mω2 R/π2) ∫ sinΨdΨ = 0,32mω2 R/π2. (6.10)

0

(.6.3) . , . ,

F = 4F = 1,28mω2 R/π2. (6.11)

m = 0,1; ω =2πf, f = 10/; R = 0,5, F = 25,6.

 

7. . .

 

[19, .76-82] m (.7.1) R v. F, m m v2/R, .

F׀׀ = (m v2/R) sin α. (7.1)

w , , v = wt,

F׀׀ = (m w2t2/R) sin α, (7.2)

t .

 

. 7.1.

 

- , .

, , , , . , , [8, .258-259]. , ω. F (.7.2) , () ( α = 180). , F . , , , .

 

. 7.2.

 

[8, .257]

dα /dt = M / IZ ω, (7.3)

: ; IZ ; ω .

(, F)

= F, (7.4)

: F ; F , .

(7.4) (7.3)

dα /dt = F / IZ ω, (7.5)

(7.5) , IZ , ω , F, t, - (.7.3).

 

. 7.3.

 

,

v = R (dα /dt). (7.6)

(7.6)

w = R (d2α /dt2). (7.7)

(7.5) (7.7)

w = (R / IZ ω) (dF/dt). (7.8)

, F, .

, ω dα /dt << ω , F, [8, .260].

F , (.7.4). F . , F, F׀׀ .

 

. 7.4.

 

, , .

, - , , . , , , (.7.5).

 

. 7.5.

 

[1, .451] F׀׀ (.7.2) , 0 π F

π

F = (1/ π ) ∫ (m w2t2/ R) sin α dα = 2m w2t2 / Rπ. (7.9)

0

F :

F = 8m w2t2/ Rπ. (7.10)

, m = 1. w = 5/2, , . t = 1. () R = 0,5. (7.10) F = 8∙ 1∙ 52 ∙12 /0,5 π ≈ 127.

 

 

1.                  . . , 14- ., .: , , 2001, 864.

2.                  . . . .1. . 5- ., . .: ., 2010, 560.

3.                  .. . . 2- ., .:, 1996, 456.

4.                  .. : . 4- ., . .: , 2009, 576.

5.                  / .., .., ... 8- .,. . .: , , 2008, 1056.

6.                  .. , 2- ., . . . - . .: , 1971, 752.

7.                  .. . 1. . 2-, . . .: , 1997, 554.

8.                  .. .. . . . .-. . . -. ., , 1978, 416.

9.                  . . ( ): - - , 1973., 512.

10.              : / .., ... 15- ., . .: , 2010, 608.

11.              .., , , , 3(29), 2007, ISSN 1684-2626.

12.              .. : . 2- ., . .: . - , 1980, 368.

13.              .., .. : . 2 . .1. , . / . ... 5- ., . .: . 2003. 576.

14.              ., ., . : : . ./ . .. ... 3- ., . .: . - . 1983. ( , 1). 448.

15.              . ., , . . , 1977, 99.

16.              .. , , 3 (18), 2004, ISSN 1684-7288.

17.              .. , , 5 (71), 2012, ISSN 1991-3087.

18.              .. , , 4, 2012, ISSN 2077-3153.

19.              .. , , 11 (65), 2011, ISSN 1991-3087.

20.              .., .. . .: , 2007. 332.

 

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