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, , , . , t ! v , t t+dt dm , ! ! dv . F!" ! t f dt dm, : ! ! ! ! ! ! (1) (F!" + f)dt =M(v + dv) - M v =Mdv

! v' dm! t, ! dt v + dv ! dt - f

! , f, : ! ! ! ! ! ! ! ! - fdt = dm(v + dv) - dmv' ! - fdt = dmv +dmdv - dmv'

! dmdv , :! ! ! ! ! ! - fdt = dmv - dmv' ! fdt = - dmv + dmv' (1) (2) : (2)

! ! ! ! ! ! ! ! F!" dt - dm(v - v') =Mdv ! F!" dt =Mdv + dm(v - v') !! ! ! dv dv ! ! ! dm ! ! dm ! M F!" =M + (v - v') = F!" - (v - v') dt dt dt dt

(3)

! ! ! v'- v v!" dm dm/dt dM/dt , (3) : ! dv ! ! dM M = F!" + v#$ dt dt

(4)

H (4) , d/dt . , . (4) , ! v!" (dM/dt) ! v!" . ! v!" (dM/dt). : H (3) . . :! ! dv ! dM ! d(Mv) ! ! dM ! dM ! ! M +v = F!" + v' = F!" + v' dt dt dt dt dt ! dP ! ! dM (5) = F!" + v' dt dt ! dP/dt , ! . v' ! ! ( v'= 0 ),

, :! ! d(Mv) ! dP ! = F!" = F!" dt dt

m L . ! F . x ! F. . : , . ! t v :

.d dm (L - x)v = T(x) v ! dt dt

[

]

L

dv dx dv dx - v - x = T(x) - v ! dt dt dt dt

L-x

(

) dv - v dt

2

= T(x) - v2 ! L - x a = T(x)

(

)

(1)

x

! ! F, m/L a t. t :

d dm xv = F - T(x) v ! dt dt

(

)

x x

dv dx dx + v = F - T(x) - v ! dt dt dt dv = F - T(x) ! xa = F - T(x) dt

(2)

(1) (2) : L-x a xa

(

)

=

T(x) ! F - T(x)

L-x T(x) ! = x F - T(x)

LF - xF - LT(x) + xT(x) = xT(x) ! (L - x)F = LT(x) ! T(x) = F(1 - x/L)

(3)

! (3) F, L-x .

: (1) (2) t . . , ! ! . , L , m, , , a. N x . ! m* g . YH: x , L-x m, . T ! ! . T mg , m! g

! fx x, x . E N, : (1) fx - mg - m! g = (m + m! )a ! fx = (m + m! )(g + a) , L-x, m=(L-x)m*, m* , (1) :

.

. (2)

fx = [m + (L - x)m* ](g + a)

H (2) fx x, () f0 =(m+Lm*)(g+a) f* =m(g+a). P.M. fysikos

M, , , L , . ! v0 . : i) x

ii) x . , . : i) - t o x. T x/2 x. E , , , :

d (M + x)v = 0 ! (M + x)v = ct ! dt

[

]

(M + x)v = Mv0 ! v =

Mv0 v0 = M + x 1 + x / M

(1)

! (1) v x.ii) (1) :

dx Mv0 = ! (M + x)dx = Mv0dt dt M + x (2) :

(2)

Mx + x2/2 = Mv0t + C

(3)

C t=0 x=0, (3) C=0 :

Mx + x2/2 = Mv0t ! x2 + 2Mx - 2Mv0t = 0

(4)

H (4) x :

M2 + 2Mv0t M M M ! x=x=+ + 1 + 2v0t / M ! m m m mx= M m

(

1 + 2v0t / M - 1

)P.M. fysikos

L m, . . i) N ! g , ! v :

v2 = 2gx x . ii) x . iii) . : i) E t x, o x/2. E , :d dm (L - x / 2)v = (L - x / 2)g v ! dt dt

[

]

L

! dv x dv v dx x$ dx = g # L - & - v ! dt 2 dt 2 dt 2% 2dt "dv =g dt

! ! x$ dv v dx x$ v dx = g# L - & ! #L- & 2% dt 2 dt 2% 2 dt " "

(1)

(1) ! g . :(1) dv dv dx dv dv = = v ! g= v ! gdx = vdv dt dx dt dx dx

(2)

(2) :

gx = v2 / 2 + C

(3)

t=0 x=0 v=0, C (3) :

v2 = 2gx

(4)

ii) , t :0= 0= gx dm - T(x) + v ! 2 dt gx dx - T(x) + v ! 2 2dtgx v2 - T(x) + 2 2

0=

!(5)

(2)

T(x) =

gx 3gx + gx = 2 2

(x) ! . vC t, :

! LvC = # L "

x$ x 0 ! &v + 2% 2

! x$ vC = # 1 &v ! 2L% "dvC d '! x$ * = )# 1 & v, ! dt dt ( " 2L% +

dvC dv x dv v dx ! = dt dt 2L dt 2L dt!

! ! dvC dv ! x $ v2 (1),(2) dvC x $ 2gx 3x $ = 1= g# 1 = g# 1 ! # & & & dt dt " 2L% 2L dt 2L% 2L 2L% " "

! 3x $ aC = g# 1 & 2L% "

! aC .iii) H :2 E!"# = gL(L/ 2) = gL / 2

H :

E!"# = -gL(L/ 2) = -gL2 / 2 -, :!E = E"#$ - E%&' = gL2 = mgL

P.M. fysikos

2m m L. H , , . . N : i) ii) ! . g . : i) E t x. :d (2m + x)v = (2m + x)g ! dt

[

]

2m

dv dv dx + x + v = (2m + x)g ! dt dt dt

( 2L + x) dv + v dt

2

= (2L + x)g

(1)

! v t, dv/dt o ! v m/L. , t :

d (L - x)v = - (L - x)g ! dt

[

]

L

dv dv dx - x - v = - (L - x)g ! dt dt dt2

( L - x) dv - v dt3L

= -(L - x)g

(2)

(1) (2) :dv = (L + 2x)g dt

(3)

:dv dv dx dv = = v dt dx dt dx

(3) :3L dv g(L + 2x)dx ! v = (L + 2x)g ! vdv = dx 3L 2g(L + 2x)dx 3L

d(v2 ) =

(4)

(4) :

v2 =

2g Lx + x2 + C 3L

(

)

(5)

t=0 v=0, C (5) :v2 = 2g Lx + x2 3L

(

)

(6)

x=L, ! v* :

v* =

2g gL LL + L2 = 2 3L 3

(

)

ii) H (6) :

dx = dt

2g 3Ldx

(

Lx + x2

)

!

dx Lx + x2 3L 2g ! 0L

=

2g dt ! 3L dx

!

L

0

Lx + x

2

=

2g t ! t* = 3L *

Lx + x2

(7)

t* . (7) :

!

L

dx Lx + x2dx Lx + x2 dx Lx + x2

=

0L

!

L

dx (x + L/2)2 - (L/2)2

0

=!

L

d(x + L/2) (x + L/2)2 - (L/2)2

!

0

! !

= ln x + L/2 + (x + L/2)2 - (L/2)2

0

(

)

L 0

!

L

= ln x + L/2 + x2 + Lx

0

(

)

L 0

= ln 3 + 2 2

(

)

(8)

(7) (8) :t* = 3L ln 3 + 2 2 2g

(

)P.M. fysikos