MexLekcii2010prn

. 3

m

d2 r1

= F

,

F12 = f(r2-r1);

(13)

dt 2

1

12

d

2r

m

2

2

= F

,

F21 = f(r2-r1),

(14)

dt2

21

(f) r=r2-r1

r (

). (13) (14) r1

i r2

r i r – :

r

=

m1r1 + m2r2

,

r = r

- r

.

(15)

c

m1 + m2

2

1

r1 i r2 r i r .

r2 = r1+r, rc:

r =

m1r1 + m2 (r1 + r)

=

(m1 + m2 )r1

+

m2r

= r +

m2

r Þ

c

m1

+ m2

m1 + m2

m1 + m2

1

m1 + m2

:

Þ r1 = rc -

m2

r .

(16)

m1 + m2

10

: r1 = r2 – r

(15) rc r2:

r =

m1(r2 - r) + m2r2

=

(m1 + m2 )r2

-

m1

r Þ

r

= r

+

m1

r . (17)

c

m1 + m2

m1 + m2

m1 + m2

2

c

m1

+ m2

m

d2r

m m d2r

= F

(r) ,

m

d2r

m m d2r

= F

(r) .

c

-

1 2

c

+

1

2

(18)

m + m

dt 2

1

dt2 m + m dt2

12

2 dt2

21

1

2

1

2

,

(m + m )

d2r

d2r

dv

c

c

= F

+ F

= 0

Þ

c

=

= 0 ,

(19)

dt2

1

2

dt2

12

21

dt

m m

d2r

= F21(r) - F12 (r) = F21(r) + F21(r) = 2F21 (r) .

2

1

2

(20)

m + m

dt 2

1

2

(13 14)

(13) (14), 1 2

d2

(m1r1 + m2r2 ) = F12 (r) + F21(r) = 0

dt 2

d2r

d2

æ m r

+ m r

ö

0

ç

1 1

2 2

÷

=

Þ

c

= 0

(19 )

dt 2

m

+ m

m

+ m

ç

÷

2

dt 2

è

1

2

ø

1

d2

F (r)

F (r)

F (r)

F (r)

m + m

(r - r ) =

21

-

12

=

21

+

21

=

1

2

F

(r)

Þ

dt 2

m

m

m

m

m m

2

1

21

2

1

2

1

1

2

m m

d2r

= F(r)

d2r

= F(r)

Þ

(

1 2

)

Þ

m

(20 )

m

+ m

dt2

dt2

1

2

(20),

m

r1¢ = -

m2

r r2¢ =

m1

r ;

r1¢

=

m2

.

(21)

r2¢

m1 + m2

m1 + m2

m1

:

1)

1 = 2 = ,

m = /2

– 2

; r1¢ = r2¢ – ;

2)

m1 >> m2

m =

m1m2

=

m2

» m

–

m + m

1 +

m2

2

1

2

m1

F = G

mM

= G

mM

1

» mg0 (1 - 2

h

+ 4

h2

- ...) .

(22)

(R + h)2

R2

(1 +

h

)

2

R

R2

3

3

3

3

R3

E =

mv2

- G

mM

= const ,

(23)

)

=R3 ,

, 1¢, h1=

x1¢–R3.

=U

1,

=0

).

>U

1¢

,

. 4

)

h

v2

v2

(F=ma, =a )

mg = m

= m

.

(24)

R + h

r

3

r =R +h

GM

GM

g0

v = (R3 + h)g

= (R3 + h)

=

= R3

.

(R

3

+ h)2

(R

3

+ h)

R

+ h

3

= 6,4 ×106

v

=

R

3

× g

0

×9,8 » 7,9 ( ).

(25)

1

10

mv2

mM

M

v

= 2G

= 2g

R

2 × v

= G

Þ

2

3

=

» 11 . (26)

2

R3

R3

0

1

v3 = 30g0 R3 = 30 × v1 = 15 × v2

» 43 .

(27)

E =

mv

2

- G

mM3

- G

mMC

mv

2

- G

mM3

æ

MC R3

ö

ç

÷

=

ç1

-

÷

= 0 .

2

R

R

2

R

M R

3

3−C

3

è

3 3−C ø

v2

æ

2 ×1030

×6,4 ×106 ö

v2

2

- g0R3

ç

+

÷

=

- g0R3 (1 +1,42 ×10) = v

- 2g0R3 ×15,2 = 0 .

24

11

2

ç1

6 ×10

÷

2

è

×1,5 ×10

ø

§6.

: a1>a0>a2 ( .

. 5). ,

( „ ”

!), ,

. 5

M

a

0

= F = G

MM M3

Þ a =

GMM

,

R0 – .

R2

3

0

R2

0

0

æ

1

1

ö

R0>>R3,

ç

÷

(R

- R )2 - R2

»

1- 0 » 0- 2 » GMM ç

÷

è

0

3

0

ø

GM

M

(R2

- R2

+ 2R R - R2

» GM

2R

(R0R3>>R3 ).

0

0

0 3 3

3

R2

× R2

M R3

0

0

0

DaM = 6,7 ×10

−11

× 7,3 ×10

22

2 ×6,4 ×106

»

626

×1017

= 11×10

−7

2

(

).

(3,844 ×108 )3

56,8 ×1024

DaC = 6,7 ×10

−11

×2

×10

30 2 ×6,4 ×106

»

5×10

− 7

2

(

).

(1,5×1011)3

r(t) =

p

,

p =

N 2

- ,

1 + e ×cos(j - j )

Gm2M

0

C

2

2 &

N = rmv = mr(wr) = mr

w = mr j;

m4G2 M2

+ 2mEN2

e = -

C

- ( ):

m2GM

e<1 –

C

e=0 –

e=1 –

e>1 –

a

U(r) =

,

a = GmM .

mv2

P2

N 2

GmM

r

C

E =

+ U (r) =

+

(

-

) = const - ;

2

2mr

2

r

N 2

2m

- “ ”

N 2

=

m2r 4w2

=

mr 2w2

(

2mr 2

2mr 2

2mr 2

2

).

Iw2

mr 2w2

E

=

=

.

2

2

a =

p

, b =

p

;

1 - e2

1 - e2

2)

N>0,

E = Emin, e = 0

– : rmin= rmax= p = a = b;

3)

N>0,

E > 0, e = 1 –

: rmin=p/2;

' = 'v' ,

′

=

m0

( 0 – ).

(1)

m

1 −

′

/ c)

2

(vy

;

: '

', ' d '=0:

z = z'

dz = dz'

y = y'

dy = dy'

x = (x' + Vt')

dx = (dx' + Vdt') = Vdt'

t = (t' + Vx'/c2)

dt = (dt' + Vdx'/c2) = dt'

:

′

dy

dy′

′

V

2

′

V

2

vy

vy =

=

=

= vy

1 −

,

py = mvy = mvy 1 −

.

(2)

dt

γdt′

γ

c2

c2

( )

v2 = V2 + vy2.

m =

m0

=

m0

=

( vy (2))

1 - (v2 / c2 )

1 - (V 2 / c2 + vy2 / c2 )

=

m0

=

( “ ”)

(1 -V

2

/ c

2

¢2

(1 -V

2

/ c

2

) / c

2

) - vy

m0

¢

=

=

m

( (1)