MexLekcii2010prn

16

dR=v'dt, dϕ.

, , ,

, , dϕ:				′	′			,
				dv	= v dϕ,
		′′			′
dR: v = ω(R + dR) , dv = ωdR = ωv dt .
v			′		′			,
	dv = v dϕ + ωv dt .

v0 dϕ dt,

, , : v|| = v0 dϕ = ωRdϕ . :

a|| = dv|| = ωR dϕ = ω2R = an ≡ a – ,

dt dt

, ,

v′ ;

a = dv = dϕ v′ + ωv′ = 2ωv′ – ,

dt dt

, , , ,

v′, “ ” :

′	× ) .	(5)
F = −ma = 2m(v

( “

”: v' ω

).

	) (R v' ω).
			',
	ω,
	R v'.
	,
	(±v')	( . 5,	ω
	).
	. 5					v	′2
	'			′
					=			.
				a

R

) :

v = ωR ± v',

v

2

′

2

′2

a =

(ωR ± v )

2

′

v

2

′

′

(6)

=

= ω

+

= ω

R

R

R ± 2ωv

R

R ± 2ωv

+ a .

(2),

– ' = 0 = ω2R ± 2ωv'.

( 3) :

131

		16
2	¢	(7)
F = -ma0 = -(mw	R ± 2mwv ) .

– , –

, :

	¢	¢	× ].	(5 )
	F = -2m[ ×v ] = 2m[v
: ) ,		v'	wR = v ;
	) , v' v .
	§4. ,
	,			,

. ,

, ,

. 5, – . 2.

1. . v .

F = 2m[v¢× ] = 2mvw = 4pmv . j

T

900, 900–j ( . . 2).

F =| 2m[v × ] |= 2mvwsin(90 - j) ( ).

, ,

. F =0.

2..

: ,

.

, .

( )

, (

) – . ( ),

–

(“ ” “ ” ).

3.. . 5. (

– , F )

( ),

, ,

). ( ),

.

(6) (7), F , j

,

F .= w2r = w2R ×cosj.

132

16

.

(

, F ).

( 10 ) “ ” .

,

. ,

,

,

,

,

. ,

, .

. 6.

,

,

( ).

j

×sinj, , ,

	. 6		F = 2mvw×sinj.

, ( ).

. .

!!! !

133

17

17.

. (

). .

. . . .

. .

§1.

, ,

.

), .

,

, , ,

( k

), :

	F = –kx.	(1)
	,	,
.

, , ( . 1). – ,

.

( 2- ma = F = – k ):

	ma + kx=0	m	d2x		+ kx = 0	,	(2)
			dt	2

– , k – , –

, –

. 1 .

(2) ,

:

x = A×sin(wt+j0) x = A×cos(wt+j0),

(2 )

– t,

– ( ), j=wt+j0 – ,

j0 – , w – (w=2p/ , w=2pn,–

, 2p ; n=n/t=1/T – ( -1

– “ ”), , – ,

).

: ,

.

( ) ,

(

).

134

17

, .

w0 , (2 )

(2), :

dx

= Aw cos(w t + j ) ;

d2x

= -Aw2 sin(w t + j )

(2)

dt

0

0

0

dt2

0

0

0

- mw2 + k = 0 ,

- mAw2 sin(w t + j ) +kA×sin(w0t+j0) = 0

Þ

:

0

0

0

0

1

w

=

k

n

0

=

k

;

T = 2p

m

.

(3)

0

m

2p

m

k

(2)

,

w0,

:

d2x

k

d2x

2

+

x = 0

+ w x = 0

.

(4)

dt2

m

dt2

0

,

,

l . .

, (4).

F ( . 2),

:

F . . 2 ,

F = -mg sin j = -mg	x	= -	mg	x .	(5)
	l
			l

, (5) (1) –

(5) ( ) (1)

k=mg/l. 2- :

	ma = -	mg	x		d2x	+	g	x = 0 .	(6)
		l			dt2		l
	. 2

, (2).

, (2) ,

(6) – . , F

S

	, ,	a =	d2S		,	a =	d2x		,
			dt	2			dt	2

. j,

S .

( .

(4), )

w	=	g	,	T = 2p	l	.	(7)
0		l			g

135

17

§2.

, , ,

, ,

( , )

.

.

,

– , :

= cos(wt+j1), y = Bcos(wt+j2).

(8)

, ,

, ( ,Y). ?

y=f(x) ? , (8)

t. .

x

=cos(wt+j1) = coswt×cosj1 - sinwt×sinj1

|×cosj2

( ×sinj2 )

A

y

= cos(wt+j2) = coswt×cosj2 - sinwt×sinj2

|×cosj1

( ×sinj1 )

B

x

×cosj2 = coswt×cosj1×cosj2 - sinwt×sinj1×cosj2

(a)

A

y

×cosj1 = coswt×cosj2×cosj1 - sinwt×sinj2×cosj1

( )

B

( ) ( ),

x

×cosj2 -

y

×cosj1 = sinwt×(sinj2×cosj1 - sinj1×cosj2) = sinwt×sin(j2-j1). (9)

A

B

sinj2 sinj1,

( ¢) ( ¢) – :

x		×sinj2 = coswt×cosj1×sinj2 - sinwt×sinj1×sinj2	(a¢)
A
y	×sinj1 = coswt×cosj2×sinj1 - sinwt×sinj2×sinj1		( ¢)
B

x

×sinj2 -

y

×sinj1 = coswt×(cosj1×sinj2 - cosj2×sinj1) = coswt×sin(j2-j1). (10)

A

B

(9) (10) :

x2

cos2 j

2

+

y2

cos2 j - 2

xy

cos j cos j

2

= sin2 wt ×sin 2 (j

2

- j ) ,

A2

B2

1

AB

1

1

x2

sin 2 j

2

+

y2

sin 2 j - 2

xy

sin j sin j

2

= cos2 wt ×sin 2 (j

2

- j )

A2

B2

1

AB

1

1

136

														17
, , cos2j+sin2j=1,
		x2	+	y2	- 2	xy	cos(j	2	- j ) = sin 2	(j	2	- j )	.	(11)
		A2		B2		AB			1			1

.

.

1). j2–j1 = 0 – ( ) , sin0=0, cos0=1 :

x2

y2

xy

æ x

y ö

2

y =

B

x

+

- 2

= 0

ç

-

÷

= 0

.

A2

B2

AB

è

A

B ø

A

tga= / ( . 3).

. 3

w.

,

S =

x2 + y2 . j2–j1=0, j2=j1=j,

S =

A2 cos2 (wt + j) + B2 cos2 (wt + j)

=

A2 + B2

cos(wt + j) = C cos(wt + j) ,

,

S

w

C =

A2 + B2

.

2). j2

– j

1

=

– 180 , cos180 = –1,

sin180 = 0. (11) :

x2

y2

xy

æ

x

y ö2

+

+ 2

= 0

+

= 0

y = -

B

x

.

ç

÷

A2

B2

AB

è

A

B ø

A

, a /2 (a>90 ).

3). j - j =

p

(=90 ) – , j - j = 3

p

(=270 ) –

2

1

2

2

1

2

p

p

;

cos

= 0 , sin

=1.

(11)

2

2

x2

y2

x 2

B

+

=1

y = ±B 1-

= ±

A

2

- x

2

.

(12)

A2

B2

A2

A

Y –

, .

( = ), .

(8) :

	ìx = Acos wt
		, Þ 2 + 2 = 2 y = ± A2 - x2 .
	í
	îy = Acos(wt + p/ 2)	= -Asin wt

:

w

,

/2 3 /2 ( ).

137

17

( ,

)

, .

, j1=0, j2=- /2, w =2w ( (8)

: x=Acoswt, y=Bsin(2wt) ),

	. 4				Bx
				y = ±2			A2 - x2
					A2
	, . 4 (t1=0,		t2 /4w, t3= /2w).

§3. . .

j0

,

(

). ,

w1=w2=w. ,

) x1=A1cos(wt+j1) i x2=A2cos(wt+j2) = 1+ 2

( . . 5, j2-j1=a)

		A = A2		+ A2	+ 2A A cos a ,
		1		2	1	2
	tgj =		A1 sin j1	+ A2 sin j2		=	A2 sin j2	.
. 5	0		A1 cosj1	+ A2 cos j2			A1 + A2 cos j2

w1 w2 , Dw<<(w1, w2).

1 = cosw1t, x2 = Acosw2t; w2 = w1+Dw.

, :

x = x1+x2	= A(cosw1t + cosw2t) = 2A cos	w1 - w2	t ×cos	w1 + w2	t .
	2		2

.

(w1+w2»2w):

x = A	coswt ,		A	= A (t) = 2A cos	Dw	t = 2A cos	w	t . (13)
p			p	p	2	2

(13) , (

»w) ,

w =Dw/2.

(t) 0 |±2 |.

138

17

(t), w =Dw. ( .6).

	2A	x
		Ap(t)
	0	t
	-2A

. 6

§4.

,

( , ). ,

( ) (“ ”).

, , ,

, .

												= sin(wt+j)					,
:
	1		2		1	æ dx ö			2	1		2	2		2		2
E =		mv		=		mç		÷	=		mw A			cos		(wt + j) = E )cos		(wt+j). (14)
	2				2					2
						è dt ø

( , )

x

kx2

1

2

2

E =

1 mg

A

2

2

(wt + j) .

E = -ò(-kx)dx =

=

kA

sin

(wt + j) .

sin

(15)

2

2

2

l

0

k=mw2

( . (3) ; (7)),

E

= E + =

1

mw2 A2

(16)

2

:

=const

.

~

( ) (E )= E )= E ).

. 7

(t),

E (t)

i E (t).

, . , ,

v

.

0. ,

, .

, .

139

17

.

.

	x			A
		1	T	T	3T
	,					T
		0	4	2	4		.
			π/2	π	3π/2	ω0t
		0
		-1

0			π/2		3π/2	ω t, 2π t
	0			π
						0	T
		Ek
mv	2
2	0
	0		π/2	π	3π/2	ω0t
				. 7

, . 8,

(U, ) ,

.

E	E,E ,	E
E		E
2
0		t
	2

.8

140