16
dR=v'dt, dϕ.
, , ,
, , dϕ: |
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dv |
= v dϕ, |
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dR: v = ω(R + dR) , dv = ωdR = ωv dt . |
v |
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dv = v dϕ + ωv dt . |
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v0 dϕ dt,
, , : v|| = v0 dϕ = ωRdϕ . :
a|| = dv|| = ωR dϕ = ω2R = an ≡ a – ,
dt dt
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v′ ;
a = dv = dϕ v′ + ωv′ = 2ωv′ – ,
dt dt
, , , ,
v′, “ ” :
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× ) . |
(5) |
F = −ma = 2m(v |
( “
”: v' ω
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', |
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ω, |
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R v'. |
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(±v') |
( . 5, |
ω |
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v |
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a |
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R
) :
v = ωR ± v',
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a = |
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(ωR ± v ) |
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(6) |
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R |
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R ± 2ωv |
R |
R ± 2ωv |
+ a . |
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(2), |
– ' = 0 = ω2R ± 2ωv'. |
( 3) : |
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16 |
2 |
¢ |
(7) |
F = -ma0 = -(mw |
R ± 2mwv ) . |
– , –
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× ]. |
(5 ) |
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F = -2m[ ×v ] = 2m[v |
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v' |
wR = v ; |
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§4. , |
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. ,
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. 5, – . 2.
1. . v .
F = 2m[v¢× ] = 2mvw = 4pmv . j
T
900, 900–j ( . . 2).
F =| 2m[v × ] |= 2mvwsin(90 - j) ( ).
, ,
. F =0.
2..
: ,
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, .
( )
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) – . ( ),
–
(“ ” “ ” ).
3.. . 5. (
– , F )
( ),
, ,
). ( ),
.
(6) (7), F , j
,
F .= w2r = w2R ×cosj.
17
17.
. (
). .
. . . .
. .
§1.
, ,
.
), .
,
, , ,
( k
), :
, , ( . 1). – ,
.
( 2- ma = F = – k ):
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ma + kx=0 |
m |
d2x |
+ kx = 0 |
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(2) |
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2 |
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– , k – , –
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. 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
– “ ”), , – ,
).
: ,
.
( ) ,
(
).
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17 |
, , cos2j+sin2j=1, |
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cos(j |
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- j ) = sin 2 |
(j |
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- j ) |
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(11) |
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.
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1). j2–j1 = 0 – ( ) , sin0=0, cos0=1 :
x2 |
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y ö |
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è |
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tga= / ( . 3).
. 3
w.
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x2 + y2 . j2–j1=0, j2=j1=j, |
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S = |
A2 cos2 (wt + j) + B2 cos2 (wt + j) |
= |
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A2 + B2 |
cos(wt + j) = C cos(wt + j) , |
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C = |
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2). j2 |
– j |
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– 180 , cos180 = –1, |
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sin180 = 0. (11) : |
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y ö2 |
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A |
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, a /2 (a>90 ). |
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3). j - j = |
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(=90 ) – , j - j = 3 |
p |
(=270 ) – |
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; |
cos |
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(11) |
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+ |
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y = ±B 1- |
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- x |
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(12) |
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Y –
, .
( = ), .
(8) :
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ìx = Acos wt |
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, Þ 2 + 2 = 2 y = ± A2 - x2 . |
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í |
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îy = Acos(wt + p/ 2) |
= -Asin wt |
:
w
,
/2 3 /2 ( ).
17
( ,
)
, .
, j1=0, j2=- /2, w =2w ( (8)
: x=Acoswt, y=Bsin(2wt) ),
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. 4 |
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Bx |
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y = ±2 |
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A2 - x2 |
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, . 4 (t1=0, |
t2 /4w, t3= /2w). |
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§3. . .
j0
,
(
). ,
w1=w2=w. ,
) x1=A1cos(wt+j1) i x2=A2cos(wt+j2) = 1+ 2
( . . 5, j2-j1=a)
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A = A2 |
+ A2 |
+ 2A A cos a , |
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1 |
2 |
1 |
2 |
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tgj = |
A1 sin j1 |
+ A2 sin j2 |
= |
A2 sin j2 |
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. 5 |
0 |
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A1 cosj1 |
+ A2 cos j2 |
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A1 + A2 cos j2 |
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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 . |
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2 |
2 |
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(w1+w2»2w):
x = A |
coswt , |
A |
= A (t) = 2A cos |
Dw |
t = 2A cos |
w |
t . (13) |
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p |
2 |
2 |
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(13) , (
»w) ,
w =Dw/2.
(t) 0 |±2 |.