(), (.17). .
.17
m, :
: t=0,
: .
:
:
1-
:
- ;
.18
- . /.
- ( ).
- (.18).
- (1 /c=1 )
- .
, .
. 19.
.19
2-
:
(.39).
.20
3- ( )
:
(.20).
.
: , - ( ), . - .
F0 - .
- .
.21
m, (.21), :
: .
- .
-
.
(.22).
.22
( ) , ( ).
1. =∙sinωt T=12 . , t1 , x=A/2. t2 .
: t1-? t2-?
. t1 c /2: /2=∙sinωt1, sinωt1=1/2, .. ωt1=π/6, (2π/)t1=π/6.
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t1=T/12=1 c.
t=T/4. , t2=T/4- T/12= 2 c.
2. , , , ?
. x=Acos(ω0t+α). t = 0 = 0, φ π /2, .. :
x=A/2, , ( , .. ).
3. x=5cosω0t (), ω0= 2 1. , 8 /.
.
α=0
, . , α=0,
4. , , 10 /, 100 /2. , .
.
x=Acos(ω0t+α),
v=-Acos(ω0t+α)=-vmaxsin(ω0t+α),
a=-A cos(ω0t+α)=- amaxcos(ω0t+α),
, xmax=A; vmax=Aω; amax=Aω2.
ω0=10 1.
5. = 0,02 , W=3∙107 . F= 2,25 ∙ 105 ?
.
, F=-kx,
6. , . 1 ?
. , , , :
, , x= l /2,
7. x1=A1∙sinω0t x2=A2∙cosω0t, 1 = 1 ; 2 = 2 ; ω0 = 1 1. , v α. .
. , , x=A∙cos(ω0t+α)
:
=1+4+2∙2∙cos0,5π=5 2.
:
α=arctg(-0,5)=-26,6=-0,46 .
x=2,24∙10-2cos(t-0,46) .
8. (.23), x1=A1cosω(t+τ1) x2=A2cosω(t+τ2), 1=1 ; 2=2 ; τ1=1/6 ; τ2=1/2 ; ω=π /. φ1 φ2 ; φ .
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.23
. :
=∙cos(ωt+φ) (1)
, , :
x1=A1cos(ωt+ωτ1) x2=A2cos(ωt+ωτ2) (2)
(2) (1) : φ1=ωτ1=π/6 φ2= ωτ2=π/2 .
, .23.
, :
φ2-φ1=π/3 .
1, 2 φ2-φ1 (3), : =2,65 .
41.1:
φ=arctg(5/ )=70,9=0,394π .
, ω.
=∙cos(ωt+φ),
=2,65 , ω=π /, φ=0,394π .
9. m=10-2 =10 =0,2 =4 . t=0: =. t= 1 .
: -? -?
:
=cos(ωt+φ0), ω=2π/.
.. t=0 =, s(ω∙0+φ0)=A, sφ0=1, φ0=0.
, =0,2cos[(2π/4)t]= 0,2cos[(π/2)t] ().
: =mv2/2, v=dx/dt=-Aω∙sinωt.
=[mA2ω2∙sin2ωt]/2=5∙10-3 .
:
=kx2/2=[k2cos2ωt]/2=[k2cos2(π/2)]/2,
=0.
10. l= 1 mc=3m1 d= l/2 m=m1. Z (. 24). T -?.
.24
.
J - , m - , lc - . J1 J2:
J=J1+J2. (2)
, , J1=mc l 2/12, .. J1=m1 l 2/4.
, J=Jo+m a 2. ,
J2=m1(l /4)2+ m1(3 l /4)2 = (5/8)m1 l 2.
J1 J2 (2), :
J= m1 l 2/4+(5/8)m1 l 2=(7/8)m1 l 2.
l c
l c=(Σmixi)/Σmi=(3m1∙0 + m1(3 l /4))/(3m1+m1)=(3/16) l.
(1) J, Jc (m=3m1+m1=4m1), :
=2,17 .
11. (.25), x=2cosω0t () y=sinω0t (). , , ω0=π/3 (1).
.25
. y=cos(ω0t+α) :
, α= -π/2 , : . , , .
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, , , . . , , .
t=0: x1=2cos0=2; y1=1cos(-π/2)=0;
t=1 c: x2=2cos(π/3)=1; y2=1cos(π/3-π/2)=1cos(-π/6)=0,86.
, 1 (2; 0), 2 (1; 0,86). , .
12. 1 10 2 . , , . , 5 .
. :
( A0=5 t = 10 = 600 ):
, βt=ln2,
N, t, , t=NT, , , βNT=ln2,
:
δ=βT=2∙10-3.
, , . : x=5∙10-2e-0,001tcosπt ().
13. , ( k = 10 /, m = 100 ) r = 0,02 /. β , F0 = 10 .
. :
:
:
14. D mD = 10 , = 30 , A , = 36.1 / (. 26). D m = 15 . B , O1 B ( ). v, , = 100 () / . D E.
.26
. Ox , (. 27). O Ox D E, , , B (). , D E.
.27
, x ( M). , (). , .. . , x M : . . , x
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,
. , ,
,
, (1)
(1)
, (1)
(2)
x 2 (1)
. (3)
(2)
. (4)
(3)
. (5)
(3),
, . , A1 A2
1 = 0.7472 , 2 = 0.0034 .
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C1 C2 : 1 = 1.2928 , 2 = 0.2181 .
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- p / k?
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- , , , , ?
- n?
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- p n?