Tổng hợp dao động điều hòa Chuyên đề ôn thi ĐH CĐ Vật Lý Thầy Đoàn Văn Lượng

1


A.

   
1
= A
2






)cos(
)cos(
22
11


tax
tax


1
+ x
2
= acos(t + 
1

) + acos(t + 
2
)

2
cos.
2
cos2
baba 











2
cos.
2
cos2
1212



tax



2
cos2
12


 aA


2
12













2
cos0
12




2
12





0
2
cos
12










 




<
2
12








2
12



6
7
2
12








6
5


)

B.







)cos(
)cos(
222
111


tAx
tAx
.

1
+ x
2
.




AtAx

 )cos(


cóA






  
1
= A
1
cos(t + 
1
   







)0(),(
11
11
1
toxA
AA
A





;
x
2
= A
2
cos(t + 
2








)0(),(
12
22
2
toxA
AA
A





quay

1

2


1
A

2
A

A


2

21
AAA



t + ).
B

 cos 2
21
2
2
2
1

AAAAA
,

2211
2211
coscos
sinsin
tan



AA
AA



.


)cos()cos(
222111

 tAxvàtAx


21




21
0


 

21
0




0; 2K





)12(,  K


,(2 1)
22
K


   
:Hai dao 
 = k2 

1
+ A
2

 = (2k+1) 
1
 A
2


 
2
12


 k
uông pha 
2
2
2
1
AAA 
.

2121
AAAAA 
.

C. :

:

)cos()cos(
222111

 tAxvàtAx


21
xxx 


1111
)cos( AtAx



;
2222
)cos( AtAx





2121
AAAxxx



Cho vecto A
1
và A
2
 
 
hòa
)cos(

 tAx




2211
coscos

AAA
x

và
2211
sinsin

AAA
y



22

yx
AAA 
hay

 cos2
21
2
2
2
1
AAAAA

 :
2211
2211
coscos
sinsin
tan



AA
AA
A
A
x
y







A


1
A


2
A


y

x

O


3

 

2,0 K

21max
AAA 

  :

)12(,  K

21min
AAA 


  :
2
)12(,
2


 K

2
2
2
1
AAA 


  :
2
cos2
1



 AA
và
2
21





  :
3
3
1
AA 


;

2
2
1
AA




1
3
2
AA





maxmin
AAA 


2:
:
 x = Acos(t + ) 
A

: z =
a + bi z =A(sin +i cos) A=
22
ab
) hay Z = Ae
j(t + ).


 z = Ae
J
,
Trong các máy tính CASIO fx- 570ES, ESPlus : r    A  ).

+ -180
0
<  < 180
0

hay -< <  
trên. 
 
b. Gi pháp th hin phép công và tr s phc:

1 1 2 2
A A A
  
    


2 2 1 1
A A A
  
    
;
1 1 2 2
A A A
  
    


D
1/.  :
- 
- 





 :
:
21
AAA



+
1 1 2 2
A A A
  
    


4
MODE 2 CMPLX
a CASIO fx  570ES, 570ES Plus


- 
 toán
SHIFT MODE 1
Math.

MODE 2
CMPLX
r (ta hiêu:A)
SHIFT MODE  3 2
r 
D)

SHIFT MODE 3
D
R)
SHIFT MODE 4
R

SHIFT (-).


 rad
rad
rad. 

/2).
 rad.
 (Rad)=
φ(D).π
180




15
30
45
60
75
90
105
120

135
150
165
180
360

1
π
12

1
π
6

1
π
4

1
π
3

5
π
12

1
π
2

7
π
12

2
π
3

9
π
12

5
π
6

11
π
12


2

bhép tính : a +bi A



).
-a + bi A  SHIFT 2 3 =
: 8 SHIFT (-) (:3 -: 4+ 4

3
i .Ta SHIFT 2 3 =   8
1
π
3

-A  a + bi SHIFT 2 4 =
8 SHIFT (-) (:3 -8
1
π
3
, SHIFT 2 4 =  :4+4
3
i








c. 

 :
+: MODE 2 CMPLX.
-SHIFT MODE 3 D
 SHIFT MODE 4 R )
:
1 1 2 2

A A A
  
    

-A
1
SHIFT (-) 
1
+ A
2
SHIFT (-) 
2
= 
SHIFT 2 
 3 = r   )
 4 = a+bi )


5
a+bi SHIFT 2 3 = : A)
+ : MODE 2 CMPLX.
:
1 1 2 2
A A A
  
    

-A
1
SHIFT (-) 

1
+ A
2
SHIFT (-) 
2
=
SHIFT + =  A. SHIFT = : 
+ 
= ,  
SHIFT = SD .

d.
 
x
1
= 5cos(

t +

/3) (cm); x
2
= 5cos


A. x = 5
3
cos(

t -


/4 ) (cm) B.x = 5
3
cos(

t +

/6) (cm)
C. x = 5cos(

t +

/4) (cm) D.x = 5cos(

t -

/3) (cm) 




22
1 2 1 2 2 1
2. .cos( )   A A A A A



: tan  =



A=
22
5 5 2.5.5.cos( /3) 5 3  

(cm)
tan  =
5.sin( / 3) 5.sin0 5. 3 / 2 3
1
5cos( /3) 5.cos0 3
5. 1
2






=>
 = x = 5
3
cos(

t +

/6) (cm)

: MODE 2
-DSHIFT MODE 3
5 SHIFT (-) (60) + 5 SHIFT (-)  0 =
 5

3
30
x = 5
3
cos(

t +

/6) (cm)
:
15 5 3
22
 i
thì
SHIFT 2 3 =  5
3
30 ).


ng Rad (R): SHIFT MODE 4
MODE 2 CMPLX. 
5 SHIFT (-). (/3) + 5 SHIFT (-)  0 =  5
3

1
π
6
Hay: x = 5
3
cos(


t +

/6) (cm)
 
x
1
= 3/2) cm, x
2

A. - /3) cm B/3)cm - /6) cm
Cách 1:
2211
2211
coscos
sinsin


AA
AA




6
 
22
1 2 1 2 2 1
1 1 2 2
1 1 2 2

2 cos 2
2
3sin 1.sin
:
sin sin
2
3
2
tan 3
cos cos 3
3cos 1.cos
2
3
A A A A A cm
HD
AA
AA















    









      













Cách 2: Dùng máy tính:MODE 2 CMPLX
 theo  SHIFT MODE 3

3
 SHIFT (-). (90) + 1 SHIFT (-).  180 = 120

 nh:
x
1
= 3cos(t - /2) cm, x
2
= cos(
A. x = 2cos(t - /3) cm B.x = 2cos(t + 2/3)cm C.x = 2cos(t + 5/6) cm D.x = 2cos(t - /6) cm
Cách 1:
 
22
1 2 1 2 2 1
1 1 2 2
1 1 2 2
2 cos 2
2
3sin 1.sin 0
:
sin sin
3
2
tan 3
s s 3
3cos 1.cos0
2
3
A A A A A cm
HD
AA
Aco A co













    










       















Cách 2: Dùng máy tính:MODE 2 CMPLX
radian(R): SHIFT MODE 4

3
 SHIFT (-). (-/2) + 1 SHIFT (-)  0 = -/3

 
1
= 2
3

3

)
cm, x
2

6

) cm ;x
3
-

2



A. 
6


rad . B. 
3

rad. C. 
6

rad. D. 
6


rad.
HD: Cách 1:  x
2
vµ x
3
có:







       





23 23
4sin 8sin
62
tan 3
3
4 cos 8cos
62



        


22
23 23
A 4 8 2.4.8.cos 4 3 x 4 3 sin 2 t
3

 x
23
vµ x
1
có:






   





2 3 sin 4 3 sin
1
33
tan
3
2 3 cos 4 3 cos
33

   
    
22
A 2 3 4 3 2.2 3.4 3 cos 6
 


           


max
x 6 cos 2 t cm v A 12 ; rad

66

Cách 2: : MODE 2 ;DSHIFT MODE 3
2
3
 SHIFT (-) 60 + 4 SHIFT (-)  30 + 8 SHIFT (-)  -90 =  6-30

7
: 3
3
-3i thì bSHIFT 2 3 =  6 -30 ) => vmax= A =12 (cm/s) ; =/6

 
x
1
= cos(2t + )(cm), x
2
=
3
.cos(2t - 
A. x = 2.cos(2t - 2/3) (cm) B. x = 4.cos(2t + /3) (cm)
C. x = 2.cos(2t + /3) (cm) D. x = 4.cos(2t + 4/3) (cm)
 MODE 2 CMPLX
rad (R): SHIFT MODE 4
-1 SHIFT(-)   +
3
 SHIFT(-)  (-/2 =  2-
2
π
3

. 

 
)()
2
2cos(
3
4
))(
6
2cos(
3
4
cmtcmtx






A.
.
3
;4 radcm

B.
.
6
;2 radcm


C.
.
6
;34 radcm

D.
.
3
;
3
8
radcm


 MODE 2 CMPLX
radian(R): SHIFT MODE 4

4
3
SHIFT (-).  (/6) +
4
3
SHIFT (-).  (/2 = 
1
π
3

 Degre(D): SHIFT MODE 3

4

3
SHIFT (-).  30 +
4
3
SHIFT (-).  90 =  60

 
1
= 4 cos(t - /2) (cm) ,
x
2
= 6cos(t +/2) (cm) và x
3
=2cos(
là
A. 2
2
cm; /4 rad B. 2
3
cm; - /4 rad C.12cm; + /2 rad D.8cm; - /2 rad
 X570ES : MODE 2 CMPLX
tính rad (R). SHIFT MODE 4 
4 SHIFT(-) (- /2) + 6 SHIFT(-) (/2) + 2 SHIFT(-) 0 = 2
2
 /4. 

 
x
1
= a

2
cos(t+/4)(cm) và x
2
= a.cos(t +  
A. x = a
2
cos(t +2/3)(cm) B. x = a.cos(t +/2)(cm)
C. x = 3a/2.cos(t +/4)(cm) D. x = 2a/3.cos(t +/6)(cm) 
: MODE 2 CMPLX
 (D) SHIFT MODE 3 

2
 SHIFT(-)45 + 1 SHIFT(-)180 = 1 90,






8

( A
2
và

2
 :
:
12
A A A ;

21
A A A ;

+
2 2 1 1
A A A
  
    
;
1 1 2 2
A A A
  
    

x
2
: x
2
=x - x
1
 x
2
= A
2
cos(t + 
2
)

2
và 

2
?
 : MODE 2 CMPLX
- SHIFT MODE 3 D
Radian SHIFT MODE 4 R )
 :
1 1 2 2
A A A
  
    

2 2 1 1
A A A
  
    

A SHIFT (-)  -  A
1
SHIFT (-) 
1
= .
SHIFT 2 3 =  A
2
 
2


 : MODE 2 màn hCMPLX
 :
1 1 2 2

A A A
  
    

2 2 1 1
A A A
  
    

A SHIFT (-)  - A
1
SHIFT (-) 
1
=
SHIFT + =  A
2.


SHIFT = : 
2



 
2
cos(t+5

1
=A
1

cos(t + 
1
) và x
2
=5cos(t+/6)(cm)

A. 5cm; 
1
= 2/3 B.10cm; 
1
= /2 C.5
2
(cm) 
1
=  /4 D. 5cm; 
1
= /3
 MODE 2 CMPLX
- là rad (R): SHIFT MODE 4 . 
5
2
 SHIFT(-)  (5/12)  5 SHIFT(-)  (/6 =  
2
π
3

 
1
=
2

3
cos(2t + /3) (cm), x
2
= 4cos(2t +/6) (cm) và x
2
= A
3
cos( t + 
3

t - 
A. 8cm và - /2 . B. 6cm và /3. C. 8cm và /6 . D. 8cm và /2. 
: MODE 2 CMPLX
 (R) SHIFT MODE 4 x
3
= x - x
1
x
2

6 SHIFT(-)  (-/6) - 2
3
 SHIFT(-)  (/3) - 4 SHIFT(-)  (/6 =  -
1
π
2
.

Câu 1: 
1

=
8cos(2t + /2) (cm), x
2
= 2cos(2t -/2) (cm) và x
3
= A
3
cos(t + 
3


2
cos(2t +  
A. 6cm và 0 . B. 6cm và /3. C. 8cm và /6 . D. 8cm và /2.
Câu 2: 
1
=
a.cos(2t + /2) , x
2
= 2a.cos(2t -/2) và x
3
= A
3
cos(t + 
3

a
2
cos(2t - 
A. a và 0 . B. 2a và /3. C. a

2
và /6 . D. 2a
2
và /2.

9
2/. 
1
, A
2
hay A:

- 
AAA

,,
21

- 
C
c
B
b
A
a
sinsinsin



- 

 







6
cos
11


tAx








2
cos6
2


tx









sin
3
sin
sin
3
sin
2
2
A
A
AA

.
A
min
khi sin =1  = /2
3



.
 :           
 

11
cos10

 tx
 







2
cos
22


tAx

cos( )
3
x A t





2



max


6
sin
sin
sin
6
sin
11




A
A
AA


21max
2
AAA




=>A
2
= A

1
tan/3 =
310
.


B :           
 
11
cos10

 tx
 







2
cos
22


tAx

cos( )
3
x A t






2

A. 10
3
cm B. 20cm C. 20 /
3
cm D. 10/
3
cm

4: 
1
=
2
3
sin
t

(cm), x
2
= A
2
cos(
t



2

t




2



=
/3



2


2


/3


B. 2
3

/4



C. 4
3

/2



/6



Bài 5: 
1
= A
1
cos(

t -

/6) cm
và x
2
= A
2
cos(

t -




t -


2
có giá 

1

1
A

2
A

A


3




10
A. 15
3
cm B. 9
3
cm C. 7 cm D. 18

3
cm

Bài 6: 

1
= 10 cm, 
1
=
6

; A
2

2
= -
2



A. 10 cm. B.5
3
cm. C. 0. D. 5 cm

7: 
1
/6 và

2
-

2

là bao nhiêu?
A. A = 2
3
(cm) B. A= 5
3
(cm) C. A = 2,5
3
(cm) D. A=
3
(cm)

8: 
x
1
= A
1
cos(t+/3)(cm) và x
2
= A
2
cos(t- 
t+ 
2
 
2max
?
A /3; 8cm B  /6;10cm C. /6; 10cm 

9: 

1
= 4cos(4t +
3

) cm
và x
2
= 4
2
cos(4t +
12


A. 4cm B. 6cm C. 8cm D. ( 4
2
- 4)cm

Bài 10: 
11
cos( t + )( )
3
x A cm



và
22
os( t - ) ( )

2
x A c cm




6cos( t + )( )x cm

1

Tha
1

2

2max
?
A. 16 cm. B. 14 cm. C. 18 cm. D. 12 cm

Bài 11: 
11
cos( t + )( )
3
x A cm



và
22
os( t - ) ( )

2
x A c cm




6cos( t + )( )x cm

1


1

2

2max
?
A. 16 cm. B. 14 cm. C. 18 cm. D. 12 cm.

Bài 12: 
1
= A
1
cos (

t) cm và x
2
= 2,5
2
cos (


t +

2
).

2


2
A. - /4 B. - 3/4 C. -2 /3 D. 3/4
Bài 13. 
cmtBxcmtAx )
2
cos(.;)
3
cos(.
21






cmtx )cos(.2


. 





11
A. 4cm và
6/

B.
32
cm và -
6/

C.
3
cm và
3/

D. 2cm và
12/



Bài 14: 
1
= A
1
cos(t
3
2



) cm
và x
2
= A
2
cos(t
6


t+
2




A.

=
rad
4

B.

=
rad

C.
.
3
rad




D.

=
rad
6



Bài 15:  
x
1
=2
3

2
=A
2

2

2
-

2
 
 B. 2
3

 C. 4
3
 
Bài 16:  là
1
x 10cos(2 t )cm   
;
2
x cos(2 t )cm
2
2
A

  

x cos(2 t )cm
3
A

  
. Khi

2

A.
10 3cm
B. 20cm C.
20
cm
3

D.
10
cm
3

Bài 17: 

1
, A
2
,
1
rad
3

  
,
2
rad
2


. Dao 
2

thì A
1
và A
2


A.
9 3cm
1
A 
; A
2
=18cm B. A
1
=18cm; A
2
=9cm
C.
9 3cm
1
A 
; A
2
=9cm D. A
1
=9cm;
2
9 3cmA 


Bài 18: 

1
x 6cos(4t )cm
3




và
2
x 6cos(4t )cm
12



A. 4cm B. 6cm C. 8cm D. (
42
4)cm
Bài 19 :
1
= A
1
cos10t; x
2
=
A
2
cos(10t +
2

1
3
cos(10t +
2
-  =
6



2



A.
2
1

4
3
B.
3
1

3
2
C.
4
3

5
2
D.
3
2

3
4

12
ng dn chi tit:
3:
1:

6
sin
sin
sin
6
sin
11




A
A
AA


21max

2
AAA





=>A
max
=
20
3
cos
1


A
. 
max
/2 = 10 thi
310
6
sin2
12


AA
.
 Ta có:
 

 
2 2 2
1 2 1 2 1 2 2 2
2 2 2 2 2 2
2 2 2 2
A A A A A A A A A 2AA cos
10 A A AA 3 A AA 3 A 10 0 *
          
        


 
2 2 2
3A 4A 4.10 0 A 20 cm      


 
2
A 10 3 cm

 3:


1
OA A


10
.sin
sin

6
A



 A
max

0

* Khi A=A
max
/2 =10 cm  
1
OA A

 
10 3A cm

: : x
1
= 2
3
sint = 2
3
cos(t -
2

)
: A = A

1
+ A
2


2
là
3

A=2cm;
1
23A cm

A
1
2
= A
2
+ A
2
2
 2AA
2
cos
3

= A
2
+ A
2

2
 AA
2

<=>A
2
2
 AA
2
+ A
2
 A
1
2

= 0
<=>A
2
2
 2.A
2
+ 2
2
 4.3

= 0
<=>A
2
2
 2A

2
 8

= 0 => A
2
= 4cm.
Ta th: A
2
2
= A
1
2
+ A
2

1.
Suy ra  = 0 => 
2
=
3

. 
5:  1
Khi A
2

21
21
2
sin / 2 sin /3

3
AA
AA

  
(1)
Tam giác OAA
2

2 2 2
12
9AA
(2)

2 2 2
11
4
9
3
AA
=> A
1
=9
3
cm. 
1
A

2
A


A

3/



1
A

A

2
A

10cm

/3
/6
/6


O




/6
A
O


A
2


/3
A
1

/6
A
1
A
2
A


13
 2:





sin
6
sin
2
AA
A

max
khi
cmAAAcmA 3918
2
22
21





6: 

3
sin

A
=

sin
1
A
=> A =

sin
1
A
sin
3


A = A
min
khi sin = 1 => A
min
= A
1
sin
3

= 5
3


7:   qauy nh  bên:

A min khi 
A= A
1
cos (/6) =10
3
/2 = 5
3
(cm) .
Và A
2
= A
1
sin (/6) =10.1/2 = 5 (cm)

8:  c t qauy nh  bên:
A
2

1
,A
2
,A là góc vuông
 mà A
2

Theo 
A
Sin
A
Sin


2
=>


Sin
A
SinA .
2

.
 =5cm, = /6. Nên A
2

.

2
 =/2 =>
cm
Sin
A
A 10
2
1
5
6
.1
max2



 = / - 
1
/= / /2 - /3 / = /6
Vì  <0 =>  = - /6 . 

Bài 9: 
Cáh 1: 


1
A
2



3

-
12

=
4



1
= 4cm ,OA
2
= 4
2
cm , và góc A
1
OA
2
=/4

1
A
2
= /2 và tam giác OA
1
A
2


1
.

1
=A
1
A
2

A
1
A
2

Khi  A
1
A
2


Cách 2: 
1

2


2
M
1
là x = x

1
- x
2
= 4cos(4t +5/6) ( cm) .
Suy 
1
và M
2
là x
max


III
I
A
1

/4
O
IV
x
II
A2
Hình
1
A






A

2
A


1
A

/6
A

2
A

O
M
O

A

A
2
A
1


3





1
A

2
A

A

O
x


14
Bài 10:  
5
.
6
rad






Ta có:
2

2
sin
.
sin sin sin
AA
AA

  
  

Vì

i nên A
2
s ln nht khi sin𝛃 ln nht tc là góc 𝛃 = 90
0
.

2max
6
12( )
sin
sin
6
A
A cm


  


Bài 11: 

2
2
0
2 sin
sin sin30
A
A
AA


  

Ta có A
2max
khi sin

=1 => A
2
= 2A = 12cm


Bài 12: 

Khi A
2

2
2

2,5 2
sin
sin / 2 sin 2
2,5 2
A
AA
A


    


Hay  = /4 =>.

Tam giác OAA
2


2
= -( /2 + /4 ) = - 3/4

Bài 13: 
22
ax
4 2 2 3
4
22
sin sin30 sin sin 1
6
M

A
B
B

  


  



   








Bài 14: 
- 
- 
1
A
và
2
A
là
2

à -
36
v


nên
2
3 6 6
  

   

- 
2
2
12
.sin sin 24.sin
sin sin sin 1/2
A
AA
A
  
  
    
.

2
max khi sin =1

0

90


mà
2
3




1
).Nên
6





Bài 15 1: Ta có x
1
=2
3

3
-
2

)(cm)

2

và 
2
A
1
A
2


A

x




O
A
2

A

2
B
x


A
O

2

A
2


/4
A
1

1
A

A

2
A



0
30

5
6


2
A

α

β
1
A

A


15
6

O
1
A
2
A
A

3


O
2
A
1
A
A
3




6



A
1
2
= A
2
2
+ A
2

- 2AA
2

2
 
A
1
2
= A
2
2
+ A
2

- 2AA
2
cos(

3

)
=> A
2
2
- 4A
2
cos
3

- 8 = 0
=> A
2
2

- 2A
2
 8 = 0 => A
2
= 4 cm
A
2
2
= A
1
2
+ A
2

- 2AA
1
cos 
 = góc A
1
OA =
2


=> 16 = 12 + 4 - 16
3
cos => cos = 0
=>
2


2

+ k =>  
2
=
3

. 
2

2
=
3



 2:


)(4082)cos(2
22
2
222
22
2
2
1
cmAAAAAAAA 


)(
3
0
2;32;4
2
22
1
2
212
rad
AAAcmAcmAcmA








Bài 16: 
- Áp 
1
A
sin
s
6
A
in



 A max khi sin=1 
A
max
=20cm
- 
max
A
A 10cm
2


- 
2

Bài 17: 
- 
A
s
sin
6
2
A
in




 A
2
max khi sin=1  A
2
=18cm
Bài 18: 
12
5
d |x x | 6
6

   


Bài 19 :
Xét tam giác OA

1
A
O

2
=
3



A
A
2

A
1

A
2

A
A
1

O

2
- 

A

A
2

A
1


16

sin
2
A
=
6
sin
1

A
=> sin =
1
2
2A
A
(1)
A
2
2
= A
1
2

+ A
2
 2AA
1
cos = 4A
1
2
- 2
3
A
1
2
cos (2)
sin =
1
2
2A
A
=
2
cos324


=>4sin
2
 = 4 - 2
3
cos
2
3

cos = 4(1- sin
2
) = 4cos
2
 => 2cos (2cos -
3
) = 0 (3)
=> cos  =
2
3

=>  =
2

=> 
2
=
2

+
6

=
3
2

=>
2



=
4
3

  =
6

=> 
2
=
6

+
6

=
3

=>
2


=
2
1
A

3/.:
Bài 1. hòa cù  
trình:

1
x 4cos(10t )
4


(cm)và x
2
= 3cos(10t +
4
3

) (cm).
A. 50cm/s; 10 m/s
2
. B. 7cm/s; 5 m/s
2
. C. 20cm/s; 10 m/s
2
. D. 50cm/s; 5 m/s
2
.
Bài 2. 
 là x
1
=5cos(10

t) cm, x
2
=10cos(10




2

A. 1125J B. 0,1125J C. 0,225J D. 1,125J
Bài 3. 
trình là
11
cosx A t


và
22
cos
2
x A t







A.
2 2 2
12
2E
AA



B.
2 2 2
12
E
AA


C.
 
2 2 2
12
E
AA


D.
 
2 2 2
12
2E
AA



Bài 4                 



A.
10



m/s B.
10


cm/s C.


m/s D.


cm/s
Bài 5: 
là: x1=4cos(10t+

/4) cm; x2=3cos(10t-3


A. 10cm/s B. 7cm/s C. 20cm/s D. 5cm/s
Bài 6: 
là: x1=4cos(10t+

/4) cm; x2=3cos(10t-3


A. 10cm/s
2
B. 1cm/s
2

C. 10m/s
2
D. 1m/s
2
Bài 7: 

t+

/2)
cm, x2=2cos(5


A. 10
2

cm/s B. 10
2
cm/s C.10

cm/s D. 10cm/s
8: ( 




O
A



A
2
A
1

17


A.
4
3
. B.
3
4
. C.
9
16
. D.
16
9
.
9: 
 


 
A.
4
3
. B.

16
27
. C.
27
16
. D.
16
9
.
10: 
cm)t2cos(10x
11

và
cm)2/t2cos(Ax
22


cm)3/t2cos(Ax 

2

nhiêu?
A.
310
cm B.
3/10
cm C.
3/20
cm D. 20cm

Bài 11: 
1
x 4,8cos 10 2t cm
2





cm,
 
22
x A cos 10 2t cm  


0,3 6 /ms

2
.
A.7,2 cm B. 6,4cm C.3,2cm D.3,6cm
Bài 12: 

1

A
2
= 4
3



hai là:
A. 3W/4. B. 2W/3. C. 9W/4. D. W

ng dn chi tit:
Bài 1. 
Cách 1: Ta có: A =
0
21
2
2
2
1
90cos2 AAAA 
= 5 cm
 v
max
= A = 50 cm/s = 0,5 m/s; a
max
= A = 500 cm/s
2
= 5 m/s
2
. D
Cách 2: MODE 2 CMPLX
 (D) SHIFT MODE 3
4 SHIFT(-)45 + 3 SHIFT(-)135 = 5 81,869,
Suy ra A = 5cm  v
max
= A = 50 cm/s = 0,5 m/s; a
max

= A = 500 cm/s
2
= 5 m/s
2
. D
Bài 2.  1:  
2 2 2 2
11
.0,01.(10 ) .(0,15) 0,1125
22
W m A J

  

 2
2 2 2
11
22
kA m A


. Do


=. 

Bài 3.

18


2
2
2
1
AAA 
suy ra :
 mAAmE )(
2
1
2
2
2
1
2

 
2 2 2
12
2E
AA




Bài 4.  
22
v A x

  



=2

f=20


2 2 2
1 2 1 2
2 os
2
A A A A A c

  
. 


m/s. C


Bài 5.  Qua VTCB thì V=V
max
=
A





=-3


/4-

/4=-

nên 2
v=10cm/s. 
Bài 6.  Qua VTB thì a=a
max
=
A


2



=-3

/4-

/4=-

nên 2 dao
 pha. 
2
=1m/s
2.
B

Bài 7.  V=V

max
=
A





=


A=
22
12
AA
=2
2

2

cm/s. 
8  1:


ng A
1
A
2

1

A
2
= 10 cm
A
1
= 6 cm; A
2

: x
1
= 6cos(t +
2

); x
2
= 8cost


= W
tM
=
2
1
W
> x
1
=
2
2
A

1
= 3
2
(cm)
6cos(t +
2

) = 3
2
> -6sint = 3
2
> sint = -
2
2


2
= 8cost = ± 4
2
cm = ±
2
2
2
A
=> W
t2
=
2
2
W

=> W

= W
t2
=
2
2
W


1
= m
2
và f
1
= f
2

=>
2
1
đ
đ
W
W
=
2
1
W
W

=
2
2
2
1
A
A
=
16
9
C.
 2:

12
xx 

12
xx 


2
2
2
1
2
AAMN 



= W

t

2
2
1
1
A
x 

16
9
8
6
2
2
2
2
2
2
2
1
22
2
2

A
A
W
W
W

W
WW
A
x
N
M
đN
đM
tđ
. C.
9: 

12
xx 

12
xx 


2
2
2
1
2
AAMN 


=
W
1

 
2
3
1
1
A
x 
.
2
2
2
A
x 
=> W

= 3W
2
/4
16
27
4
3.33
2
2
2
2
2
1
2
1


A
A
W
W
đ
đ
. C.
O
M
N
O
M
N
A
1
A
2
O
x

19
10: 




Sin
A
Sin

A

)6/(
1
 

1
= 20cm
Ta có A
2
2
= A
2
 A
1
2
= 20
2
 10
2
= 300
Hay A
2
= 10
3
cm. 

Bài 11: Giai: W= Wd + Wt = 3Wd + Wt = 4 Wt
Hay
22

11
4
2 2 2
A
kA kx x   


0,3 6v 
m/s=
30 6 /v cm s

 :
2
2
2
v
Ax


 :
22
900.6
27
4 200 4
AA
A    
=> A = 6cm
 :
2 2 2 2 2 2 2 2
1 2 2 1

2
6 4,8
3,6
A A A A A A
A cm
      
 


Bài 12
x
1
= 4cost (cm); x
2
= 4
3
cos(t + ) (cm)

1
A
2
và vecto A = A
2
 A
1


2
 x
1

x = Acos(t + 

2
= A
1
2
+ A
2
2
 2A
1
A
2
cos = 64 - 32
3
cos

khi cos(t + > A
2
= 16
64 - 32
3
cos = 16 =>cos =
2
3
=>  =
6




2
= 4
3
cos(t + ) = x
2
= 4
3
cos(t +
6

)
Khi W

= W

=
2
2
1
kA

1
= 0 => cost = 0 ;sint = ± 1

2
= 4
3
cos(t +
6


) = 4
3
cost cos
6

- 4
3
sint sin
6

= ± 2
3
cm = ±
2
2
A

W

=
2
2
2
kA
-
2
2
2
kx
=

4
3
2
2
2
kA
=>
1
2
đ
đ
W
W
=
W
W
đ 2
=
2
24
3
2
1
2
2
kA
kA
=
4
3

2
1
2
2
A
A
=
4
9
=> W

=
4
9







A
1
A

A
2

O


O 

A

A
2
A
1

20
4/.  

Bài 1

1
= 3cos(
3
2

t -
2

) và x
2
=3
3
cos
3
2


t (x
1
và x
2

1

= x
2

A. ± 5,79 cm. B. ± 5,19cm. C. ± 6 cm. D. ± 3 cm.

Bài 2: Hai v

là
 
1
55
3cos
36
x t cm





và
 
2

20 2
5cos
33
x t cm





           

A. 0,1s. B. 0,05s. C. 0,5s. D. 2s.

Bài 3
x
1
=6cos(10t +
3

) (cm),x
2
=6
3
cos(10t -
6



- -
 -

Bài 4

và vuôn


A
3
4

4
3

3
4

4
3

Bài 5t là
 
1
6cos 10 3x t cm


và
 
2
8cos 10 6x t cm




8x cm


1

A.  B. 
C.  D. 
ng dn chi tit:
Bài 1 
x = 6cos(
3
2

t -
6

) (cm); 3cos(
3
2

t -
2

) =3sin(
3
2

t )
x

1
= x
2
=> 3cos(
3
2

t -
2

) = 3
3
cos
3
2

t
=> tan
3
2

t =
3
= tan
3

=>
3
2


t =
6

+ k => t =
4
1
+
2
3k

x = 6cos(
3
2

t -
6

) = x = 6cos[
3
2

(
4
1
+
2
3k
) -
3


]
= 6cos(k -
6

) = ± 3
3
cm = ± 5,19 cm
/6

A
1
A
2
A
X
1

x
2


/6
X

=x
1
+ x
2

0

X

=x
1
- x
2

-x
2


/6

21
 



(
2
3

t


5
6
) (
2
3


t


6
)

1
= x
2
thi x
1
-x
2


= x
1
-x
2



= x
1
+x
2
/6.
Nên ta có x = 6cos (/6) = 3
3

= 5,19cm ; x = 6cos (5/6)= -3
3
= -5,19cm . 
 


= x
1
-x
2
= 6cos(
2
3

t


5
6
) (
2
3

t


6
)
Khi x
hi

= 0 thì cos(
2
3

t


5
6
) = 0 =>
2
3

t


5
6
=


2

(
2
3

2.



6
) = 6cos(

4
3


6
) =6cos(

7
6
) = -3
3
= -5,19cm
(
2
3

0,5.


6
) = 6cos(

3


6
) =6cos(


6
) =3
3
= 5,19cm 
 4: Ta có x
1
= 3cos(
2
3

t -
2

)
2
3sin
3
t



x1 = x2
2 2 2 2 1 3
3sin 3 3 os tan 3 ;k Z
3 3 3 3 3 2 2
k
t c t t t k t
    


          


A =
22
12
6A A cm
;
1
2
1
tan
6
3
A
A


      


2
)
36
t cm


 5,19cm. 
Bài 2: :


)(
6
9
3
15
radt



.
)(
10
1
15
3
)
6
9
1)1.(2(
15
3
)
6
9
12()12(
minmax
stktkx 


 Sau:

)(
10
1
312
21
s
TT
t 


Bài 3 : x
1
= 6cos(10t +
3

) (cm); x
2
= 6
3
cos(10t -
6

) (cm)

1
+ x
2
= 12cos10t (cm)

1

AA
2

Khi x
1


3
2

vé
góc
3
2


3
2

= -  D
A
2
A
1
A

/6

A
1

A
2
A

22
Bài 4: 
- 
1 1 1
x A cos( t )   

2 2 2
x A cos( t )   

-  
1 2 2 2
d | x x | | ) | | Acos( t ) |
11
A cos( t )+A cos( t )             

22
max 1 1 1 2 2 1
d A A A 2A A cos( )         

- 
1
=A
2
=6cm 
21
3


     
 x
1
và x
2

3


- 
1
=3cm  3cm và 6cm

Bài 5:

* tan  = A
2
/A
1
=>  = 53,13
0
_  = (
A
,
1
A

* Khi
8x cm


+ cos = 8/10 =>  = 36,87
0

+  +  = 90
0
=>
1
A
x
1









5/.  :
Câu 1. 
1
=2cos (4t +
1

)cm và x
2
=2cos( 4t +
2



0


12
2 cos ( 4t +
6


1

là
A.
2

B. -
3

C.
6

D. -
6


Câu 2. 
2



trình x
1
= 6cos(

t-
2

) cm, x
2
= 6 cos(

t-



A. (3/40)s. B. (1/40)s. C. (1/60)s. D. (1/30) s.
Câu 3: 

t+

/6),
x2=2Acos(10

t+5

/6) và x3=A(10

t-




A. x=Acos(10

t+

/2) cm B. x=Acos(10

t-

/2) cm
C. x=Acos(10

t+5

/2) cm D. x=Acos(10

t-5

/2) cm
Câu 4:  
)2/cos(
11

 tAx
;
)cos(
22
tAx



;
)2/cos(
33

 tAx
. 
1
t

310
1
x
cm
A

2
A

1
A

/3

-8
10
-10
x


0

23
,
cmx 15
2

,
330
3
x
cm. 
2
t

1
x
cm,
2
x
= 0cm,
3
x
= 60cm

A. 50cm. B. 60cm. C.
340
cm. D. 40cm.

Câu 5:
1

, x
2
, x
3
.

12
= 4
2
cos(5t  
23
= 3cos(5t)cm; x
13
= 5 sin(5t - 
2

A. x
2
= 2
2
cos(5t -  B. x
2
= 2
2
cos(5t + /4)cm.
C. x
2
= 4
2
cos(5t + /4)cm. C. x

2
= 4
2
cos(5t - /4)cm.

Câu 6: 
1
, x
2
, x
3
.

12
x 6cos( t )cm
6

  
;
23
x 6cos( t )cm
3

  
;
13
x 6 2 cos( t )cm
4

  


1


3
là:
A. 0cm B. 3cm C.
32
cm D.
36
cm
Câu 7. 
1
=
A
1
cos(2

t +
2
3

) cm; x
2
= A
2
cos(2

t)cm; x
3

= A
3
cos(2

t -
2
3


1

1
= -
20cm, x
2
= 80cm, x
3
= -
2
= t
1

1
= - 20
3
cm, x
2
= 0cm,x
3
= 40

3
cm.

A. x
2
= - )cm. B. x
2
= 40
2
cos( t + /4)cm.
C. x
2
= 4cos( t + /3)cm. C. x
2
= 4
2
cos( t - /4)cm.

Câu 8:
1
, x
2
, x
3

12
= x
1
+ x
2

; x
23
= x
2
+ x
3
; x
13
= x
1
+ x
3
; x=x
1
+ x
2
+ x
3
   
12
   
23
=6cos(t +2/3)cm;
x
13
=6
2
cos(
2
=x

1
2
+x
3
2

A. 6
2
cm B. 6cm C. 24cm D. 6
3
cm
Câu 9: 
1
; x
2
và x
3
t là
12
2cos(2 /3)x t cm


;
23
2 3cos(2 5 /6)x t cm


và
31
2cos(2 )x t cm





A
23
cos (

t +
2

) cm B
3
cos (

t -

) cm
C 2
3
cos (

t +

) cm D
3
cos (

t +
2


) cm
Câu 10. Da
12

x
23
=2
3

31

A. 1 cm. B. 3 cm. C.
3
cm. D. 2
3
cm.

ng dn chi tit:
Câu 1. Do A
1
=A
2
=2 nên
1
2 cos 2 2.2cos cos cos
2 2 2 2 3
th
AA
   

  
     



24
Vì 0


12

21
2
0
2 2 2 3 3
    
  

        
(1)

Do A
1
=A
2

362
21
21









(2)

1
= -
6

và 
2
=
2

.
Câu 2. 

1
 x
2
= Acos(wt + )
=> x
1
= x
2


=> A = 6
2
cm ;  = - /4 => x = 6
2
cos(wt  /4)
x
max
=> cos(wt  /4) =  1
=> 10t  /4 = k  => t = 1/40 + k/10
1/40s .B

Câu 3. 
1 1 2 2 3 3
sin sin sin
x
A A A A
  
   
A
1 1 2 2 3 3
s s s
y
A Aco A co A co
  
   
0

TT
=
22

xy
AA


=
x
y
A
A
nên

=

/2 .V
Câu 4.
 x
1
và x
2
vuông pha nên:
22
12
12
1
   

   
   
xx
AA

X
2
và x
3
vuông pha nên:
2
2
3
2
23
1







x
x
AA


2

22
1
12

20 0
1 20
   

   
   
   
A cm
AA


1

2
2 2 2
12
2
1 2 2
10 3 15
1 1 30
20

     

      

     

     


xx
A cm
A A A


2
2
2
2
3
2
3
2 3 3
15 30 3
1 1 60
30




      










x
x
A cm
A A A


22
2 3 1
( ) 50   A A A A cm
.
Gi 2 :

2
:
22
12
22
12
1
xx
AA

=>
2
1
2
1
1
x
A


=> A
1
= 20cm. 
3
= 60cm

1
:
 
2
2
22
2
10 3
15
1
20 A


=> A
2
= 30cm

A

A
1
A
2

25

 
 
2
2
22
3
30 3
15
1
30 A

=> A
3
= 60cm
A = 50cm.  A
Câu 5.  1:

 
  
  
   
12 1 2
23 2 3 2
13 1 3
3
x x x 4 2cos 5t cm
4

3
x x x 3cos 5t cm 2x 5cos 5t 3cos 5t 4 2 cos 5t
4
x x x 5sin 5t 5cos 5t cm
2


   







       







     





 

2
x 2 2cos 5t cm
4






Câu 5. : Ta có :
12 23 13
2 13 13
2 2 cos 5
24
x x x
x x x x t cm



     


.  A
Câu 6: :
12 13 23
1
x x x
x 3 6
2 12



  

13 23 12
3
x x x
x 3 2
2 12


  

- 
3

1
góc
2

 x
1
max thì x
3
=0.  A

Câu 7.  


/2 nên x
1


1
vuông pha

1
2
= x
1
2

1
2
= (- 20)
2
+ (- 20
3
)
2
suy ra A
1
= 40cm
 : A
2
= 80cm ; A
3
= 80cm
 ! x = x
1 +
x
2 +

x
3

 : 40

120 + 80

0 +80

-120 = 40

-60
 -. : -.  A

Câu 8. :

x=x
1
+ x
2
+ x
3
=
12 23 13
25
6 6 6 2
55
6 3 12
6 2 6 2 os
2 2 12 12

x x x
x c t cm
  


    


     




1 23
2 13
3 12
6cos
6
0
2
6cos
3
x x x t cm
x x x
x x x t cm







   





  




   





* Theo bài x
2
=x
1
2
+x
3
2
và x=x
1
+ x
2

+ x
3
= x
1
+ x
3
 x
1
x
3
=0
1
3
0
62
0
2
32
tk
x
x
tk





  










  

