103
155 A 12 F = m . ’ . ( ) . = = . • , . • . • , . • , - . • , . • , = 0, = 0, , . - . ( = . 0 ). • - . . , , . , : – – – .
| Date post: | 02-Feb-2016 |
| Category: |
Documents |
| View: | 231 times |
| Download: | 0 times |
155
A 1 2
F = m .
.
( ) .
== .
, . . , . , - .
, .
, = 0, = 0, ,
.
- .
(
= . 0 ).
-. . , , .
, :
.
156
12
.
- , :
. F = 0 (1) ( ). . . = (2) .
.
-, - - , . = 0 (4)
= 0 (3)
-, - .
= . (9)
- - .
- .
- - .
= t (10)
= 12t
2(11)
= 0+ t (14)
= 0t +
1
2t
2(15)
= 0 t (18)
= 0t 1
2t
2(19)
= . 0 (5) = t (6)
= (7)
= (8)
= (12) = (13)
= (16) = (17)
= (20) = (21)
0 = 0
0 0
0 = 0
0 0
0
0 0
0
157
1. , m = 2 kg, R = 1 m, - - . t = 0 F
4 . :
) ) t
1= 5 s
) t = 0 t
1
) t1 t
2= 13 s,
t1 F
.
cm
=
1
2mR
2.
m = 2 kg , R = 1m , F = 4 N , I
cm=
1
2mR2
) = ;
t = 0 F
.
: F
w ,
. . , :
=
cm F =
1
2mR2 FR =
1
2mR2 =
2F
mR = 4 rad/s
2
.
) t1= 5 s ,
1 = ;
, - , :
1= t1 1 = 4 5 rad/s 1 = 20 rad/s
1
, .
) N1= ;
t1:
1=
1
2t1
2 1=
1
2 4 52 rad 1 = 50 rad
1=
1 2
1=
1
2
1=
25
.
) t1= 5 s , t
2= 13s , N = ;
t1, F
, , = 0.
= cm = 0.
t1 1 = 20 rad/s.
t = t2 t
1= 8 s :
= 1 t = 20 8 rad = 160 rad
:
= 2 =
2 = 80
158
12
2. m = 10 kg, R = 1 m . t = 0 F
1= 30 N F
2= 20 N. -
cm
=
1
2mR
2.
) . ) t
1= 4 s.
) t1 F
1, t
2.
i) . ii) -
t = 0 t2.
iii) t = 0 . iv)
t2= 3 s t
3= 5,5 s.
m = 10 kg , R = 1m , F
1= 30 N , F
2= 20 N , I
cm=
1
2mR2
) ,1 = ; t = 0 : F
1, F
2
w , .
F
1 F
2:
F1= F
1R = 30 N m
F2
= F2R = 20 N m
F1>
F2
F1 . .
, :
= I
cm ,1 +F1R F2R =
1
2mR2 ,1 ,1 =
2 F1 F
2( )mR
,1 =2 30 20( )
10 1rad/s2 ,1 = 2 rad/s
2
,1
, -
.) t
1= 4 s ,
1= ;
,1, - , = ,1t = 2t (S.I.).
t1= 4 s
1= 2 4 rad/s
1= 8 rad/s.
) i) ,2 = ; ,
ii) = f (t) = ; ,
iii) N = ; ,
iv) t2= 3s , t
3= 5,5 s , = ; , = ;
F
1 F
2 .
i) , ,2.
= cm ,2 F2 = Icm ,2 F2R =
1
2mR2 ,2 ,2 =
2F2
mR ,2 =
2 2010 1
rad/s2 =
,2 = 4 rad/s2
,2 -
=
F2, .
ii) t1 0 = 1 = 8 rad/s , = 1 ,2 t t1( ) = 8 4 t 4( ) = 24 4t (S.I.). t
2, = 0 24 4t
2= 0 4t
2= 24 s t2 = 6 s. :
= 2t, 0 t 4 s24 4t, 4 s t 6 s
(S.I.)
t = 0: = 0 t = 4 s: = 8 rad/s t = 6 s: = 0
}
:
159
iii) ( 0 6 s) = f (t).
= =6 8
2rad = 24 rad
= 2 = 2 = 12
.
iv) t2= 3s 0 t 4 s,
- ,1 = 2 rad/s
2 , - = ,1R = 2 m/s
2 .
.
t3= 5,5 s 4 s t 6 s,
- ,2 = 4 rad/s
2 , = ,2R = 4 m/s
2 .
.
3. m = 6 kg, R = 0,2 m - . - . t = 0 F, , . t = 4 s d = 8 m.
) . ) t = 4 s; ) F. )
t1= 2 s;
I = 12
mR2.
m = 6 kg , R = 0,2 m , d = 8 m , I =
1
2mR2
) = ; , F
A= 0 F =
, = (-) } F = T (1). : w
F
. ,
( )
, ,
- .
= t (2) =12t
2 (3).
d S
, d = S = R, .
d = R = dR = 8
0,2rad = 40 rad.
(3) = 12t
2 =2t2 =
2 4042
rad/s2 = 5 rad/s2
- -
,
, .
160
12
) t = 4 s , = ; (2) = t = 5 4 rad/s = 20 rad/s ) F = ; , :
= R =
1
2mR2 =
1
2mR =
1
2 6 0,2 5 N T = 3 N
(1) F = 3 N ) t
1= 2 s , = ; , = ;
t1
1= t1 1 = 5 2 rad/s 1 = 10 rad/s.
, .
= = .() = 1R = 10 0,2 m/s = 2 m/s
- , .
= () = R = 5 0,2 m/s
2 = 1 m/s2
4. () M = 1 kg , R = 0,4 m 12 m = 0,1 kg L = R = 0,4 m . -
f0=
50
Hz
. t = 0 , t
1= 1 s. :
) - ) ) ) . -
Icm
=
1
12mL
2.
= 1kg , R = 0,4 m , m = 0,1kg , L = R , f
0=
50
Hz , t
1= 1s , I
cm=
1
12mL2
) . = ; H - : I .() = .() + 12 .() (1) () m
1, m
2, ,
. m1+ m
2+ = . , -
R , . , :
I.() = m1R
2 + m2R2 + = m
1+ m
2+( )R2 = MR2 I.() = R2 (2)
R2 .
Steiner, :
.() = cm + m
R
2
2
=
1
12mR2 + m
R2
4=
1
3mR2 I.() =
13
mR2 (3)
(1) (2)
(3)I. =R
2 + 12 13
mR2 I. = MR2 + 4mR2 I. = 1 0,4
2 + 4 0,1 0,42( ) kgm2 . = 0,224 kgm2 ) = ; t = 0 -
0= 2f
0 0 = 100 rad/s.
,
. = 0 t
t = t1
= 0 0 = 0 t1 t1 = 0 =
0
t1
= 100 rad/s2 .
- - , :
= t xA =1
2t
2
161
) N = ; :
=
0t1 1
2t1
2 = 100 1 12100 12
rad = 50 rad
= 2 = 2 = 25
.
) T = ; -, :
= . R = I. =.
R T = 0,224 100
0,4N T = 56 N
5. , L = 2 m m = 3 kg, - z z . t = 0 F
F = 12 N
-. = 8 m.
) . ) . ) t
1
= 9.
) t1 F. :
i) t2
ii) t1 .
Icm
=1
12ML
2.
L = 2 m , m = 3 kg , F = 12 N ,
T= 8 Nm , N = 9
) = ;
, L2 .
Steiner, :
IA= I
cm+ m L
2
2
=
1
12mL2 + m L
2
4=
1
3mL2 I
A=
1
3 3 22 kgm2 I
A= 4 kgm2
) = ; : w
z z ,
F. ,
= 8m
F L z z .
, :
= IA FL T = IA =FL
T
IA
=12 2 8
4rad/s2 = 4 rad/s
2
162
12
) t1= ; , = ;
- .
= t (1) =12t
2 (2).
1 , = 9
, :
1= 2
1=
9
2 rad 1 = 18 rad
(2) 1=
1
2t1
2 t1=
21
t
1=
2 184
s t1= 3 s
(1) 1= t1 1 = 4 3 rad/s 1 = 12 rad/s
= L2
,
, = 1L
2 = 12 m/s .
) i) t2= ; , ii) = ;
i) F . , t1.
= = = =
8
4rad/s2 = 2 rad/s
2
t1
0 = 1 = 12 rad/s , :
= 1 t (3) = 1 t
12 t
2 (4)
t2 t = t
2 t
1. -
= 0, (3) 0 = 1 t t =
1
t = 6 s. t
2 t
1= 6 s t
2= 9 s.
ii) (4) t = 6 s :
= 1 t 1
2 t
2 = 12 6 12 2 62
rad = 36 rad
= 2 = 2 = 18
.
6. , M = 14 kg R = 0,5 m, - . m = 1 kg. t = 0 F
F = 16 N.
) -. ) t = 0,5 s : i) ii) , -
iii) m iv) v) .
Icm
=
1
2MR
2.
163
= 14 kg , m = 1kg , R = 0,5 m , F = 16 N , Icm
=
1
2MR2
) = ; t = 0 - F
.
- . w
, w
1
F. .
- :
I.() = () + .() =1
2MR2 + mR2 =
1
2M + m
R2 I.() = 2 kgm2
, :
() = .() F(O) = I. FR = I. =
FR
I. = 4 rad/s
2
) t = 0,5 s , - .
i) = ; = t = 2 rad/s .
ii) () = ; , () = ; , = ; () = R () = 2 m/s
2
() = 2R () = 2 m/s
2
= ( )2 + ()
2 = 2 2 m/s2
iii) . = ;
. = m . = 2 2
iv) s= ;
T. = sN T. = sw1 T. = smg s =T.mg
s= 0,2 2 .
v) . () = ;
. ,
.,x
.,y .
E .,x = m = 1 2 = 2 .,y = m = 1 2 = 2
. () = .,x + T .,y = T.,xR + 0 = 2 0,5 Nm T. () = 1m
164
12
.
: . . F = m. F = 0 = . , .
-
= 0 = . = t
= t
= 12t
2
= 0+ t
= 0t +
1
2t
2
= 0 t
= 0t 1
2t
2
t . =
0
. =
0
2
2
-
= 0 = .x = t
= t
x =1
2t2
= 0+ t
x = 0t +
1
2t2
= 0 t
x = 0t 1
2t2
t . =
0
x . =
0
2
2
:
:
165
.1 () ()
.
= .() = R .
= () = R - .1. ( I). t
0= 0 -
, : w
-
. ( II). w
T
.
w.,
F. .
, T = (-) , :
. , = . = R -
, = = R x = R.
:
= (1) = . = R (2) = = R (3)
m .
FZ= m
w T = m mg T = m (4) = t (5)
x = 12t2 (6)
.
= .
R = 12
MR2(1)
= 12R
(3)
= 12
(7)
F = 0 F. w . = 0
(1)
F . = g + T (8)
= t (9) =12t
2 (10)
, x L , x = L = R
(11).
. . F
.
. .
166
12
2. ( I). --. w
., w
1, w
2
F. .
t0= 0
w
1 w
2 :
w1= w
1R = m
1gR
w2
= w2R = m
2gR
m1> m
2 w1>
w2.
1 1 , 2 - 2 - .
( )
1 w
1
1 .
2 w
2
2 .
w.,
1, 2 F
.
. ,
1=
1
2=
2(-).
, : 1 , 2 .
, 1 = 2 = . = R 1 ,2
, 1 = 2 = = R x1 = x2 = R.
:
1 = 1
(1)
2 = 2 (2)1 = 2 = == . = R (3)1 = 2 = == = R (4)
1 , 2 .
1F = m11 w
1 T
1= m
1
1
(4)
m1g T1 = m1 (5)1 = = t (6)
x1 =12t2 (7)
2
F = m22
2 w
2= m
2
2
(4)
2 m2g = m2 (8)2 = = t (9)
x2 =12t2 (10)
.
= .
1R
2R =
=
1
2MR2
(1)
(2)
1
2=
1
2R
(4)
1 2 =12 (11)
F = 0 F. 1 2 w . = 0
(1)
(2)
F . = 1 + 2 + g (12)
= t
(13) = 12t
2 (14)
, x1
1
x2
2, L -
, x1 = x2 = L = R (15).
1,
2
. . h h = x
1+ x
2 x
1= x
2.
.
167
3. ( ). ---. w
1, w
2, w.
F. .
t0= 0 -
w
1 w
2 :
w1= w
1R
1= m
1gR = 2mgR
w2
= w2R
2= m
2g 3R = 3mgR
w2>
w1
2 2 , 1 1 - .
( )
1 w
1
1 .
2 w
2
2 .
w ,
1, 2 F
. ,
1=
1
2=
2(-).
, : 1 1 .() -
, 2 2 .() , 1 = .() = R1 2 = .() = R2
1 1 () , 2 2 () , 1 = () = R1 2 = () = R2
x1 = R1 x2 = R2.
:
T1 = 1 (1)T2 = 2 (2)1 = R1 (3)2 = R2 (4)1 = R1 (5)2 = R2 (6)
1 , 2 .
1
FZ= m
1
1
1 w
1= m
1
1
1 m1g = m11 (7)1 = 1t
(8)
x1 =121t
2 (9)
2
FH= m
2
2
w2 T
2= m
2
2
m2g T2 = m22 (10)2 = 2t (11)
x2 =122t
2 (12)
.
= . 2R
2
1R
1=
= I.(1)
(2)
2R2 T1R1 = I= I. (13)
F = 0 F. 1 2 w . = 0
(1)
(2)
F . = 1 + 2 + w. (14)
= t (15) =12t
2 (16)
: x
1
1 L
1 , x1 = L1 = R1 (17)
x2
2 L
2 x2 = L2 = R2 (18)
h h = x1+ x
2 x
1 x
2.
.2 - (I)
, : . = 0 . = 0
= . 1R T2R = 0 T1 = T2
168
12
E
1 m
1= 4 kg,
2 m
2= 6 kg
. 2
. t0= 0 . :
) 1,
2
) ) ) t = 4 s.
, R = 0,1 m, g = 10 m/s2.
m1= 4 kg , m
2= 6 kg , g = 10 m/s2 , I. = 0 , R = 0,1m
) 1= ; ,
2= ;
t0= 0 ,
1
w1 -
1,
2
2
.
1 w
1
1.
2 w
2, N
2.
1, 2 F
.
. ,
1=
1
2=
2(-).
, : 1 , 2 1 , 2 . -
, 1 = 2 = . = R 1 , 2 1 , 2
, 1 = 2 = = R.
:
1 = 1 (1)2 = 2 (2)1 = 2 = = R (3)1 = 2 = = R (4)
1 , 2 .
1
F = m11(4)
w1 T
1= m
1
m1g T
1= m
1
40 1= 4
1 = 40 4 (5)
2
F = m22(4)
2= m
2
2 = 6 (6)
.
= .
1R
2R = 0
1=
2
(1)
(2)
1 = 2 (7)
(7) (4)
(6) 40 4 = 6 10 = 40 = 4 m/s2
1=
2= 4 m/s
2
) = ;
(4) =R = 40 rad/s
2
) 1= ; , T
2= ;
(6) (7)
1=
2= 24
) t = 4 s , 1= ; ,
2= ; , = ;
1,
2 , :
1=
2= t
1=
2= = 16 m/s
(3) = R = 160 rad/s
: , = t = 160 rad/s.
169
7. , M = 4 kg R = 0,2 m, , . , m = 2 kg. t
0= 0 -
, - .
) - .
) F. ;
) t1= 2 s -
. ) 0 t
1 -
. I
cm=
1
2MR
2 g = 10 m/s2.
= 4 kg , R = 0,2 m , m = 2 kg , Icm
=
1
2MR2 , g = 10 m/s2 .1
) = ; , = ; ( ). t
0= 0 -
, : o w
-
-
. ( ). w
T
. , w
.,
F. .
, T = (-). :
. , = . = R -
, = = R x = R.
:
= (1) = . = R (2) = = R (3)
m .
F = m w T = m mg T = m = mg m = 20 2 (4)
= t (5)
x = 12t2 (6)
.
= .
R = 12
MR2(1)
= 12R
(3)
= 12
= 2 (7)
F = 0 F. w . = 0
(1)
F. = g + T F . = 40 + (8) = t (9)
= 12t
2 (10)
. . (4), (7) 20 2 = 2 4 = 20 = 5 m/s2 .
(3) = R =R =
5
0,2rad/s2 = 25 rad/s
2
170
12
) T = ; , F. = ; (4) T = 20 2 5( ) N T = 10 N (8) F. = 40 + 10( ) F. = 50 F
. -
.) t
1= 2 s ,
1= ; ,
1= ;
(5) 1= t
1
1= 5 2 m/s
1= 10 m/s (9)
1= t1 1 = 25 2 rad/s 1 = 50 rad/s
1 : (2) 1 = 1R 1 =
1
R
1=
10
0,2rad/s
1= 50 rad/s
) L = ; , N = ; , x L
0 t1= 2 s.
L = x(6)
L =1
2t
1
2 L = 12 5 22 m L = 10 m
t1 (10) :
= 1
2t1
2 = 12 25 22 rad = 50 rad
N = 2 = 25
.
8. 1
2, m
1= 4 kg m
2= 1 kg ,
M = 20 kg R = 0,1 m.
1,
2
. t0= 0 -
, (-) .
) . )
1,
2
. )
F. . ) t
1
1
2 h = 8 m :
i) 1,
2
ii) iii) iv) t
0= 0.
cm
=1
2MR
2 g = 10 m/s2.
m1= 4 kg , m
2= 1kg , M = 20 kg , R = 0,1m , I
cm=
1
2MR2 , g = 10 m/s2 .1
) ( ). ---. - w
1, w
2, w. F
.
. t0= 0
w
1 w
2 :
w1= w
1R = m
1gR = 4 N m
w2
= w2R = m
2gR = 1 N m
w1>
w2
, 1 1 2 2 .
L - - L = R.
171
) 1= ; ,
2= ; , = ;
( ). 1 w
1
1 .
2 w
2
2 .
w.,
1, 2 F
.
. ,
1=
1
2=
2(-).
, : 1 , 2 .
, 1 = 2 = . = R 1 , 2
, 1 = 2 = = R x1 = x2 = R.
:
1 = 1
(1)
2 = 2 (2)1 = 2 = == . = R (3)1 = 2 = == = R (4)
1 , 2 .
1
F = m11 w
1 T
1= m
1
1
(4)
m1g T
1= m
1
1= m
1g m
1
1 = 40 4 (5)1 = = t (6)x1 =
12t2 (7)
2
F = m22
2 w
2= m
2
2
(4)
2 m
2g = m
2
2= m
2g + m
2
2 = 10 + (8)2 = = t (9)
x2 =12t2 (10)
.
= .
1R
2R =
=
1
2MR2
(1)
(2)
1
2=
1
2R
(4)
1
2=
1
2
1 2 = 10 (11)
F = 0 F. 1 2 w . = 0
(1)
(2)
F. = 1 + 2 + g F . = 1 + 2 + 200 (12)
= t (13) =12t
2 (14)
1,
2 .
H . (11)
(5)
(8) 40 4 10 + ( ) = 10 40 4 10 = 10 15 = 30 = 2 m/s2
(4) 1=
2= = 2 m/s2
(4) = R =R =
2
0,1rad/s2 = 20 rad/s
2
) 1= ; , T
2= ; , F. = ;
(5) T1= 40 4
1= 40 4 2( ) 1 = 32
(8) 2= 10 +
2= 10 + 2( ) 2 = 12
(12) F. = 1 + 2 + 200 F. = 32 + 12 + 200( ) F. = 244 . F. - .
) h = 8 m i)
1= ; .
2= ;
, x1
1 -
x2
2, L
, x1 = x2 = L =12t2 ( (7) (10)).
h h = x1+ x
2 x
1= x
2, x1 = x2 =
h2 x1 = x2 = 4 m.
x1=
1
2t
1
2 t1=
2x1
t
1=
2 42
s t1 = 2 s.
(6), (9) 1=
2= t
1
1=
2= 2 2 m/s
1=
2= 4 m/s
172
12
ii) 1= ;
1 (13)
1= t1 1 = 20 2 rad/s 1 = 40 rad/s
2 (3)
1=
1R
1=
1
R
1=
4
0,1rad/s
1= 40 rad/s
iii) L = ;
L = x
1= x
2= 4 m
iv) N = ;
t1 (14) :
= 12t1
2 = 12 20 22 rad = 40 rad
N = 2 = 20
.
9. (1) (2) , R
1= 0, 1 m R
2= 0,3 m M
1= 9 kg
M2= 1 kg . -
, , . , - , . .
1,
2
m1= 2 kg m
2= 1 kg.
1,
2
. t0= 0 -
, . ) . )
. )
1,
2 F.
. ) t
1
1,
2 h = 16 m.
) t1;
Icm
=
1
2MR
2 g = 10 m/s2.
R1= 0,1m , R
2= 0,3 m , M
1= 9 kg , M
2= 1kg , m
1= 2 kg , m
2= 1kg , I
cm=
1
2MR2 , g = 10 m/s2 .1
) ; ( ). ---. - w
1, w
2, w.
F. .
t0= 0
w
1 w
2 :
w1= w
1R
1= m
1gR
1= 2 Nm
w2
= w2R
2= m
2gR
2= 3 Nm
w2>
w1
- , 1 - 1 2 - 2 .
L - - L = R.
173
) = ; , 1 = ; , 2 = ; ( )
1 w
1
1 .
2 w
2
2 .
w.,
1, 2 F
.
. ,
1=
1
2=
2(-).
, : 1 1 .() -
, 2 2 .() , 1 = .() = R1 2 = .() = R2
1 1 () , 2 2 () , 1 = () = R1 2 = () = R2
x1 = R1 x2 = R2. :
I. = (1) + (2) =1
2
1R
1
2 +1
2M
2R
2
2=
1
2 9 0,12 + 1
21 0,32
kgm2 = 0,09 kgm2, . = 0,09 kgm2.
:
T1 = 1 (1)T2 = 2 (2)1 = R1 (3)2 = R2 (4)1 = R1 (5)2 = R2 (6)
1 , 2 .
1
FZ= m
1
1
1 w
1= m
1
1
(5)
T1= m
1g + m
1R1
1 = 20 + 0,2 (7)1 = 1t (8)
x1 =121t
2 (9)
2
FH= m
2
2
w2 T
2= m
2
2
(6)
2= m
2g m
2R2
T2 = 10 0,3 (10)2 = 2t (11)
x2 =122t
2 (12)
.
= . 2R
2
1R
1= I.
(1)
(2)
2R
2 T
1R
1= I.
0,32 0,1
1=
1= 0,09 32 1 = 0,9 (13)
F = 0 F. 1 2 w . = 0
(1)
(2)
F. = 1 + 2 + 1 +2( )gF . = 1 + 2 + 100 (14)
= t (15) =12t
2 (16)
(13) (7)
(10) 3 10 0,3( ) 20 + 0,2( ) = 0,9 30 0,9 20 0,2 = 0,9 2 = 10 = 5 rad/s2
(5) 1= 5 0,1m/s2
1= 0,5 m/s
2
(6) 2= 5 0,3 m/s2
2= 1,5 m/s
2
) 1= ; , T
2= ; , F. = ;
(7) T1= 20 + 0,2 5( ) N T1 = 21 N
(10) T2= 10 0,3 5( ) N T2 = 8,5 N
(14) F. = 21 + 8,5 + 100( ) N F. = 129,5 N ) h = 16 m , t
1= ;
x1+ x
2= h
(9)
(12)
1
2
1t1
2 +1
2
2t1
2= h 1
2
1+
2( )t12 = h t1 = 2h1+
2
t1= 4 s.
) 1= ; ,
2= ;
(8) 1=
1t1
1= 0,5 4 m/s
1= 2 m/s
(11) 2=
2t1
2= 1,5 4 m/s
2= 6 m/s
174
12
10. - m1= 2 kg, =
= m2= 4 kg R
1= 0,1 m, R
2= 0,2 m -
. m = 2 kg. . . t
0= 0 .
) ,1 , , 2 - (1) (2).
) (2). ) F.,1 F.,2 . ) t
1= 2 s :
i) ii) (1) (2) iii) t
0= 0.
Icm
=
1
2MR
2 g = 10 m/s2.1.088 = 33, 1.664 = 40,8.
m1= 2 kg , m
2= 4 kg , m = 2 kg , R
1= 0,1m , R
2= 0,2 m , g = 10 m/s2 , I
cm=
1
2MR2 , 1.088 = 33
1.664 = 40,8
) = ; , ,1 = ; , ,2 = ; --. w
1, w
2, w
F.,1 , F.,2
. t0= 0
w
.
,1 ,2 , - .
w
1 .
(1) w
1,
1, 2
F.,1 -
. (2) w
2,
2
F.,2 .
, 1=
1
2=
2(-).
, : .
, = .(1) = 1R1 = .(2) = 2R2
, = (1) = ,1R1 = (2) = ,2R2 x1 = R11 x2 = R22 .
175
:
T1 = 1
(1)
T2 = 2 (2) = 1R1 = 2R2 (3) = ,1R1 = ,2R2 (4)
.
FH= m
w T1= m
1= mg m
1 = 20 2 (5) = t (6)
x = 12t2 (7)
(1) (2) .
(1)
1= .(1) ,1
1R
1
2R
1=
1
2m
1R
1
2 ,1
(1)
(2)
1
2=
1
2m
1R
1 ,1
(4)
1
2=
1
2m
1
1 2 = (8)1 = ,1t (9)
1 =12 ,1t
2 (10)
(2)
2= .(2) ,2
2R
2=
1
2m
2R
2
2 ,2
T2=
1
2m
2R
2 ,2
(4)
2=
1
2m
2
2 = 2 (11)2 = ,2t (12)
2 =12 ,2t
2 (13)
(8) (5)
(11) 20 2 2 = 5 = 20 = 4 m/s2
(4) = ,1R1 ,1 =R
1
,1 =4
0,1rad/s2 ,1 = 40 rad/s
2
(4) = ,2R2 ,2 =
R2
,2 =4
0,2rad/s2 ,2 = 20 rad/s
2
) 1= ; , T
2= ;
(5) T1= 20 2 4( ) N T1 = 12 N
(11) T2= 2 4 N T
2= 8 N
) F.,1 = ; , F.,2 = ; , F
= 0.
(1): F = 0 F
x= 0
Fy= 0
F.,1x = 2F.,1y = w1 + 1
(1)
(2)
F.,1x = T2
F.,1y = m1g + T1
F.,1x = 8 NF.,1y = 32 N
F.,1 F.,1 = F.,1x2 + F.,1y
2= 82 + 322 N = 1088 N = 33 N F.,1 = 33 .
(2): F = 0 F
x= 0
Fy= 0
F.,2x = 2F.,2y = w2
F.,2x = 8 NF.,2y = 40 N
F.,2 F.,2 = F.,2x2 + F.,2y
2= 82 + 402 N = 1644 N = 40,8 N F.,2 = 40,8 .
) t1= 2 s ,
i) = ; ,
ii)
1= ; ,
2= ; ,
iii) N
1= ; , N
2= ;
i) (6) = t1 = 4 2 m/s = 8 m/s
ii) 1 (9)
1= ,1t1 1 = 40 2 rad/s 1 = 80 rad/s
(12) 2= ,2t1 2 = 20 2 rad/s 2 = 40 rad/s
2
(3) = 1R
1=
2R
2
1=
R
1
2=
R
2
. 1= 80 rad/s,
2= 40 rad/s.
iii) (10) 1=
1
2 ,1t1
2 1=
1
2 40 22 rad 1 = 80 rad. 1 =
1
2
1=
40
.
(13) 2=
1
2 ,2t1
2 2=
1
2 20 22 rad 2 = 40 rad. 2 =
2
2
2=
20
.
176
12
11. , 1,
2,
, . 1
m1= 5 kg,
2 m
2= 4 kg M = 2 kg
R = 0,1 m. - . = 30.
1,
2 . -
t0= 0
. ) . )
1,
2
. )
1,
2.
) 2
t1
1,
2 h = 9 m.
) t1
1. :
i) 2
ii) t1
.
cm
=
1
2MR
2 g = 10 m/s2.
m1= 5 kg , m
2= 4 kg , M = 2 kg , R = 0,1m , = 30 , I
cm=
1
2MR2 , g = 10 m/s2
) ; ( ). --. w
1, w
2, w.,
F
. .
t0= 0 w
1 w
2,x
:
w1= w
1R = m
1gR = 5 Nm
w2,x
= w2,x
R = m2g 30 R = 2 Nm
w1 > w2,x
, 1 1 2 2 .
177
) 1= ; ,
2= ; , = ;
( )
1 w
1
1 .
2 w
2,
2
.
w.,
1, 2 F
. -
. ,
1=
1
2=
2(-).
, : 1 , 2 .
, 1 = 2 = . = R 1 , 2
, 1 = 2 = = R x1 = x2 = R.
:
1 = 1 (1)2 = 2 (2)1 = 2 = = R (3)1 = 2 = = R (4)
1 , 2 .
1
F1= m
1
1
(4)
w1 T
1= m
1
T1= m
1g m
1
1 = 50 5 (5)1 = = t (6)
x1 =12t2 (7)
2
F2= m
2
2
(4)
2 w
2x= m
2
2= m
2g 30 + m
2
2 = 20 + 4 (8)2 = = t (9)
x2 =12t2 (10)
.
= .
1R
2R =
1
2MR2
(1)
(2)
1
2=
1
2R
(4)
1
2=
1
2
1 2 = (11) = t
(12)
= 12t
2 (13)
(11) (5)
(8) 50 5 (20 + 4) = 30 9 = 10 = 30
= 3 m/s2 ,
1=
2= = 3 m/s2.
(4) = R =R =
3
0,1rad/s2 = 30 rad/s
2
) 1= ; , T
2= ;
(5) T1= 50 5 3( ) N T1 = 35 N
(8) T2= 20 + 4 3( ) N T2 = 32 N
) h = 9 m , 2= = ; ,
1= ;
(7), (10) 1,
2 x
1= x
2=
1
2t2.
1 h
1= x
1.
2 h
2= x
2 30 = 1
2x
2.
h1+ h
2= h x
1+
1
2x
2= h
(7)
(10)
1
2t
1
2 +1
2 1
2t
1
2= h 3t
1
2= 4h t
1
2=
4h
3 t
1=
4h
3 t
1=
4 93 3
s t1 = 2 s.
(9) 2= = t
1
2= 3 2 m/s
2= 6 m/s
(12) 1= t1 1 = 30 2 rad/s 1 = 60 rad/s
178
12
) i) = ; , = ; t
1= 2 s -
,
2
. , T = (14).
2
, = = R (15).
-
0=
1= 60 rad/s.
:
= (14) = R (15)
2 .
F = m2
w2x T = m
2
T = m2g 30 m
2
= 20 4 (16)
.
= .
R = 12
MR2 (14)
= 12R
(15)
= 12
=
(17)
(16), (17) 20 4 = 5 = 20 = 4 m/s2 .
(15) = R =
R =
4
0,1rad/s2 = 40 rad/s
2
ii) N = ;
t1:
= 0 t (18)
= 0 t
12 t
2 (19)
, = 0, :
(18) 0 = 0 t t =
0
t = 60
40s t = 1,5 s
(19) = 60 1,5 12 40 1,52
rad = 45 rad
= 2 = 22,5
.
179
. T ( ) cm
:
.
: xOy. x x cm . y y .
. Fx Fx = mcm (1). Fy :
Fy= 0
(2),
cm .
Fy= m (3), cm .
.
: .
(cm)
= Icm (4)
, - .
Fx = mcm (1) (cm) = Icm (4). cm . cm , , .
.
180
12
.3
1. (I) R. . t = 0 - , -. .
cm
=
1
2MR
2.
( ) - ( ).
( ). w T
.
.
cm -
. ,
= cm .() = cm R (1). , = = 0 (1) cm = R (2).
cm
= R d
cm( )dt
=
d(R)dt
= Rddt
= R cm = R (3)
.() , .() = = 0.
cm
() . :
.() = cm () 0 = cm R cm = R (3)
:
= cm R (1)cm = R (2)cm = R (3)
F = Mcm
w T = Mcm
T = Mg Mcm (4)cm = cmt (5)
xcm =12cmt
2 (6)
(cm)
= Icm
R = 12
MR2
= 12R
(3)
= 12cm
(7)
= t (8)
= 12t
2 (9)
L xcm , :
L = xcm = R (10).
181
2. (I) R. - . t = 0 , - F
.
, . - . -
Icm
=
1
2MR
2.
, - ( ) ( ).
( II). w T
.
. 0 = 0, :
= t (1) , x =12t
2 (2)
cm
. , ,
, = .() cm = R cm (3). , , : = = R cm (4)
, .() > cm R > cm .
= R cm
ddt
= R ddt
d cm( )
dt = R cm (5)
.() , .() = .
cm
() . :
= .() = ( ) cm = R cm = R cm (5)
:
= t (1)
x =12t
2 (2)
= R cm (3) = = R cm (4) = R cm (5)
F = Mcm
w T = Mcm
T = Mg Mcm (6)cm = cmt (7)
xcm =12cmt
2 (8)
(cm)
= Icm
(cm)
= Icm
R = 12
MR2
= 12R (9)
= t (10)
= 12t
2 (11)
L xcm
. L = xcm + x =12cmt
2 +12t
2= R (12)
cm Rcm Rxcm R
182
12
.4
, , :
.
,
.
, -
. . . , -
, .
1. , R . t
0= 0 F
-
. . . F.
cm
=
1
2MR
2.
. = ;
w
, F ,
N
. ,
. -, :
cm
. , .
F
.
F
, w
, N , .
. ,
. (
.
, ).
, cm = R (1) cm = R (2).
, :
Fx= M
cm
F T . = cm (3)
, :
(cm)
= Icm .R =
1
2MR2
. =1
2R
(2)
. =12cm (4)
(3), (4) :F T..
=
cm
1
2M
cm
F T..
= 2 F T. = 2. F = 3T. . =F
3
cm = cmt (5) xcm =12cmt
2 (6).
= t (7) =12t
2 (8).
- - .
183
2. , R . t
0= 0
0 F
. . . - F.
cm
=
2
5MR
2.
. = ;
w
, F K,
N
. ,
. -, : -
cm
. , .
F
.
F
, w
, N , ,
. . ,
.
( .
, -
). , cm = R (1) cm = R (2).
, :
Fx= M
cm
F T . = cm (3)
, :
(cm)
= Icm .R =
2
5MR2
. =2
5R
(2)
. =25cm (4)
(3), (4) :F T..
=
cm
2
5M
cm
F T..
=
5
2 2F 2T. = 5. 2F = 7T. . =
2
7F
cm = 0 cmt (5) xcm = 0t 12cmt
2 (6).
= 0 t (7) = 0t 12t
2 (8).
:
.
.
, .
T. - . .(max) = sN,
s .
0 T. T.(max) = sN . = .(max) = sN, . T.(max) T..
- - - - - .
184
12
3. R - = 30. . . w .
cm
=
2
5MR
2.
. = ;
w
, w
x
w
y, N
. , .
- , :
cm
.
, .
wx.
w
, N ,
. - . ,
.
( .
,
). , cm = R (1) cm = R (2).
, :
Fx= M
cm
wx T. = cm
Mg 30 . = cm (3)
, :
(cm)
= Icm .R =
2
5MR2
. =2
5R
(2)
. =25cm (4)
(3), (4) :
Mg 30 T..
=
cm
2
5M
cm
g 30 T.
.=
5
2
5T. = 2g 30 2. 7T. = 2g 1
2 . =
Mg
7 T. =
w
7
cm = cmt (5) xcm =12cmt
2 (6).
= t (7) =12t
2 (8).
.
185
4. R t
0= 0 -
= 30, 0.
0 -
. . w .
cm
=
2
5MR
2.
. = ;
w
, w
x
w
y, N
. , .
- , :
cm
.
, .
w
x.
w
, ,
. - .
. (
.
, ).
, cm = R (1) cm = R (2).
, :
Fx= M
cm
wx T. = cm
Mg . = cm (3)
, :
(cm)
= Icm .R =
2
5MR2
. =2
5R
(2)
. =25cm (4)
(3), (4) :
Mg T..
=
cm
2
5M
cm
g T.
.=
5
2
5T. = 2g 2. 7T. = 2g 1
2 . =
Mg
7 T. =
w
7
cm = 0 cmt (5) xcm = 0t 12cmt
2 (6).
= 0 t (7) = 0t 12t
2 (8).
186
12
5. , R - . t
0= 0
F F = 9 N. -
. F
, . -
. . -
cm
=
1
2MR
2.
F = 9 N , I
cm=
1
2MR2 ,
. = ;
w
, F , -
N
. ,
. -, : -
cm
. , .
- F
( F
). -
. .
( ). , cm = R (1) cm = R (2).
, :
Fx= M
cm
F + T . = cm (3)
, :
(cm)
= Icm FR .R =
1
2MR2
F . =1
2R
(2)
F . =12cm (4)
(3), (4) :F + T.F .
=
cm
1
2M
cm
F + T.F .
= 2 F + T. = 2F 2. 3T. = F . =F
3 T. = +3 N
, . F
-
R3
, :F
x= M
cm
F + T . = cm (5) F R
3 .R =
1
2MR2
F
3 . =
1
2R
(2)
F
3 . =
1
2
cm F 3T . =
32cm (6)
(5), (6) :F + T.F 3.
=
cm
3
2M
cm
F + T.F 3.
=
2
3 3F + 3T. = 2F 6. 9T. = F . =
F
9 T. = 1 N
, .
187
6. , R . . t
0= 0 -
F -
. . . - .
Icm
=
1
2MR
2.
. = ;
, FA= 0 F = .
, T = (-). } F = T (1)
w
, T ,
N
. , .
-, : -
cm
.
, .
.
w
,
, ,
cm .
.
. .
. , cm = R (2) cm = R (3).
, :
Fx= M
cm
T . = cm (4)
, :
(cm)
= Icm R .R =
1
2MR2
T . =1
2R
(1)
(3)F . =
12cm (5)
(4), (5) :
T.F .
=
cm
1
2M
cm
T.
F .= 2 T. = 2F 2. 3T. = 2F . =
2F
3
cm = cmt (6) xcm =12cmt
2 (7).
= t (8) =12t
2 (9).
- - .
188
12
.5
= 0 cm = R cm = R xcm = R 0 cm R cm R xcm R
( ). , . . .
1. , R . t
0= 0 ,
d = R2
, F .
. -
cm
=
1
2MR
2.
d =
R
2, I
cm=
1
2MR2 ,
;
w
, F -
N .
F :
-
cm
-
.
, :Fx = Mcm F = Mcm (1)
, :
(cm) = Icm F R2
=
12
MR2 F = MR (2)
(1), (2) cm
= MR cm = R . , = 0, cm = R, xcm = R.
2. , R . t
0= 0 -
F .
x 0 x 2R. x . = f (x).
-
cm
=
1
2MR
2.
189
0 x 2R , I
cm=
1
2MR2 , T. = f (x) = ;
w
, F ,
N
. ,
. , - : -
cm
.
- F
( F
).
. . ( ).
, cm = R (1).
, :
Fx = Mcm F + T . = cm (2)
, :
cm
= Icm F x R( ) T.R = 1
2MR2
(1)
F x R( ) T .R = 12 MRcm (3) (2), (3) :
F + T.F x R( ) .R =
cm
1
2MR
cm
F + T.
F x R( ) .R =2
R FR + T.R = 2Fx 2FR 2T.R
3T.R = F 2x 3R( ) T. = F 2x 3R( )3R
0 x 2R
2x 3R = 0 x = 3R2
, . = 0, .
2x 3R > 0 x > 3R2
, . > 0, . .
2x 3R < 0 x < 3R2
, . < 0, . .
: x = 3R2
: T . = 0, x >3R2
: T . > 0, x 2R., x
cm=
0 t =
0 >
0RT =
2T
RT = 2R xcm
> 2R.
0= 0
0 0
0=
0R
0>
0R
0<
0R
191
.7
m R t
0= 0
. K -
0 -
0.
, :)
0= 14 m/s
0= 0
) 0= 0
0= 21 rad/s
) 0= 4 m/s,
0= 4 rad/s
) 0= 10 m/s
0= 3 rad/s
) 0= 2 m/s
0= 16 rad/s;
R = 1 m, = 0,2 ( )
Icm
=
2
5mR
2 g = 10 m/s2.
R = 1m , = 0,2 ,
cm=
2
5mR2 , g = 10 m/s2 , t = ;
) 0= 14 m/s ,
0= 0
t0= 0
- = 0 , , : -
cm
, 0. :F
y= 0 w = 0 N = mg, = = mg (1)
Fx= m
cm T = m
cm
(1)
mg = m
cm
cm= g cm = 2 m/s
2 () :
cm
= Icm TR =
2
5mR2 =
2
5mR
(1)
mg = 2
5mR =
5g2R = 5 rad/s
2
cm = 0 cmt (2). = t (3).
= 0, cm = R ( ).
cm= R
(2)
(3)
0
cmt = tR 14 2t = 5t t = 2 s.
t 2 s ( ).
t = 2 s
.
: .
, , () (IV).
192
12
) 0= 0 ,
0= 21rad/s
t0= 0 -
, = 0R ,
, :
cm
-
, 0. , :
F
y= 0 w = 0 N = mg, = = mg (1)
F
x= m
cm T = m
cm
(1)
mg = m
cm
cm= g cm = 2 m/s
2 () :
cm= I
cm TR =
2
5mR2 =
2
5mR
(1)
mg = 2
5mR =
5g2R = 5 rad/s
2
cm = cmt (2). = 0 t (3). = 0, cm = R ( ).
cm= R
(2)
(3)
cmt =
0 t( )R 2t = 21 5t t = 3 s.
t 3 s ( ).
) 0= 4 m/s ,
0= 4 rad/s
t0= 0
,
0= 4 m/s
-
0R = 4 m/s .
= 0 0 R = 0, - t
0= 0 :
., .
) 0= 10 m/s ,
0= 3 rad/s
t0= 0
, -
0= 10 m/s
-
0R = 3 m/s
. = 0 0R = 7 m/s ,
, :
cm
, 0.
193
:F
y= 0 w = 0 N = mg, = = mg (1)
Fx= m
cm T = m
cm
(1)
mg = m
cm
cm= g cm = 2 m/s
2 () :
cm
= Icm TR =
2
5mR2 =
2
5mR
(1)
mg = 2
5mR =
5g2R = 5 rad/s
2
cm = 0 cmt (2). = 0 + t (3). = 0, cm = R ( ).
cm= R
(2)
(3)
0
cmt =
0+ t( )R 10 2t = 3 + 5t t = 1 s.
t 1 s ( ).
) 0= 2 m/s ,
0= 16 rad/s
t0= 0
-, -
0= 2 m/s
0R = 16 m/s -
. = 0R 0 = 14 m/s ,
, :
cm
-
, 0. :F
y= 0 w = 0 N = mg, = = mg (1).
Fx= m
cm T = m
cm
(1)
mg = m
cm
cm= g cm = 2 m/s
2 () :
cm
= Icm TR =
2
5mR2 =
2
5mR
(1)
mg = 2
5mR =
5g2R = 5 rad/s
2
cm = 0 + cmt (2). = 0 t (3). = 0, cm = R ( ).
cm= R
(2)
(3)
0+
cmt =
0 t( )R 2 + 2t = 16 5t t = 2 s.
t 2 s ( ).
194
12
.8 + . . .
m, R - k. - - , . d = A . , ... .
cm
=
1
2mR
2.
cm=
1
2mR2 , ... ; , T = ;
- , x . w
, N ,
F.
. ,
. F
. .
. .
. , cm = R (1) cm = R (2).
, :
Fx= m
cm
F . T . = mcm (3)
, :
(cm)
= Icm .R =
1
2mR2
. =1
2mR
(2)
. =
12
mcm (4)
(3), (4) :F. T..
=
mcm
1
2m
cm
F. T..
= 2 2. = F. T. 3T. = F. . =F.3
(5)
x .. :
Fx= T. F.
(5)
F =
F.3 F. F =
2
3F. F =
2
3kx F = 2k
3 x
F x, ...
D =2k
3 T = 2 m
D T = 2 3m
2k.
, : T. ( , )
Fx= F. = kx, ... D = k = 2
m
k.
195
12. M = 0,9 kg R = 16
m.
- . t
0= 0 ,
. .
) - .
) . ) t
1
1= 120 rad/s :
i) ii) iii) . ) t
1 .
i) . ii) t
2= 7 s ,
t
1.
cm
=
1
2MR
2 g = 10 m/s2.
= 0,9 kg , R = 16
m , Icm
=
1
2MR2 , g = 10 m/s2 .3
) cm
= ; , = ; (- ) - (- ).
( ). - w T
-
. - - .
cm
. , :
= cm .() = cm R (1) , = = 0. (1) cm = R (2)
cm
= R d
cm( )dt
=
d(R)dt
= Rddt
= R cm = R (3)
196
12
, :F = M
cm w T = M
cm
T = Mg Mcm (4)
cm = cmt (5)
xcm =12cmt
2 (6)
, :
(cm)
= Icm R =
1
2MR2
= 12R
(3)
= 1
2cm (7)
= t (8)
= 12t
2 (9)
(4), (7) Mg Mcm
=
1
2M
cm g = 3
2
cm
cm=
2g
3
cm=
20
3m/s
2.
(3) cm
= R =
cm
R = 40 rad/s
2
) T = ;
(7) T = 12 0,9 20
3N T = 3 N
) 1= 120 rad/s , i) L = ; , ii) N = ; , iii) = ;
i) L xcm , L = x
cm.
(8) 1= t1 t1 =
1
t
1=
120
40s t1 = 3 s
(6) xcm
=
1
2
cmt1
2=
1
2 20
3 9 m = 30 m xcm = 30 m
L = 30 m.
ii) (9) = 12t1
2=
1
2 40 32 rad = 180 rad = 180 rad
= 2 = 90
.
iii) :
.() = cm (1) = 1R = 120 1
6m/s = 20 m/s .() = cm(1) = 20 m/s.
= .()2 +
cm(1)
2= 202 + 202 m/s = 20 2 m/s.
= 20 2 m/s. ) i) ; t
1 w
.
-
cm.
F = M cm w = M
cm Mg = M
cm cm = g = 10 m/s
2.
cm
= cm (1)
+ cm t (9).
xcm
= cm(1) t + 1
2cm t2 (10)
1
t1, w
.
w , (
). ii) t
2= 7 s ,
2= ; ,
cm(2)= ; , x
cm= ;
t1= 3s t
2= 7 s t = t
2 t
1 t = 4 s.
,
2=
1= 120 rad/s.
(9) cm(2)
= cm (1)
+ cm t = 20 + 10 4( )m/s = 60 m/s cm(2) = 60 m/s
(10) xcm
= cm(1) t + 1
2cm t2 = 20 4 + 1
210 42
m = 160 m xcm = 160 m
197
13. M = 0,2 kg R = 0,2 m. . t
0= 0
, F
.
,
cm, = 2cm .
. : ) , -
) F ) -
0 t1, t
1= 2 s
) t1.
-
cm
=
1
2MR
2 g = 10 m/s2.
= 0,2 kg , R = 0,2 m , = 2cm , Icm =
1
2MR2 , g = 10 m/s2 .3
) cm
= ; , = ; , = ; , -
( ) - ( ).
( ). w
T
. .
, : F = 0 F = = (-). F = T (1). 0 = 0, :
= t (2) x =12t
2 (3)
cm -
. ,
= .() cm = R cm (4).
, : = = R cm (5) , .() > cm R > cm .
= R cm d ( )
dt= R d
dt
d cm( )
dt = R cm (6)
, :F = M
cm w T = M
cm
T = Mg Mcm (7)
cm = cmt (8)
xcm =12cmt
2 (9)
, :
(cm)
= Icm R =
1
2MR2
= 12R (10)
= t (11)
= 12t
2 (12)
198
12
(7), (10) Mg Mcm
=
1
2MR g cm =
1
2R 2g 2cm = R (13).
(6)
(13)
= 2g 2cm cm 2cm = 2g 3cm 5cm = 2g cm =
2g
5
cm= 4 m/s
2
(13) =2g 2
cm
R =
2 10 2 40,2
rad/s2 = 60 rad/s2
= 2cm = 8 m/s2 .
) F = ; (7) T = Mg M
cm T = 1,2 N
(1) F = T F = 1,2 N
) 0 t1= 2 s , L = ; , N = ;
L xcm -
.
L = x
cm+ x
(3)
(9) L =
1
2
cmt1
2 +1
2t1
2 L = 12
cm+ ( )t12 L = 1
2 4 + 8( ) 22 m L = 24 m
t1 :
= 1
2t1
2 = 12 60 22 rad = 120 rad
:
L = R = LR = 24
0,2rad = 120 rad
= 2 = 60
.
) = ; (5) = R cm
(8)
(11) = t1R cmt1 = 60 2 0, 2 4 2( )m/s = 16 m/s = 16 m/s
: (5) =
(2)
= t1 = 8 2 m/s = 16 m/s
14. M = 4 kg, R = 0,2 m . t
0= 0 -
F 12
. . )
. ) , -
. ) t
1= 5 s :
i) ii)
d = R
iii) . ) t
1= 5 s , -
. t2= 8 s
.
cm
=
1
2MR
2.
199
= 4 kg , R = 0,2 m , F = 12 N , Icm
=
1
2MR2.
.4
) . ;
w
, F K, N
. , .
, :
cm
. , .
F
.
F
, w
, N
, , . - .
. (
T .
, ).
) cm
= ; , = ; , T. = ; , cm = R (1) cm = R (2).
, :
Fx= M
cm F T. = Mcm
. = F Mcm T. = 12 4cm (3)
cm = cmt (4)
xcm =12cmt
2 (5)
, :
(cm)
= Icm R =
1
2MR2
. =1
2R
(2)
. =
1
2
cm
T . = 2cm (6)
= t (7)
= 12t
2 (8)
(3), (6) 2cm
= 12 4cm 6
cm= 12
cm= 2 m/s
2 .
(2) cm
= R =
cm
R =
2
0,2rad/s2 = 10 rad/s
2
(6) . = 4 ) t
1= 5 s
i) 1= ; ,
cm(1)= ;
(7) 1 = t1 1 = 10 5 rad/s 1 = 50 rad/s (4)
cm(1)=
cmt1
cm(1)= 2 5 m/s
cm(1)= 10 m/s
: (1) cm(1)
= 1R
cm(1)= 50 0,2 m/s
cm(1)= 10 m/s
ii) d = R , = ;
cm -
, d = R t
1
=
cm(1)+ .() .
cm(1) = .() = 1R = 10 m/s.
= cm(1)2 + .()
2= 102 + 102 m/s = 10 2 m/s = 10 2 m/s .
200
12
iii) = ;
cm -
,
,
=
cm+ + .
cm
= = R = 2 m/s2.
= 12R = 50
2 0,2 m/s2 = 500 m/s2.
x x=
cm = 0.
= , = = 500 m/s
2.
,
.) t
1= 5 s , t
2= 8 s ,
= ;
( IV). . - , F
cm
.
:
cm
cm( 1 ) = 10 m/s 1 = 50 rad/s. , :
F
x= M
cm F = M
cm
cm=
F
M cm = 3 m/s
2
t = t2 t
1= 3s, :
cm
= cm(1)
+ cm t
cm= 10 + 3 3( )m/s cm = 19 m/s
( V). : .() = 1R .() = 10 m/s ,
cm
. ,
=
cm + ..
= cm .() = 9 m/s .
cm .
15. M = 2,5 kg, R = 0,1 m . t
0= 0
0= 100 rad/s F
7 K . F
. - .
) . ) . ) . )
t0= 0 t
1 .
Icm
=
2
5MR
2.
201
M = 2,5 kg , R = 0,1m , 0= 100 rad/s , I
cm=
2
5MR2 , F = 7 N
.4
) . ;
w
, F K, -
N
. ,
. ,
:
cm
-
. , .
F
.
F
, w
, N , -
. .
.
(
T .
,
).)
cm= ;
, cm = R (1) cm = R (2).
, :F
x= M
cm F T. = Mcm
. = F Mcm T. = 7 2,5cm (3)
cm = 0 cmt (4)
xcm = 0t 12cmt
2 (5)
, :
(cm)
= Icm R =
2
5MR2
. =2
5R
(2)
. =
2
5
cm
T . = cm (6)
= 0 t (7)
= 0t 12t
2 (8)
(3), (6) cm
= 7 2,5cm 3,5
cm= 7
cm= 2 m/s
2 .
) . = ; (6) . = 2 ) x
cm= ; N = ;
t1, = 0.
(7) 0 = 0 t1 t1 = 0 t1 =
0
(9)
cm
= R =
cm
R =
2
0,1rad/s2 = 20 rad/s
2.
(9) t1 = 5 s
0=
0R 0 = 10 m/s.
(5) xcm
= 0t1 1
2
cmt1
2= 10 5 1
2 2 52
m = 25 m xcm
= 25 m
:
(8) = 0t1 1
2t1
2= 100 5 1
2 20 52
rad = 250 rad
= 2
=
125
= 125
.
202
12
16. M = 1,4 kg R = 17
m t0= 0
, = 30, 0= 50 m/s
0. .
) . ) . ) . ) ; ) ;
cm
=
2
5MR
2 g = 10 m/s2.
= 1,4 kg , R = 17
m , 0= 50 m/s , = 30 ,
cm=
2
5MR2 , g = 10 m/s2
.4
) . ;
w
, w
x
w
y, N
. , .
-, :
- -
cm
-
. , .
w
x.
w
, ,
. .
. ( -
.
, ).
) cm
= ; , = ; , cm = R (1) cm = R (2).
, :
Fx= M
cm w
x T. = Mcm
mg . = Mcm T. = Mg cm (3)
cm = 0 cmt (4)
xcm = 0t 12cmt
2 (5)
, :
(cm)
= Icm .R =
2
5MR2
. =2
5R
(2)
. =
25cm (6)
= 0 t (7)
= 0t 12t
2 (8)
(3), (6) Mg 30 cm
=
2
5
cm g 30 = 7
5
cm
cm=
5
7g 30
cm=
25
7m/s
2 .
(2) cm
= R =
cm
R = 25 rad/s
2
) . = ;
(6) . =2
51,4 25
7 . = 2
203
) h = ; h .
30 = hx
cm
12
=
h
xcm
h = xcm2
(9)
t1,
cm= 0. (4) 0 =
0
cmt1
cmt1=
0 t
1=
0
cm
t1 = 14 s
(5) xcm
= 50 14 12 25
7142
m xcm = 350 m. (9) h = 175 m.
) s= ;
, .
.(max) , . . .(max) = T..
y , F
y= 0 N w
y= 0 N = Mg 30.
.(max) = sN T.(max) = sg 30 (10).
. T.(max) (6)
(10)
2
5M
cm
sg 30
s
2cm
5g 30
s
2 257
5 10 32
s 2 3
21.
. = .(max) = sN , s =2 321
, .
17. M = 4 kg R = 0,2 m . t
0= 0 F
-
. t = 2 s L = 4 m.
) -.
) F. ; ) t = 2 s; ) 0 2 s;
cm
=
1
2MR
2.
M = 4 kg , R = 0,2 m , L = 4 m , t = 2 s , I
cm=
1
2MR2
.7 )
cm= ; , = ;
, FA= 0 F = T = (--
). F = T (1). w
, T
, N
. , .
, :
-
cm
. , .
204
12
- (
).
.
. (
). , : cm = R (2) cm = R (3) .() , . = .() .
cm
= () = R, .() = cm + () = 2cm. = 2cm (4).
(4) : = = 2cm ddt
= 2 d
cm
dt = 2cm (4).
:
= t (5)
xA =12At
2 (6)
:cm = cmt (7)
xcm =12cmt
2 (8)
: = t (9)
= 12t
2 (10)
0 t = 2 s:
xA=
1
2t
2 xA = 2
xcm
=
1
2
cmt2 xcm = 2cm.
:
L = x
A x
cm L = 2 2cm
(4)
L = 4
cm 2
cm
cm=
L
2
cm= 2 m/s
2
(3) cm
= R =
cm
R =
2
0,2rad/s2 = 10 rad/s
2
) . = ; , F = ;
, :F
x= M
cm F + T. = Mcm
F + T . = 8 (11)
, :
(cm)
= Icm R .R =
1
2MR2
(1)
F . =1
2R
(3)
F . =
1
2
cm
F T . = 4 (12)
(11), (12) F + T. + F T. = 12 2F = 12 F = 6 N. (11) T. = +2 , .) t = 2 s , = ; = 2cm . (7)
cm=
cmt
cm= 2 2 m/s
cm= 4 m/s
= 8 m/s. ) 0 2 s , N = ; 0 t = 2 s :
= 1
2t
2 = 1210 22 rad = 20 rad
= 2 N = 10
.
205
18. M = 2 kg, R = 0,2 m - . . t
0= 0 -
F 12 -
. .
) - . F.
) - .
) ; ) t
1= 4 s :
i) ii) iii) L . ) ;
Icm
=
1
2MR
2 g = 10 m/s2.
= 2 kg , R = 0,2 m , F = 12 N , I
cm=
1
2MR2 , g = 10 m/s2 .4
) . ; , T. = f (F) = ;
, FA= 0 F = T = (-).
F = T (1). w
, T
,
N
. ,
. -
, :
cm
-
.
, .
.
w
,
, -
,
cm .
.
. . .
, : cm = R (2) cm = R (3)
, :
Fx= M
cm T. = Mcm (4)
cm = cmt (5)
xcm =12cmt
2 (6)
, :
(cm)
= Icm TR .R =
1
2MR2
(1)
F . =1
2R
(3)
F . =
12cm (7)
= t (8)
= 12t
2 (9)
206
12
(4), (7) :
T.F T.
=
cm
1
2M
cm
.
F T.= 2 . = 2F 2T. 3. = 2F T. =
2F
3 (10). . = 8 .
) cm
= ; , = ;
(4) . = cm cm =T.
cm= 4 m/s
2
(3) cm
= R =
cm
R =
4
0,2rad/s2 = 20 rad/s
2
) = ; , , = .( ) . .() () = R = cm () = 4 m/s
2 . = .() = () = 4 m/s
2.
) t1= 4 s ,
i) = ; , ii) = ; , iii) L = ;
i) , - .. = . = R.
(8) = t1 = 20 4 rad/s = 80 rad/s. = 80 0,2 m/s = 16 m/s ii) : = cm + . = 2cm = 2R = 32 m/s = 32 m/s iii) 1 = . = cm , L xcm
. L = xcm
(6)
L =
1
2
cmt1
2 L = 12 4 42 m L = 32 m
2
L = 12t1
2 L = 12 4 42 m L = 32 m
) s= ;
y , :
F
y= 0 N + T w = 0 N = Mg T
(1)
N = Mg F
.(max) = sN T.(max) = s Mg F( ) . ,
.
.(max) , .
. .(max) (10)
2F
3
sMg F( ) 2 12
3
s2 10 12( ) 8s 8 s 1.
s = 1, . = .(max) , .
19. M = 2 kg R = 0,25 m.
r = R4
. t0= 0 -
F 12
. . :
) cm -
) )
) 0 t
1= 2 s.
cm
=
1
2MR
2.
207
= 2 kg , R = 0,25 m , r = R4
, F = 12 N , Icm
=
1
2MR2
) cm
= ; , = ; , F
A= 0 F = T = (-).
F = T (1). w
, T
,
N
. , .
-, :
-
cm
-
.
,
. -
( ).
.
. ( ).
, cm = R (2) cm = R (3).
, :F
x= M
cm + T. = Mcm
(1)
F + T. = cm T. = cm F T . = 2cm 12 (4)
cm = cmt (5)
xcm =12cmt
2 (6)
, :
(cm)
= Icm Tr .R =
1
2MR2
(1)
F R4 .R =
1
2R2
(3)
F
4 . =
1
2
cm
T. =F
4 1
2M
cm T. = 3 cm (7)
= t (8)
= 12t
2 (9)
(4), (7) 2cm 12 = 3
cm 3
cm= 15
cm= 5 m/s
2 .
(3) cm
= R =
cm
R =
5
0,25rad/s2 = 20 rad/s
2
) . = ;
(7) . = 3 5( ) . = 2 , .
) .() = ; , = ;
cm -
() .
.() =
cm
+ () .
cm
= 5 m/s2 () = R = cm = 5 m/s2.
.() = cm + () = 2cm = 10 m/s2 .() = 10 m/s
2 .
, - .() . = .() .
cm
() .
.() =
cm
+ () .
cm
= 5 m/s2 () = r = R
4= 20 0,25
4m/s2 = 1,25 m/s2.
.() = cm + () = 6,25 m/s2 , = .() = 6,25 m/s
2.
208
12
) 0 t1= 2 s , = ; , L = ;
(9) = 12t1
2 = 12 20 22 rad = 40 rad
r = R4
.
L L = r L = R4 L = 0,25
4 40 m L = 2,5 m.
L : xA =
12t1
2 .
xcm =12cmt
2 .
L :
L = x
A x
cm=
1
2 cm( )t12 = 1
2 6,25 5( ) 22 m = 2,5 m L = 2,5 m
20. , , , , . m = 5 kg, m
1= 4 kg -
R1= 0,25 m, m
2= 2 kg
R2= 0,5 m. T t
0= 0 -
, , . :
) ) ,1 ,2 ) ) ) h t
0= 0 t
1
20 m/s. -
cm(2)
=
1
2m
2R
2
2
cm(1)
=
1
2m
1R
1
2. g = 10 m/s2 .
m = 5 kg , m
1= 4 kg , R
1= 0,25 m , m
2= 2 kg , R
2= 0,5 m , I
cm(1)=
1
2m
1R
1
2 , Icm(2)
=
1
2m
2R
2
2 , g = 10 m/s2
) = ; ---. - t
0= 0 ,
w . -
,1, ,
cm -
,2. w
1 .
w
1( ),
1, 2 F
. (
). w
2,
2 -
K,
2 -
. , .
209
2.
w
2,
2,
2 , .
.
. , .
, 1=
1
2=
2(-).
, : .(1) -
cm , :
= .(1) = 1R1 = cm = 2R2 (1)
cm , :
= ,1 = ,1R1 = cm = ,2R2 .
: T1 = 1 (1) T2 = 2 (2) = cm = 1R1 = 2R2 (3) = cm = ,1R1 = ,2R2 (4) , :
F = m w T
1= m
mg T1 = m (5) = t (6)
x = 12t2 (7)
(cm)=
cm(1) ,1
1R
1
2R
1=
1
2m
1R
1
2 ,1(1)
(2)
1 T
2=
1
2m
1R
1 ,1
(4)
T1 2 =12
m1 (8)
1 = ,1t
(9)
1 =12 ,1t
2 (10)
F
x= m
2
cm
(4)
T2 T . = m2 (11)cm = cmt (12)
xcm =12cmt
2 (13)
(cm)
= cm(2) ,2
T.R2 =1
2m
2R
2
2 ,2
T. =1
2m
2R
2 ,2
(4)
. =12
m2 (14)
2 = ,2t
(15)
2 =12 ,2t
2 (16)
(5), (8), (11), (14) :
mg T1+ T
1 T
2+ T
2 T. + . = m +
m1
2+ m
2+
m2
2
mg = m + m1
2+
3m2
2
50 = 10 = 5 m/s2 ) ,1 = ; , ,2 = ; (4) = ,1R1 ,1 =
R
1
,1 =5
0,25rad/s2 ,1 = 20 rad/s
2
(4) = ,2R2 ,2 =
R2
,2 =5
0,5rad/s2 ,2 = 10 rad/s
2
) . = ;
(14) T. =1
2m
2 . =
1
2 2 5 . = 5
) 1= ; ,
2= ;
(11) T2 T. = m2 2 = . + m2 2 = 5 + 2 5( ) 2 = 15
(8) T1 T
2=
1
2m
1
1=
2+
1
2m
1
1= 15 +
1
2 4 5
1= 25
) = 20 m/s , h = ;
h = x = 12t2 (7).
= 2cm cm =2
cm= 10 m/s.
(3) = cm = 10 m/s
(6) = t t = t = 2 s
(7) h = 12 5 22 m h = 10 m
210
12
21. , , , , .
m = 53
kg, m1= 2 kg -
R1= 0,2 m, m
2= 4 kg
R2= 0,4 m.
. t
0= 0 ,
, . :
) ) ,1 ,2 )
) t
0= 0 t
1= 4 s.
-
cm(2)
=
1
2m
2R
2
2
cm(1)
=
1
2m
1R
1
2. g = 10 m/s2 .
m =5
3kg , m
1= 2 kg , R
1= 0,2 m , m
2= 4 kg , R
2= 0,4 m , I
cm(1)=
1
2m
1R
1
2 , Icm(2)
=
1
2m
2R
2
2 , g = 10 m/s2
) = ; ---. - t
0= 0 ,
w .
,1, - ,
cm -
,2. w
1 .
w
1( ),
1, 2 F
. ( ).
w
2,
2 ,
2
. , .
2
(
2 ). -
.
. ( ).
, 1=
1
2=
2(-).
-, :
.(1) - ( ).
.(1) = 1R1 = 2cm = 22R2 , = .(1) = 1R1 = 2cm = 22R2 .
(1) .() :
211
( 1 ) = ,1R1 .() = cm + () = 2cm , () = cm = ,2R2 . = (1) = ,1R1 = .() = 2cm = 2 ,2R2 .
:
= 1R1 = 22R2ddt
= R1 d1dt
= 2R2 d2dt = ,1R1 = 2,2R2
: T1 = 1 (1) T2 = 2 (2) = 1R1 = 2cm = 22R2 (3) = ,1R1 = 2cm = 2 ,2R2 (4) , , :
F = m w T
1= m
mg T
1= m
50
3 T
1=
5
3
50 31 = 5 (5) = t (6)
x = 12t2 (7)
(cm)=
cm(1) ,1
1R
1
2R
1=
1
2m
1R
1
2 ,1(1)
(2)
1 T
2=
1
2m
1R
1 ,1
(4)
T1
2=
1
2m
1
1 2 = (8) 1 = ,1t
(9)
1 =12 ,1t
2 (10)
F
x= m
2
cm
(4)
T2+ T. =
1
2m
2
2 + . = 2 (11)cm = cmt (12)
xcm =12cmt
2 (13)
(cm)
= cm(2) ,2
2R
2T.R2=
1
2m
2R
2
2 ,2
2 T. =
1
2m
2R
2 ,2
(4)
2 . =
1
2m
2
cm
(4)
T2 . =
1
4m
2
T2 T . = (14)2 = ,2t
(15)
2 =12 ,2t
2 (16)
(11), (14) 22 = 3 2 = 1,5 (17). (8)
1 1,5 = 1 = 2,5 (18)
(5) (18)
50 3 2,5 = 5 12,5 = 50 = 4 m/s2
) ,1 = ; , ,2 = ;
(4) = ,1R1 ,1 =R
1
,1 =4
0,2rad/s2 ,1 = 20 rad/s
2
(4) = 2 ,2R2 ,2 =
2R2
,2 =4
0,8rad/s2 ,2 = 5 rad/s
2
) 1= ; , T
2= ; ,
. = ;
(18) T1= 10 N
(17) T2= 6 N
(11) T. = 2 2 . = +2 , .
) t1= 4 s , L . = ;
R2.
L . :
L . = R22 , 2 .
(16) 2=
1
2 ,2t1
2=
1
2 5 42 rad = 40 rad, :
L . = 0,4 40 m L. = 16 m
212
12
22.
m1= 6 kg, =
= m2= 4 kg R
1=
1
6m, R
2=
1
9m -
.
-. = 30. t
0= 0
. ) . ) . ) -
r = 130
m.
) ; ) ; .
Icm
=
1
2mR
2 g = 10 m/s2.
m1= 6 kg , m
2= 4 kg ,
R1=
1
6m , R
2=
1
9m ,
= 30 , Icm
=
1
2mR2 ,
g = 10 m/s2
) ; --. t
0= 0 w
1x w
2x :
w1x
= w1x
r = m1g 30 r
w2 x
= w2x
r = m2g 30 r
m
1> m
2, w1x > w2 x .
. -
1,
2.
. .
.(1)
.(2) .
) cm(1)
= ; , cm(2)
= ;
, 1 = 1 2 = 2 (-). , . = 0, . = . 1 r 2 r = 0 1 = 2 . 1 = 1 = 2 = 2 . , :
213
- , :
cm = cm(1) = 1R1 = cm(2) = 2R2 = .(.) = r
, : cm = cm(1) = ,1R1 = cm(2) = ,2R2 = (.) = r
: T1 = T2 (1) = cm(1) = cm(2) = .(.) = 1R1 = 2R2 = r (2)
= cm(1) = cm(2) = (.) = ,1R1 = ,2R2 = r (3)
, :
(1)
Fx= m
1
cm(1)
(3)
w1x T
1 T.(1) = m1
T1+T.(1) =m1g 30m1 1 + .(1) = 30 6 (4)
(cm)
= cm(1) ,1
T.(1)R1 =1
2m
1R
1
2 ,1
T.(1) =1
2m
1R
1 ,1
(3)
.(1) =1
2m
1
T .(1) = 3 (5)
(2)
Fx= m
2
cm(2)
(3)
2 w
2x T.(2) = m2
T2T.(2) =m2g 30+m2
2 .(2) = 20 + 4 (6)
(cm)
= cm(2) ,2
T.(2)R2 =1
2m
2R
2
2 ,2
T.(2) =1
2m
2R
2 ,2
(3)
.(2) =1
2m
2
T .(2) = 2 (7)
(4) (5)
1+ 3 = 30 6 1 = 30 9 (8)
(6) (7)
2 2 = 20 + 4 2 = 20 + 6 (9)
(1) 1=
2
(8)
(9)30 9 = 20 + 6 15 = 10 = 2
3m/s2
(3) cm(1)
= cm(2)
=
2
3m/s
2
) r = 130
m , ,1 = ; , ,2 = ; , = ;
(3) = ,1R1 ,1 =R
1
,1 =
2
31
6
rad/s2 ,1 = 4 rad/s2
(3) = ,2R2 ,2 =
R2
,2 =
2
31
9
rad/s2 ,2 = 6 rad/s2
(3) = r =r =
2
31
30
rad/s2 = 20 rad/s2
) 1= ; , T
2= ;
(8) (1)
T1= T
2= 30 9 2
3
N T1= T
2= 24 N
) .(1) = ; , .(2) = ; (5) .(1) = 2
(7) .(2) =4
3
214
12
23. m R = 0,5 m t
0= 0
, , = 0,6 = 0,8. - 12 m = 0,5 ( ).
) 0
0 .
) ; )
cm
; ) -
m = 1 kg;
cm=
2
5mR
2 g = 10 m/s
2.
R = 0,5 m , = 0,5 , = 0,6 , = 0,8 , x = 12 m , Icm
=
2
5mR2 , g = 10 m/s2
) 0= ; ,
0= ;
() () . ( , ). t
0= 0 w
,
w
x w
y, N
.
w
, N , ,
. w
x
.
. :
F
x= m w
x= m mg = m = g = 6 m/s2
x = 12t
1
2 t1=
2x
t
1=
2 126
s t1 = 2 s.
0= t
1
0= 12 m/s.
0= . = 0.
215
) t2= ;
.7
() (V) . ( ). t
1= 2 s = 0 = 12 m/s
, .
Fy= 0 N w
y= 0 N = mg , = = mg .
wx= mg = m 10 0,6 = 6m = mg = 0,5m 10 0,8 = 4m.
wx> T, :
cm
, 0. :
F
x= m
cm w
x T = m
cm 6m 4m = m
cm cm = 2 m/s
2
cm = 0 + cm t (1). :
cm= I
cm R =
2
5mR2 =
2
5mR 4m =
2
5mR =
10
R = 20 rad/s
2
= t (2). = 0, cm = R ( V).
cm= R
(1)
(2)
0+
cm t = R t 12 + 2t = 10t t = 1,5 s.
t2 t
1= t t
2= t
1+ t t
2= 3,5 s
) cm
= ; , = ; (1)
cm=
0+
cm t
cm= 12 + 2 1,5( )m/s cm = 15 m/s
(2) = t = 20 1,5 rad/s = 30 rad/s
) m = 1kg , . = ;
(V) (VI) . ( VI). w
, w
x w
y, N
. , .
. w
x .
. -
. . (VI) (
T .
, ).
, cm = R (3) cm = R (4).
, :F
x= m
cm w
x T. = m cm
mg . = m cm (5)
, :
cm
= Icm .R =
2
5mR2
. =2
5mR
(4)
. =25
m cm (6)
(5), (6) :
mg ..
=
m cm
2
5m
cm
mg .
.=
5
2 5. = 2mg 2.
7. = 2mg . =
2mg 7
. =12
7
216
12
24. m = 1 kg, R = 0,1 m k = 150 N/m. - , . , d = 0,4 m, . :
) ... ) : i) ii) iii) . ) x
1= 0,2 3 m :
i) ii) iii) . ) , t = 0
.
cm=
1
2mR
2 g = 10 m/s2.
m = 1kg , R = 0,1m , k = 150 N/m , d = A = 0,4 m , Icm
=
1
2mR2
) A.A.T. = ; , T = ; .6
(. ). - , - x . - w
, N
, F.
. , -
. - F
..
- -
. .
.
, cm = R (1) cm = R (2).
, :
Fx = mcm F . T . = mcm (3)
, :
cm
= Icm .R =
1
2mR2
. =1
2mR
(2)
. =12
mcm (4) (3), (4) :
F. ..
=
mcm
1
2m
cm
F. .
= 2 2. = F. . 3. = F. . =F.3
(5)
x .. :
F
x= . F. ( )
(5)
F
x=
F.3 F. Fx =
2
3F. Fx =
2k
3 x
F x, ...
D =2k
3 D = 100 N/m. = 2 m
D T = 2 1
100s T =
5s.
217
) i) Vmax
= ;
, , ..., .. (x = 0) . Vmax = ., . .. .
A = d = 0,4 m D = m.2 . =
D
m . = 10 rad/s.
Vmax
= 10 0,4 m/s Vmax
= 4 m/s
ii) = ; = 2cm = 2Vmax = 8 m/s = 8 m/s .
iii) cm
= ; , = ;
cm ... -
, cm
= = 2xx = 0
cm
= 0. (2) = 0. ) x
1= 0,2 3 m
i) cm(1)
= ;
cm ..
, cm(1)
= 1. .... x
1, :
K + U = .1
2m
1
2 +1
2Dx
1
2 12
DA2 m1
2= D A2 x
1
2( ) 1 = D A2 x
1
2( )m
1= 2 m/s
cm(1)
= 2 m/s.
ii) (1) = ; (1) = 2cm(1) = 4 m/s (1) = 4 m/s.
iii) cm(1)
= ; , ,1 = ;
cm(1)
= .2 x
1
cm(1)= 102 0,2 3 m/s2
cm(1)= 20 3 m/s
2
(2) cm(1)
= ,1R ,1 =
cm(1)
R ,1 = 200 3 rad/s
2
) . = f (t) = ;
0 ...
t = 0 x = +A = +0,4 m.
x = A .t + 0( ) t = 0x = +A = 0 0 = 1 0 = 2 + 2 = 0
0 0 2
0=
2
rad.
x = A .t + 0( ) x = 0,4 10t + 2
, (S.I.) (6).
(5) . =F.3 T. =
1
3kx
(6)
T. =1
3150 0,4 10t +
2
. = 20 10t +
2
(S.I.)
s= 0,5, ...
;
Fy= 0 w = 0 N = mg, .(max) = T. = sN = smg (7). . =
F .3
=
13
kx (8).
:
. T.(max)
1
3kx
smg x
3smg
k x 3 0,5 1 10
150m x 0,1m
...
max= 0,1 m.
218
12
25. , m1= m
2= 2 kg
R1= 0,4 m R
2= 0,2 m.
. . , -. t
0= 0 , -
.
) . ) ,1 , ,2 , . ) .
) t 10
: i) ii) iii) .
cm
=
1
2mR
2 g = 10 m/s2.
m1= m
2= 2 kg
R1= 0,4 m
R2= 0,2 m
Icm
=
1
2mR2
g = 10 m/s2
) cm
= ;
, , ( ) ( ).
( ) w
1,
F
.
. w
2
.
, .
, = (1) (-). , : , = . , = .() = 1R1. H
cm -
.() , = cm .() = cm 2R2
= cm 2R2 = 1R1 cm = 1R1 + 2R2 (2).
219
cm cm =dcmdt
.
cm = 1R1 + 2R2dcmdt
= R1 d1dt
+ R2 d2dt cm = ,1R1 + ,2R2 (3).
:T = (1)cm = 1R1 + 2R2 (2)cm = ,1R1 + ,2R2 (3)
F = 0 F. w1 = 0
(1)
F . = m1g + T (4)
1 = ,1t (6)
1 =12 ,1t
2 (7)
(cm)
= cm(1) ,1
R1=
1
2m
1R
1
2 ,1(1)
T = 12
m1R1 ,1 (5)
F = m
2
cm
w2 T = m
2
cm
m2g T = m2cm (8)
cm = cmt (10)
xcm =12cmt
2 (11)
2 = ,2t (12)
2 =12 ,2t
2 (13)
(cm)
= cm(2) ,2
R2=
1
2m
2R
2
2 ,2
T = 12
m2R2 ,2 (9)
(5), (9) 12
m1R
1 ,1 =
1
2m
2R
2 ,2 R1,1 = R2,2 (14) 0,4 ,1 = 0,2 ,2 ,2 = 2,1 (15)
(3) (14)
cm
= 2 ,2R2 ,2R2 =cm2
(16)
(8) (9)
m2g 1
2m
2R
2 ,2 = m2cm
(16)
g 1
2
cm
2=
cm 4g
cm= 4
cm
cm=
4g
5
cm= 8 m/s
2
) ,1 = ; , ,2 = ;
(16) cm
= 2 ,2R2 ,2 =
cm
2R2
,2 =8
0,4rad/s2 ,2 = 20 rad/s
2
(15) ,1 = ,22 ,1 = 10 rad/s
2
) T = ;
(5) T = 12
m1R
1 ,1 =
1
2 2 0,4 10
= 4
) 1=
10
i) xcm
= ;
1=
1 2
1=
10
2 rad 1 = 20 rad.
(7) 1=
1
2 ,1t
2 t = 21 ,1
t = 2 2010
s t = 2 s
(11) xcm
=
1
2
cmt2 x
cm=
1
2 8 22 m x
cm= 16 m
ii) 1= ; ,
2= ;
(6) 1= ,1t 1 = 10 2 rad/s 1 = 20 rad/s
(12) 2= ,2t 2 = 20 2 rad/s 2 = 40 rad/s
iii) N2= ;
1 2
(13) 2=
1
2 ,2t
2 2=
1
2 20 22 rad 2 = 40 rad
2 2
1=
1
2 ,1t
2
2=
1
2 ,2
