5o Κλινικό Φροντιστήριο Λειτουργικής Ουρολογίας :: Andrology Update 2015
Ορόσημο Φροντιστήριο (Αθήνα). Βοηθήματα Γ΄ λυκείου...
193
Transcript of Ορόσημο Φροντιστήριο (Αθήνα). Βοηθήματα Γ΄ λυκείου...
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Exofilo_Fisikis.qxp 29/6/2007 2:55 Page 1
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T
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.
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........................................................................ . 7
1
...................................................................... . 27
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2
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3
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-
5
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7
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............................................................................ . 176
-
7
:
...
=
- =
1
, 0 0 = 1, 0
:
1) = + 2)
= 3) () =
4) () = 5)
=
6)
=
1) ( )2 = 2 + 2 2
2) ( )3 = 3 32 + 32 3
3) 2 - 2 = ( + )( - )
4) 3 - 3 = ( - )(2 + + 2) - = ( - )(-1 + -2 + -32 + ... + -1)
5) 3 + 3 = ( + )(2 - + 2) + = ( + )(-1 - -2 + -32 + ... + -1)
6) 3 + 3 + 3 - 3 = 1/2( + + )[( - )2 + ( - )2 + ( - )2]
7) 3 + 3 + 3 = 3 + + = 0 = =
: x R, :x, x 0
| x | = x, x 0
x|x|0
0
+
x|x|0
0
xx|x|0
0
-
8
Ma
.
=
=
=
=
.
1) || 1
2) || 1
3) 2 + 2 = 1
.
y
x
y
x
x
y
=
=
=
=
: x 0, N*. 2 1= + x = x =
2= 0 x =
: , 0 :
1) =
2)
, 0
=
3) =
4) =
5) =
6)
=
7) ||R,2=
8) ( )2
=
4)
, x ( ) 2
= +
5)
, x ( )
=
6) = 1
1=
y
x
=
=
-
9
.
.
1) ,2x,2x =x
+=+= 3) x = x = + ,
2) x = x = 2 , 4) x = x = + ,
. 1
1) 2 = (+) + (-)
2) 2 = (-) + (+)
3) 2 = (-) - (+)
1)
2 2 2
+ + =
2)
2 2 2
+ =
3)
2 2 2
+ + =
4)
2 2 2
+ =
= =
2 2 2
2 2 2
2 2 2
2
2
2
= +
= +
= +
-
10
Ma
1) (+) = +
2) (-) = -
3) (+) = -
4) (-) = +
5)
( )1
++ =
6)
( )1
=
+
7) 1
( )
+ =
+
8) 1
( )
+ =
1) 2 = 2
2) 2 = 2 - 2
= 22 - 1
= 1 - 22
3) 2
22
1 =
1) 2
1 2
2
=
2) 21 2
2
+=
3) 2
1 2
1 2
=
+
: log = x x = , > 0, > 0, 1
I: i) logx = x ii)
log = v) 1
log log =
iii) log1 = 0 iv) log
= 1
> 0, 1, 1 > 0,
2 > 0, > 0
:
1) log
1
2 = log
1 + log
2
2) log
1
2
= log
1 - log
2
3) log = log
:
log = x 10x =
:
ln = x ex =
T :
log log
log = , > 0, , 0, > 0
-
11
1)
= 2
2)
=
3)
=
4)
E
2
+=
5)
1
2=
1E
2=
&
1) :
= 4r2
V = 4/3 r3
3) :
= 2r2 + 2rh
V = h r2
4) O /
= 2( + + )
V =
5) :
(. .) h . .
2
(. .) V
3
= +
=
6) :2
2
r r
r hV
3
= +
=
2) :
= 62
V = 3
-
12
Ma
,
.
:
y = (y )
x.
y x= (y x), 0
H .
-
13
y x = + (y x), 0
( )0, . 0> .
2y x , 0=
.
2y x x, 0, 0= + >
.
-
14
Ma
y , x 0
x=
x y.
( )
y 0 , x , 0x
= > +
x
y : / / ( > 0 < 0 ).
2
y , x 0, 0
x=
-
x y.
-
15
f(x) = x > 0, 1,
0 < < 1 > 1
R R
(0,+)+)+)+)+) (0,+)+)+)+)+)
xx xx
f(x) = logx > 0, 1, x > 0
0 < < 1 > 1
(0,+)+)+)+)+) (0,+)+)+)+)+)
R R
yy yy
-
16
Ma
y x, x R=
y x, x R=
xy x , x R, x 0
x= =
-
17
xy x , x R, x 0
x= =
( )y x=
1 x 1 ( )1 x 1
0> :
( ) ( ) x f y
y
-.
y : 2
= .
O : ( ) ( )f x T f x+ =
( )y x=
( )y x , 0=
-
18
Ma
( )2y x , 0= >
( )2y x , 0= >
y x, 0, = + > >
-
19
.
,
,
,
:
() -
-
.
() -
. -
. -
. -
. y
x
= = ,
, , .
-
.
-
, -
,
t
.
A
t
= .
,
x x 1
x 2
x
,
. :
.
y
x=
-
20
Ma
1
-
. x -
t
x t= -.
. -
: ( )
( )
=
.
( ) -
. x
t
= =
2
A
-
-
1t 2s=
2t 4s=
( )
4 8x 4 2 m 12m
2 2
+ += = = =
-
21
. .
:
: s
r
=
: , s
s :
r :
.
s r= 1= . 1
(rad)
.
: 1 (1 ) 1
360 .
:
( )1 2
1 2
s s rad
r r= = .
(rad).
S
r
= S 2r= , ( )o360 , -
2= rad.
: 360 2 rad.
360 2 rad -
-
22
Ma
1.
, , , ...,
..
o18 C . .
, , , ... -
, -
, .
.
.
AB
. -
..
, x
, E
, F
...
:
.
AB
. AB
1, .
.
.
. .
,
.. AA
.
-
.. AB
AB
.
, -
..
-
.
AB
-
. -
AB//
.
-
23
.
AB =
.
.
AB =
.
=
=
.
-
( ),
( ),
0 .
0=
()
=
()
2=
()
2.
,
. -
=
-
AB =
. =
: AB = = + = +
= +
.
.
=
=
, -
+
-
.
:
2 2
2 = + +
:
=
+
-
24
Ma
. ( ) 0 0 1= =
= +
: 2 2
2 0= + +
( ) ( )2
= +
= +
:
,
. ( ) 180 180 1= =
= +
>
: 2 2
2 180= + +
( ) ( )2
=
=
: (
)
. ( ) 90 90 0= =
= +
: 2 2
2 90= + +
2 2
= +
:
=
:
( ) ( ) ( ) ( ) ( ) +
+ +
= -
-
-
.
=
=
+ =
=
-
25
A
.
( ) = = +
: ( )2 2
2 180 = + +
: ( )
( )
180
180
=
+
3.
xx ,
1
x.
i=
. x -
x. xx, OK // i
x OK x i=
. x -
.
()
xx yy -
i
j
. -
(-
), yx. yx
.
xx yy . 1
K xx -
x 2
K yy
y x y
.
(x,y). o .
-
26
Ma
1.
,
4=
3=
.
+
:
. 0=
. =
.
2
=
.
. = +
0= () :
: 7= + =
.
. = +
= () :
: 1= =
( ).
. = +
2
= () :
: 2 2
5= + =
: 3
4
= =
.
= +
2 2
2
= + +
=
+
-
.
-
27
1.
A
- - - -
t
2 1
t t t=
. S.I.: 1s.
2 1x x x=
.
. S.I.: 1m.
()
.
E . Mo S.I.: 1m/s.
2 1
2 1
x x - x = =
t t - t
-
.
.
. :
( )0 0x
= x = t x - x = t - tt
: ( )0 0
x = x + t - t
0t 0= ( )
0x 0= ( ) : x t=
-
.
=
t
S.I.: 1m / s2.
. -
. S.I.: 1m.
-
28
: ( ) =
(1): - = f(t):
0=
(2): - ( ) f t= :
=
-
x -
t.
-
.
-
.
(3) - ( )x f t= :
x = t > 0.
,
.
x x 0 x
t t 0 t
= = = =
0
x x t= + 0>
0 0
0
x x x xx
t t t t
= = = =
.=
0x x t= +
-
29
A
. -
. -
.
:
( )
( )
0 0
0 0
t t t
t
t t
= = =
= +
0t 0=
0 = :
0 t= +
0t 0=
0 0= : t=
-
: 0
t=
:
2
0
1x t t
2= +
t0 = 0 x = 0 :
2
0
1x t t
2= +
t0 = 0 x
0 = 0
0 = 0:
21
x t2
=
-
: 2
0
1x t t
2=
( ):
0 0 t
t 0
= =
=
0
t 2 2 2
2 0 0 0 0
0 0 2
1 1x t t x x
2 2 2
=
= = = 2
0
x2
=
2
0
1x t t
2= +
2
0
1x t t
2=
-
30
0 t= +
2
0
1x t t
2= +
: 0
t
=
:
2
0 0
0
1x
2
= +
2 2 2
0 0 0 0 2
x 2
+= +
2 2
0
x2
=
-
(1) - = f(t):
-
-
.
( )
2 1 t t t = = =
(2) - ( ) f t= :
0
t= + 0
0> , 0>
0 0
0
t t t t
= = = =
t= 0>
0
t t 0
= = =
- -
-
-
x .
-
31
A
(3) - x = f(t):
2
0
1x t t
2= + 2
1x t
2= ,
0 0=
. .
-
().
(1) - ( ) f t= :
E t = =
.
t
=
=
(2) - ( ) f t= :
0 0> , 0 .
.
. 21
5m/s=
-
39
A
1.12
0 -
-
= 2 m/s2
t = 8 s.
-t :
.
4
.
:
. x 9m=
. o
x 64m=
1.13
-
= 10 m/s2,
t = 0 0 = 20 m/s.
-
x = 85 m.
:
7 .
1.14
-
21
2m/s=
0 10m / s= . , -
0t 0= , -
-
18m -
22
3m/s= -
.
x 0= .
: 2 A
t 2s, x 24m= =
1.15
-
-
0t 0= -
0
x 0= . :
. .
x, .
. (t), x(t).
:
. 1 2
3 4
x 50m, x 125m
x 162,5m, x 137,5m
= =
= =
1.16
(t0=0,
0=0) -
5s -
10m/s. -
12s, -
2 m/s2, -
5s . -
,
,
10s. (t), (t), x(t).
-
40
2.
.
.
, ( ) .
21N 1kg m/s= .
-
.
.
Hooke
Hooke
.
F x=
x:
: /m.
-
.
. .
: 1 2
F F F ...F
= + +
1
.
: 1 2
F 0 F F ... 0
= + + =
0=
=
,
.
-
41
A
.
F
m =
-
.
.
2
F m=
F
m
=
- .
, .
mg
= m B
mg
= .
.
E
.
.
E -t, -t, s-t
= gt
= g
21S = gt
2
-
42
( )2g 9,78m /s=
( )2g 9,83m /s= . h . .
3 -
,
.
:
.
. . - . .
. .
-
43
A
-
1. :
:
B
: .
:B B =
.
2. ( ):
N
:
N
:
: N N=
3. :'
1F
:
1F
: - 1F
2F
:
2F
: -
2F
: 1 1F F =
2 2F F
=
() : 1 2F F= .
1 1 2 2F F F F = = = .
1F
.
,
: 1 2
F F F = +
2 2
1 2 1 2F: F F F 2FF
= + +
2
1 2
F F:
F F
=
+
F F1.
-
44
: 1 2
F F F = +
F :
2 2
1 2F F F= +
F
: 2
1
F
F=
1. ()
: 1 2
F F F = +
1 2F F F= +
.
2. ()
: 1 2
F F F = +
1 2F F F=
-
.
3. .
: 1 2 3
F F F F
= + +
-
-
.
.
. , .
-
45
A
-
:
x
x
F F F
F= =
y
y
F F F
F= =
.
()
-
.
.
.
.
.
: 1x 1F F= , 1y 1F F= ,
2x 2F F = , 2y 2F F =
. ( )x yF ' F
( ) :
2xx 1xF F F= , y 2y1y 3F F F F= +
. xF
y
F
.
:
2 2
x yF F F= +
y
x
F
F=
-
46
. ;
-
F
.
:
max T =
-
-
-
.
.
1 2 3F 0 F F F 0= + + =
1 2 3 12 3F F F F F+ = =
1 2 1 2F 0 F F 0 F F= = =
. xy . -
.
. .
. .
x 1x 2xF 0 F F ... 0= + + = (1)
y 1y 2yF 0 F F ... 0= + + = (2)
. (1), (2) .
:
x
y
F 0 0 .
F 0
= = =
= 0
=
= : x
F 0=
-
47
A
: max
0 T
:
. .
. . max
T = .
. .
.
.max T = = .
. ; -
;
.
( ).
N=
. -
.
:
.
( ).
. ( ).
. .
. ( ).
2 ;
0=
()
= (. )
xx yy
xF 0=
yF 0=
x 0=
y 0=
xF 0
yF 0=
x
x
F
m=
y 0=
xF 0=
yF 0
x 0=
y
y
F
m=
xF 0
yF 0
x
x
F
m=
y
y
F
m=
-
48
2
2
:
x x
y y
F mF m
F m
==
= :
2 2x yF F F= +
:
,
t, -
, , t .
t,
,
t.
1 2
= + 1 2
x x x
= +
h
0 ,
-
. -
:
,
( )
, (-
-
).
xx: x
0= , x 0
= , 0
x t=
yy: y
g= , y
g t= , 21
y g t2
=
-
49
A
- - - ,
: 2
2
0
gy x
2=
2
y x= , ,
.
: 2h
tg
=
: max 0
2hx =
g
:
-
: 2 2
x y = +
,
: 2 2 20
g t= + y g t=
: y
x
=
. -
-
F
.
.
.
m .
-
50
1
xx yy -
.
x T =
y =
-
:
x x
FF 0 F T 0 F T T
= = = = (1)
y yF 0 T B 0 B T= = = (2)
(2) (1): F F
B B
= =
2
F
B
T
.
F 0 F B T 0 F B T T
= + + = + = =
F 0 0 = = = (3)
: F F
BB
= =
( )3F F F T
= = =
: .
.
. ()
100=
F
-
. 30= F
.
-
51
A
xx
yy .
y
=
x =
:
x xF 0 F B 0 F B 100 1/ 2N 50= = = = =
y yF 0 0
N B 100 3 / 2N 50 3N
= =
= = =
1F
2F
1F 3N=
2F 4N= -
-
. 3F
.
xx yy. 3F
-
. -
.
:
(1) x 1 3x
F 0 F F 0= = 3x 1
FF = (
3xF
1F
).
(2) y 3y 2 3y 2
F 0 F F 0 F F= = = (
3yF
2F
)
(1) (2) : 3xF 3N= 3yF 4N=
: 2 2 2 2
3 3x 3yF F F 3 4 N 5N= + = + = ,
3y
3x
F 4
F 3= =
-
52
2.1
1F
2
F
. 1
F
30=
1
F 20N= . :
.
.
. 2
F
.
: m 2kg= , 2
mg 10
s= .
1x 1 1
3NF F F 30 20 10 3
2= = = =
1y 1 1
1F F F 30 20 N 10
2= = = =
. ( )y 1y 1yF 0 N F B 0 N mg F 2 10 10 N 10N= + = = = =
. x 1x 2 2
F 0 F F 0 F 10 3N= = =
2.2
F
, -
10.
-
;
30= , 2
mg 10
s=
xB B mg= =
yB B mg= =
x xF 0 F B 0 F mg= = =
F 10m kg 2kg
1g 10
2
= = =
y yF 0 N B 0 N mg30= = =
3NN 2 10 10 3N
2= =
-
53
A
1. -
-
.
: 2 21 2 1 2
F F F 2F F = + +
: 1
2 1
F
F F=
+
F
F m m
= =
F , -
, :
0 t= ,
2
0
1x t t
2=
E T
F
.
. F
.
yy :
yF 0 N B N mg= = = (1)
xx
. :
xF m T m N m= = =
(1) mg m g= = .
. F
.
yy: y
F 0 N B N mg= = = (1)
xx: x
F m F m= =
( )
F m F mg m1
= =
.
-
54
F >
F T=
F T< .
F
0
.
F
max
F T> .
. F
.
yy :
y yF 0 F N B N mg F= + = = (1)
( -
yy
).
xx:
xxF m F T m F N m= = =
(1)
F (mg F) m = -
( ).
F
.
yy: yyF 0 N B 0= =
yN B N mg= = (1)
xx: ( )1
xxF m F T B m= =
F mg mg m =
-
F
, x
, T
. F 0=
m mg mg g g= + = + ( ).
-
55
A
. ( )F
yy: yyF 0 N B N mg= = = (1)
xx: ( )1
xF B T m+ =
F mg mg m+ = (2)
F
-
.
-
.
y x
y y y
B B B
F 0 N B 0 N B
N B
mg
= =
= = =
=
=
max T = =
: xF 0= ()
x max
mg
mg mg
= =
= =
.
T 0= .
-
56
2.3
m 2kg= .
0t 0= F 8N=
. :
. .
. .
.
t 10s= .
: 0,6= , 0,8= , 2
mg 10
s= .
xF F 8 0,8N 6,4= = =
yF F 8 0,6N 4,8= = =
. y y
y
F 0 N F B 0
N mg F N 15,2N
= + =
= =
. 2x
xx
FF m F m 3,2m /s
m= = = =
. :
t 3,2 10m / s 32m / s= = =
2 21 1
s t 3,2 10 m 160m2 2
= = =
2.4
m 1kg= -
F
F 20= . -
;
.
:
30= , 2
mg 10
s=
xx yy
.
F
B
. :
-
57
A
2.5
F
-
30= F 2t= (S.I).
.
: m 1kg= , 2
mg 10
s= .
x
y
F F
F F
=
=
: N 0=
: yy
y
F 0 N F B 0
F B mg 10N
= + =
= = =
:
y
10F F 30 F N 20N
0,5= = =
F 2t 20 2 t t 10s= = =
x
1B B mg 30 1 10 N 5
2= = = =
y
3B B mg 30 1 10 N 5 3
2= = = =
x
3NF F 20 10 3
2= = =
y
1F F 20 N 10
2= = =
yy : y y y
F 0 N B F 0= =
( ) ( )y yN F B 10 5 3 N 5 2 3 N= + = + = +
xx xx xF (F B )
> :
xx xF F B 5(2 3 1)N
m m
= = =
-
58
2.6
m
30=
. , .
x 4m= m
4s
= . 2
g 10m/ s= ,
. ;
. , .
. h 4,5m= -
.
. 0
0= . -
:
t t
= =
21
x t2
=
2 2
2 x 2m / s2 2x
= = =
:
xF m =
mg m =
2 2 g 5m/s 2m/s= = .
y yF 0 N B mg= = =
. : x
F m =
mg T m = mg m = mg m 3
mg 5
= =
.
:
h h S 9m
s 30= = = =
:
t t
= =
21
S t2
=
2
2
1 S
2 = 2S 6m / s= =
-
59
A
2.7
m 1kg= -
0,5= . F
-
, 0,6= 0,8= .
2g 10m/ s= .
. F
.
. F ,
x 10m= .
. y yF 0 N mg F mg F= = + = +
x x
F 0 F T= =
F =
( )F mg F= +
mgF 10
= =
. F 20N= .
y yF 0 N mg F mg F 22= = + = + =
x
F m = xF T m = 2x
F 5m/s
m
= =
:
t=
21
x t2
= 2 x
t 2s
= =
-
60
2.8
1
m 4kg= 2
m 6kg= -
30= , -
3
6
= . 2
g 10m/ s= .
. F
, ,
, 4m / s= .
. ,
. -
F
, .
. 4m / s = = , :
y y F 0 N B m g= = =
x xF 0 B F T 0= + =
xF T B=
F m g=
F m g m g 25= =
F
,
.
. , -
. :
1y 1 y 1
F 0 N B m g= = =
1x 1 1 x 1 1
F m B T F m = =
21 1
1
m g m g F 15 m/s
m 4
= =
-
. :
1 0
1
t 16t s
0 15
= = =
=
2
0 1
1 32x t t x m
2 15= =
* 2
1
15 m/s
4
= .
-
61
A
2.9
1
m 1kg= 2
m 2kg=
0,5= .
, .
2g 10m/ s= :
. F
2 2m/ s=
. . 0,6=
0,8= .
.i. ( )1m :
y 1 1y 1F 0 N B m g= = =
x 1 T 1 1x 1F m F T B m = =
( )T 1 1 1F m g m g m 1 =
ii. ( )2m :
y 2 2 2F 0 N B m g= = =
x 2 2 T 2F m F T F m = =
( )2 T 2
F m g F m 2 =
iii. :
( )x 1 2F m m = +
( )T 2 T 1x 1 1 2F F T F B T m m + = +
( )2 1 1 1 2F m g m m = +
F 26N=
. (2) T 2 2F F m g m =
TF 12=
2.10
0t 0=
0
10m/ s= . 10s, -
;
2
mg 10
s= .
-
62
.
0= :
2 20
0
100 t m / s 1m / s
t 10= = = =
: xF m T m= = (1)
yF 0 N B 0 N mg= = = (2)
( ) ( )1 , 2 1
T m mg 0,1g 10
= = = = =
2. 11
m 1kg=
0,2= . 0t 0=
F 8N= .
5s,
.
2
mg 10
s= .
:
yF 0 N B 0 N mg 10N= = = =
T N 0,2 10N 2N= = =
2
x 1 1 1
F TF m F T m 6m/s
m
= = = =
1 1 t 30m / s= =
2
1 1 1
1x t 75m
2= =
: 2x 2 2 2
F m m 2m/sm
= = = =
2 2 2
2
0 t t 15s
= = =
2
2 2 2 2
1x t t 225m
2= =
: ( )1 2
t t t 5 15 s 20s= + = + =
: ( )1 2
x x x 75 225 m 300m= + = + =
-
63
A
2.12
30= .
0,2 3= .
10m / s .
: 2
mg 10
s=
yyF 0 N B 0 N mg= = =
m g = =
x xF m B T m= =
2 21 3
10 0,2 3 10 m / s 2m / s2 2
= =
: 10
t t s 5s 2
= = = =
2 2
1 1s t 2 5 m 25m
2 2= = =
2.13
m 10 3kg= 20m / s=
F 100N=
60= .
1t 10s= F
. :
.
. .
: 2
g 10m/ s=
. : F
xF
yF
x
1F F 100 50
2= = =
y
3F F 100 50 3
2= = =
.
=
-
64
: x x x
F 0 F T 0 F T T 50N= = = =
:
y yF 0 F N B 0
= + =
yN B F mg F 50 3N= = =
50 3T
350 3= = = =
. F
, -
( ) 2x ' 10 3
F m T ' m m / sm 3
= = = =
T' ' ' mg 100N= = = 02 2
t t 2 3s
= = .
2.14
-
30= 20m / s= . -
3
5
= :
. -
.
.
, .
( 3
'4
=
2g 10m/ s= ).
. -
.
: y x
B mg, mg= =
x
B
T
, -
: ( )2
0
s 12
= . y y
I yy: F 0 N B mg
T mg
= = =
= =
-
65
A
: ( ) 2x xF m T m mg mg m 8m / s
= = + = =
(1) : 2
0
s S 25m2
= = .
. , -
. :
x
1 3 3 1 3 T mg mg
2 4 2 2 8> > > > >
.
: xx x
TF m T m
m
= = =
2' 2m/s =
2
2
1 2 S 2 25mS t t t 5s
2 2m/s
= = = = t 10m / s= = .
2.15
h 1,25m= -
10m ( )
. At (
,
30= ).
x
2
F m mg m
g 5m / s
= =
= =
. :( )
( )h h
AZAZ
= =
( ) 2 2 11 h 1
AZ t g t t 1s2 2
= = =
Z 1 Z t 5m / s= =
: .
: ( ) 2 2Z t t 2s= = 1 2t t t 3s= + =
-
66
2.16
m 2kg= -
F
:
. -
t 8s=
. (, t)
. 0 4s : F 50N= -
.
2
1 1 1
FF m 25m/s
m= = =
1 1 1 t 100m / s= =
4 6s : F 0= 2
0=
2 1 100m / s= =
6 8s : F 20N= ,
-
0 2
100m / s= = .
2
3 3 3
FF m 10m/s
m= = =
3 0 3 3
t 80m / s= = 8s
. (, t)
-
67
A
2.17
-
:
. 1
m 2s
. 2
m 2s
.
: 1 2
m m 5kg= = , 2
g 10m/ s=
m1
0,25= .( )
2
: 1
m : ( )1 1 1 1F m m m g m 1= = =
2
m : ( )2 2 2 2 2F m m m g m 2= = =
( ) ( ) 22 1 1 21 2 m g m g m m 3,75m / s+ = + =
. 1
m t 2s= : 2
1 1
1S t S 7,5m
2= =
. 2 2
t 7,5m / s= =
. ( ) 1 11 T m m g 31,25N = + =
(
, 1 2
= = )
2.18
-
. , -
, ,
-
1,4m .
0,86m
i.
ii. -
;
: 2
mg 10
s=
-
68
i. O : 2
2h 2 0,86mt 0,41s
g 10m / s
= = =
xx:max 0 0
x t 1,4 0,41= = 0
3,41m / s=
ii. K: y
x 0
gt 10 0,41 1,20
3,41
= = = =
2.19
h 320m=
0
100m/ s= . -
1 10m / s= . -
s
.
2g 10m/ s= .
: 2h 2 320
t s 8sg 10
= = =
( )1 1m
x t 10 8s 80ms
= = =
:
1 0 1x S x t S t= + = +
800 S 80 S 720m= + =
:
-
- xx ... -
- yy
-
- -
-
69
A
2.20
m 2kg= .
:
. 24m/s .
. 2m / s .
. 2
4m/s .
. g=
2
mg 10
s=
:
. F 12N= . F 20N=
. F 28N= . F 0=
2.21
F
-
30= F 2t=
(S.I).
.
: m 1kg= , 2
mg 10
s= .
: t 10s=
2.22
m 1kg=
F 5N=
0,1= .
32m,
4s. :
. -
.
. .
.
.
2
mg 10
s=
:
. t 8s=
. 2
36m / s=
.
x 136m=
2.23
m 4kg=
-
F
-
:
.
t 8s=
. (, t)
:
. 3
34m / s=
-
70
2.24
1m 5kg=
2m 3kg= -
.
. .
.
d 10m= ;
2
mg 10
s=
:
. 2 2,5m / s= . t 2sec=
2.25
m 1kg= -
60=
F
-
2 2m/s= .
;
: 3
5
= , 2
mg 10
s=
: ( )F 30 3 10 N= +
-
71
3.
A
.
:
f:
,
Nf
t= , :
1f
T= , S.I.
11Hz 1s
=
.. :
:
T 1min 60s= =
1 1f Hz
T 60= =
:
T 1h 3600s= =
1 1f z
T 3600= =
:
T 12h 43200s= =
1 1f Hz
T 43200= =
.
, s
t
= ,
s: t.
SI (m / s)
-
72
t, .
t ,
s 2R= , 2R
T
= .
, - , .
-
, t
t=
:
:
2 rad, :
2=
: rad / s SI
1
fT
= 2f=
2R
T R
2
=
= =
,
() .
f
2R 1
2RT
= = 2Rf=
-
73
A
:
,
(
) ,
,
( -
). :
:
0= .
: 2
R= ,
R:
-
-
-
: ( )
22 2 2
R R
R R R= = = 2 R=
0
t t 0
= =
=
0 t= +
(t)
( rad)
( -
)
(t) -
-
t
= =
-
74
,
.
t= S.I.
2
rad1s
-
( )
:
0= >
0
t t 0
= =
2
0
1 t t
2= +
0 t= +
(t) -
-
( rad)
-
(t)
t= =
(t) -
-
t= =
(t) -
-
.
-
75
A
:
0= < 0 t= 2
0
1 t t
2=
o -
. -
(). 2
, , -
.
,
, .
F
. :
: F m=
2m
FR
=
2
mF
R=
: ( )
:
f
: ( )2 2 2 2
2
m m m 4 R fF 2Rf
R R R
= = = 2 2F 4m Rf=
-
76
3.1
. 33 (LP) 45
(single). 45 6 cm.
3s :
.
. 3s
. 3s.
. 45 45 1min,
: N 45 3
f Hzt 60s 4
= = = .
3 rad
2f2 s
= =
.
t 4,5 radt
= = =
. 2s
s R 4,5 6 10 mR
= = = s 0,27 m.=
3.2
,
. , -
:
. 900 . 1800 . 3600 . 600
:
=
t t=
1 1 2t
=
t
1 12
=
t h
1 12
1 12
=
( )
t h 111
212
=
1h= ,
T 12h=
-
77
A
.
2
= ( )3
1 t h11
=
. = ( )6
1 t h11
=
. 2= ( )12
1 t h11
=
.
3
= ( )2
1 t h11
=
3.3
200
R m
= -
3m / s= B
2m / s= .
:
.
.
.
:
A S t 3t= =
B BS t 2t= =
(2R) .
, .
( )A B200
S S 3 2R 3t 2t 6 s
= + = +
t 1200s=
.
.
( )A B200
S S 3 2R 3t 2t 6 s
+ = + = 5t 1200s= t 240s=
-
78
3.4
1R 20cm=
1
2f Hz
= .
2
R 10cm= ,
R 50cm= . :
. f2
.
.
. .
. -
-
.
1 1 2 2
1 1 1 1
2
2 2 2
R f f R
2R f R f 4f Hz
R 2R f
= =
= = ==
. -
( -
) 2
4 rad 2f 2 8
s= = =
. :
R 4m / s= =
.
() (x) -
(s)
:
( ):t
x s=
x s
t t=
4m / s= =
-
79
A
3.5
d 2m= (
)
400m / s= . -
,
,
o
45= .
.
o 45= :
2 rad
t rad
4
/ 4 Tt T s
2 8= =
:
d t 2 400 0,04s
8= = =
1 2f 2
= = 50 rad / s=
3.6
m
L. -
-
. -
. :
. -
.
.
F
yF F = , x
RF F F
L= =
y: y y
F 0 F mg F mg= = = ( )mg
F 1
=
-
80
x: 2
x
F m
R=
( )2 21m mF mg
R R= = R g =
. 22R R
2 R g
= =
R
2g
=
3.7
-
f 0,1Hz= . -
,
.
0,2=
-
d,
. 2
g 10m/ s= 2 10= .
.
: 2
max
m F mg
d
2 2
mg 4 f dm
2 2
gd
4 f d 5m
3.8
,
0,9m= .
. -
, ,
. 2
g 10m/ s= .
,
B
A
N
, -
.
:
K A 2
2
K
F N mgm
mgmF
= +
+ ==
-
81
A
A
N 0= :
2mmg g 3m / s= = =
3m/s . ( -
).
3.9
m 0,1kg= -
1m= . 2
g 10m/ s= :
.
1
4m / s=
. .
-
.
. ()
T
B
.
K 2 2
1 12
1
K
F T Bm m
T B T mg T 0,6NmF
= +
+ = = ==
.
( )T 0=
2 2m mB mg g 10 m / s= = = =
-
82
3.10
-
m 100g=
-
R 50cm= .
.
10
25 ;
. 25 -
;
. t 13s= . ( )2 10
. 10s
1 1 1 t 18 10rad 180 rad= = =
15s : 2 2 2
t 10 15rad 150 rad= = =
1 2
330 rad= + =
N 165 2
= = .
. 165 x 165 2R 165 m= =
:
x 165 33 m/s m / s
t 25 5= = =
. t 13s= 10 rad / s= 2
2
K
F m m R 50N
R= = =
3.11
8cm .
5m/s. -
.
2R 125
T f z 10Hz 2R 4
= = =
3.12
1200kg -
72km/h. 40m. -
, .
-
83
A
:
2 2
4
m 1200 20T N 1,2 10 N
R 40
= = =
km 1000m72 72 20m/s
h 3600s= =
3.13
20m/s= . -
1R 0,5m=
2R 1m= .
.
:
2
1 1
2
2 2
20f Hz, 800m / s
10f Hz, 400m / s
= =
= =
3.14
-
R 1m= , .
m / s= .
.
-
;
: 24 m/s
9=
3.15
-
-
.
R 2m=
. -
-
- 0,32= .
min
.
2
mg 10
s= 2 10
: min
.f 12
min=
3.16
-
1f 20Hz=
1R 1m= .
, -
2f 60Hz= ;
: 2
1R m
3=
-
84
4.
; -
.
:
.
,
.
1: m1 m
2 ,
.
:
F
: m1 m
2 ().
F'
: m2 m
1 ().
3 . : F F' =
:
1
B
, 2
B
.
1
N
, 2
N
.
;
.
p
.
p
: m
p
.
p
m
.
p m=
: .
S.I.: m1kgs
( 1N s )
-
85
A
...
-
, ( ), :
o p . p p= =
2
-
.
:
F m F m
t= =
m mF m
t t
= =
p pF
t
=
p
Ft
=
.
-
:p
Ft
=
.
.
.
,
:
p m =
, :
p m =
-
(...):
p . p p 0 p p= = = +
0 m m = +
m
m=
.
, o
.
-
86
.
(..) -
-
-
.
-
, 1
m , 2
m
:
1 2
1 2 1 2
F F m m
t t= =
1 1 2 2 1 2m m p p
= =
( )1 1 2 2 1 2 1 2p p p p p p p p
= + = + ( ) ( ) p p
=
.
1
()
.
. N .
.
, t.
. 0
p p p=
0
0
p m p m
p p p m m 2m
= =
= = =
.
p 2mF
t t= =
-
87
A
2
1m 60kg= 2m 40kg= , -
1
3m / s= 2 2m / s= .
-
t 0,1s= . :
.
. F1,2
2
1
. F2,1
1
2
. ...
p p
= ( )1 1 2 2 1 2 m m m m V = + V 1m /s=
. 2 :
1
1,2
pF
t=
1, 1,
1,2
p pF
t
=
1 k 1 1
1,2
m V m F
t
=
1,2F 1200=
. 2 :
2
2,1
pF
t=
2, 2,
2,1
p pF
t
=
( )2 k 2 22,1
m V m F
t
=
2,1F 1200= +
:
p p p = ( ) p p p
= +
3
m 100kg= 1
0,3m / s=
2 0,4m / s= .
.
-
t
:
1
1,2
pF
t=
, 2
2,1
pF
t=
, -
.
( )
= +
-
88
4.1
m 50g= -
300m / s= .
M 1kg= .
. .
.
200m / s= , ,
.
. -
0,5= . ;
. . -
2
g 10m/ s= .
. 3p m 50 10 kg 300m/s 15kg m/s= = = .
. ,
( ) F 0= :
( ) p p m m m m / 5m/s= = + = =
. :
2Fx M g g 5m/s= = = = = =
. : ( )2
1x t t 1
2=
( )0 t 2= .
(2)
t 1s
= = ( )1 x 2,5m =
p p p =
( ) p p p
= +
( ) ( )( )2 21 2p m m = + 2p 50kg m/s=
p 4
p 3= =
-
89
A
4.2
1
m 20kg= 2
m 5kg=
1
3m / s= , 2
2m / s=
, .
. -
.
.
.
. ;
;
. : 1 1 2 2
p m m 50kg m / s= =
. , -
( )1 2 m m 50kg m / s 25 50m / s 2m / s+ = = = .
. 1
m :
( )1 1, 1, 1 1 1p p p m m 20 2 20 3 kg m / s 20kg m / s= = = =
2
m :
( )2 2, 2, 2 2 2p p p m m 5 2 5 2 kg m / s 20kg m / s= = = + =
1 2
p p 0+ = ( ,
p 0= -
p .= ).
4.3
m 20g=
1kg= -
1m.
, -
10 m / s= .
. -
.
.
;
2g 10m/ s= .
-
90
, F 0= :
. ( )( )
m 1,02 10p p m M m m / s 51 10 m / s
m 0,02
+ = = + = = =
. -
, ( )F M m g= +
( )( ) ( )
( )2 2
M m M m T M m g m g T 20,4N+ +
+ = = + + =
4.4
1
m 6kg= 2
m 4kg=
1 8m / s=
2 9m / s=
.
. -
.
.
.
.
1
p ,
;
.
. , 1
m 1, 1 1p m 48kg m/s= =
2
m 2, 2 2p m 36kg m/s= =
,
:
2 2
1,A 2,Ap p p 60kg m / s= + =
-
2,
1,
p 48 4
p 36 3= = =
. :
( )
1 2
1 2
pp p m m p 6m / s
m m= + = = =
+
-
91
A
. ,
.
.
4.5
M 20kg= -
0 30= .
m 20kg= -
, -
10m / s= . -
-
3 / 3 .
. .
. .
. ;
. ;
2g 10m/ s= .
. :
( ) p p M m m 5m/s= + = =
. ( ) ( ) ( )x x x
F M m T M m m = + + = + + = +
( ) ( ) ( )M m g m g M m + + + = +
2 g g 10m/s = + =
. : ( )2
1x t t 1
2=
( ) t 2=
(2) 0= t 0,5s= ( )1 x 1,25m =
.
, x :
( )x
B M m g 200= + = ( ) m g 200= + = .
x
F 0= , .
-
92
4.6
m ,
. -
2.
.
-
.
yy: ( ) ( ) 1 2 y y yp p 0 p p = =
1 2 y 2
m2p p p
2
= =
( )2 2m
m 2 12
= =
xx:
( ) ( ) 2 x x
2 2
p p m p
mm 2 (2)
2
= =
= =
(1) (2): 2 2
=
1 45= =
(1) : 2
2 45=
2 2
2 2 2
2
2
= =
4.7
1
m 10tn= -
0 10m/ s= . t 6s= ,
,
2
m 30tn=
.
-
. -
0,1= . -
:
-
..
( ) ( ) x xp p=
( ) ( ) y yp p=
-
93
A
. -
.
.
.
. m1:
yy: 5y 1 1
F 0 N B m g 10 N= = = =
xx: x 1 1 1 1 1 1 1
F m T m m = = =
1
1
1
m
=
5
2
1 4
0,1 10 1m/s
10
= =
: 2 201 11 1
S t t 10 6 1 6 m2 2
= =
1
S 42m=
:
( )0 1
t 10 1 6 m/s= = 4m / s=
.
1 1 2p . p p m (m m )V= = = +
4
10
4
1 2
m 10V 4m / s 1m / s
m m 4 10= = =
+
:
yy: 4 2 5y 1 2
F 0 N (m m )g (4 10 kg)(10m / s ) 4 10 N= = + = =
xx:
x 1 2 1 2F (m m ) T (m m )= + = +
1 2
1 2
(m m )
m m
= + =
+
5
2 2
4
0,1 4 10 m / s 1m/s
4 10
= =
. -
.
:
2 2V 1S m
2 2 1= =
S 0,5m=
-
94
4.8
1
m 0,1kg= -
1
400m / s=
2
m 2kg=
.
'1 = 100m/s
1t 0,1s= . :
.
. .
. .
'
1 1 2 1 1p . p p m m V m = = = +
'
1 1 2 1 1m m V m = +
'
2 1 1 1 1m V m m =
'1
1 1
2
mV ( )
m=
V 15m/s=
. F
2 , .
( ) 2
p m V 0 30kgm / sF
t t 0,1s
= = =
F 300N=
4.9
6m=
120kg= -
.
m 60kg= , , -
.
;
.
, -
V
, ,
F 0=
-
t
:
1
1,2
pF
t=
, 2
2,1
pF
t=
-
95
A
p . p p 0 0 m MV= = + = +
( )0 m MV 1=
t x ,
x , :
xV
t=
x
t
=
(1) :
x x0 m Mx m mx
t t
= =
(m M)x m+ = m
x(m M)
=+
x 2m=
4.10
m
5m / s= -
.
-
-
.
( )12
12
=
, :
( ) ( ) ( )2 2
2 2 22 2 2 2 2 2 2
o 1 2 1 1 2 2 1 2
m mp p p m m m m
2 2
= + = + = +
2
1
2 2 2 2 2 2 2 2
2 2 2 2 21 2 1 1 1 1
m m 52m
4 4 4 4 4 16 16
= + = + = + =
2 2 2
2
1 1 1 1
16 16 16 5 m/s 4 5m/s
5 5 5
= = = =
12
2 5m/s
2= = .
-
96
4.11
0,48kg=
. -
- -
0,4= . m 20g= -
150m / s=
.
:
.
. -
.
:
. V 6m / s= . S 4,5m=
4.12
, -
1m 100g=
2m 25g= .
-
1 25cm / s=
2 50cm / s= .
,
.
: u 30cm / s=
4.13
995g= -
. m 5g=
200m/s=
.
.
: 1m / s
4.14
1
m 800kg= -
.
2
m 200kg= , -
. -
m 5kg=
800m/s= . -
S 2m= . -
- .
2g 10m/s=
:
2 2
2
1 2
m 0,4
2(m m ) gS= =
+
4.15
m 100g=
-
1 10m / s= -
2
8m / s= .
. ,
.
. t 0,09s= -
-
. (
).
:
. mp 1,8gs
=
. F 20N=
-
9 7
5.
A
1.
, ,
.
,
.
xW F x=
xF F =
W F x =
(S.I) 1Joule 1Nm=
()
, B
.
()
()
B
,
.
()
()
F
, T
, -
N
, B
.
()
-
9 8
. -
.
F
.
F
W : , ,
.
BW : -
.
W
: -
.
F B W : ,
.
W 0= , BW 0= .
:
. ,
0= 0 1= W F x= .
,
.
W F x 0= > 0
0 90 < .
. ,
90= 90 0= W 0= ,
. .. , .
. ,
180= 180 1= , W F x= .
,
.
W F x 0= < 0 090 180< .
-
9 9
A
.
F =
-
,
F(x)
F
.
.
F 0W =
.
F(x),
xx -
1
x 2
x , -
F
.
.
FW =
:
F K x=
(N. Hooke)
: .
F OABW E=
F
1W F x
2=
x: F1
W x x2
= 2
F
1W x
2=
-
100
.
()
()
. 2
1K m
2=
.
1J.
,
.. .
, , , .
.
.
AW W W= +
W h B h 0= =
:
,
.
-
101
A
.
( )
( ) ( )( )
( )
A
A
W B h W 0
h W x B B h
= = =
= = = =
A AW W=
-
, .
-
. .. ,
- .
-
, - ,
h ,
( ).
h
: U mgh= .
: U 0=
: 1
U m g h=
: 2
U m g h=
1J
-
102
U :
m (1)
(2)
( )
( ) ( )1 2 1 2 B 1 2
B 1 2
2 1 1 2
U U mgh mgh mgh WW U
U U U U U
= = =
= = =
-
-
(
). -
-
.. , .
( ) ()
. 2.
1U kx
2=
:
x
( ).
1J
.
-
-
.
oW=
o
W
:
.
-
.
,
.
U W =
W
:
.
-
103
A
.
U= +
(....)
. . W W=
( )( )2 1 2 1 2 1
2 1 2 1 1 2
1 1 2 2
W W U U
U W U U W U U
U U W W
= = =
= = = + = +=
() :2 2
1 1 2 2
1 1m mgh m mgh
2 2+ = +
:
.
.. . . . . . .E U U= + = +
(....)
.
:
W= 1 2
W W ...= + +
. . 1 2K W W ... = + + . 1 2 .K W W ... K+ + + =
.
o
2 2
0
W F x m x
x2
= =
=
2 2
2 20
0 0
1 1W m m m
2 2 2
= = = =
-
104
(...)
. . . . + = +
. . . U = + . .
. = -
. -
F, . F W= .
. = . (..
, ). ,
: . Q W= =
. . . U = + . .
-
: W
t=
: W
Pt
=
S.I. 1Joule
1Ws
=
/ 31KW 10 W=
61MW 10 W= 1HP 745,7W=
. -
, -
F
, -
x t :
F
F
W F xP F
t t
= = =
( )
W xP
t t
= = =
( ).
: QP T =
-
105
A
1. ():
2 2
0
1 1 W m m W
2 2 = =
....,
:
. ,
:
.
.
.
( )
: ( )
BB hWU B h
t t t t
= = =
: B
WU B h
t t t
= =
F =
=
: F x F = ,
xF F
. xF
.
. :
W
t= , W
.
x
F = ,
:
.
.
= , :
:
: .
-
106
( )
. ( ) .
. ....
2 2
0
1 1m m W
2 2 =
: .
: W
.
F
.
:
. :
F = : W F x =
( )F f x= : W =
,
,
.
W h=
-
107
A
2. (....)
( ) ( )2 2
1 1 2 21 2
1 1E E m mgh m mgh
2 2= + = +
: ,
( ) .
:
(, ..).
3. ( ):
( ) ( )F1 2E W W E+ + =
FW : , .
W
: (,
...), .
.
.
(
)
(
)
.
-
108
:
. . ( -
), 0
,
.
g.
-
:
( ) = 2 2
0
1 1m mg m
2 2+ =
2
0 2gH= +
.
0
, (
, ).
. (....) .... ,
.
. .
....
....
. ... ...
...: ( )1 1 2 2 1 2m m m m V = +
...: K Q= +
:
( )2 2 2 1 1 2 2 1 21 1 1
Q m m m m V2 2 2
= + +
-
109
A
,
, -
.
. -
. -
. -
:
( )
2
2
mmF
B T 1
F B T
=
+ == +
T 0= . (1) :
2
,minmmg =
,min = gl
. -
. -
: ,min 0=
.
( ).
N
B
. -
,
:
( )
2
2
mmF
B N 1RR
F B N
=
+ == +
N 0= . (1) :2
,minmmg
R=
,min gR=
-
110
5.1
1
m 10kg=
2m 20kg=
1
3m / s= ,
2 2m / s= .
.
:
( ) 1 1 2 2 1 2 p p m m m m = = +
1 m/s
3 = .
:
2 2
1 1 2 2
1 1K m m 85J
2 2= + =
: ( ) 2TE 1 2 1 5
K m m J2 3
= + =
250
J3
=
:
250 85J J
3
100 x
50x 100 x 100%
51
=
= =
-
111
A
5.2
.
, .
:
m= 2
1 m
2=
(...) p p=( )
m m m
2 + = =
( ) ( )2
2
1 1 1 m m m m
2 2 4 2= + = + =
5.3
4kg= -
0,4m= .
m 500g= 52m / s=
, -
36m/ s .
. -
.
.
600.
2
g 10m/ s= .
. ... - :
TE p p m m 2m / s= + = =
. .... ( ) ( )1 2 : TE
2
U
1 g h
2
=
=
h =
( )2
2
1 g 1 1
2 2g = =
01 11 60
2 2 = = =
-
112
5.4
m 2kg= -
0 30= .
x 3,6m= -
-
.
. -
.
.
;
.
0,3= -
. 2g 10m/ s= .
.
....:
2
1U mgh m
2= =
0 21g x 30 6m/s
2 = =
. ... :
( )TE p p m m m= + =
3m / s =
. .
:
2
1K m 36J
2= = ( ) 2
1K m m 18J
2= + =
18J=
. :
( )T m m 2m = + =
2 2mg 2m g 3m/s = = =
2 2
1x t t
x 1,5m22
0 t
=
= =
=
-
113
A
5.5
15kg
F 70N= 30 .
5m 0,3.
:
. F
,
. ,
. ,
. ,
.
. ,
5m;
2
mg 10
s=
yF 0 N F30 0 F30= + = =
115 =
0,3 115 34,5= = =
.
F
3W F30 x 70 5m 175 3 J
2= = =
. T
W T x 34,5N 5m 172,5J= = =
. N
W 0=
. B
W 0=
. F T N B
W W W W W 130,6 J= + + + =
. .... K K W =
2
1m W
2=
2W m 4,17m s
= =
5.6
60 10m
. -
45 . 0,2
;
2
mg 10
s= .
-
114
=
:
( )y
F 0 N F45 1= = +
xF 0 T F45= =
( )1 F45=
( ) F45 F45+ = F45 F45+ =
( ) F 45 45 =
( )
0,2 60 0,2 60F
45 45 2 2 20,2 1 0,2
2 2 2
= = =
30F N 15 2N
2= =
F
2W F45 x 15 2N 10m
2= =
FW 150J=
5.7
m 1kg= 30= ,
0
m 20
s= .
3
5= :
. ;
. ;
. , ;
. ;
2
mg 10
s=
.y
F 0 N = =
mg= =
.... ( ) x=
K K W =
2
0 B T
10 m W W
2 = +
2
0
1m mg x T x
2 =
-
115
A
2
0m
mg x mg x2
+ = + + ( )201 g x
2= +
( )
2 2
0 20
x m2g 1 3 3
2 102 5 2
= = +
+
x 25m=
. (),
. :
B > mg mg>
1 3 3
2 5 2>
1 3
2 10>
. , (), -
, x 25m=
....
K W = 2
1m 0 x T x
2 =
21m mg x mgx
2=
( )1 3 m m
2gx 2 10 25 102 10 s s
= = =
m 10
s=
. .
: 2 2
T 0
1 1Q m m 150J
2 2= =
5.8
m 1kg=
0
m 2
s= F
,
.
0,1= ,
F
. -
2
mg 10
s= .
:
-
116
. F
x 0= x 8m= .
. F
x 8m= x 10m= .
. F
x 0= x 10m= .
. x 12m= .
. ( )F 0 8
1W 8m 6N 24J
2
= =
. ( ) ( )F 8 10
1W 2m 3N 3J
2
= =
. ( )F 0 10W 24J 3J 21J = =
. y
F 0 N B mg= = = , T N mg 1N= = =
.... x 0= x 12m=
( ) TF 0 12K W W = +
( )2 20
1 1 1 1m m 8m 6N 4m 3N 1N 12m
2 2 2 2
= +
2 20
2 2 m m W 2 6 4
m 1 s s= + = + =
5.9
m 2kg=
0,25= .
F,
x -
F 10 5x= + (S.I).
. , -
;
. .
: 0,8= , 2
mg 10
s=
.y
F 0 N F B 0= + = mg F=
( ) ( )20 10 5x 0,8 S.I= + ( ) ( )12 4x S.I 1=
, ( )1
N 0 12 4x 0= = x 3m=
-
117
A
. 2 2 1 1 0,8 0,6= = =
( ) 0,25 12 4x 3 x= = =
( )
x 0 :T 3 x 3NF
x 0 :F 10 5x 0,6 6 3x 6N
= = = >
= = + = + =
( ) ( )x
F F 6 3x 3 x 3 4x= = + = + (S.I.)
( )x
F x :
( )( )
o 0 3
3 15W 3m 27J
2
+= =
.... 0m 3m .
( )o 0 3K W = ( )2
o 0 3
1m 0 W
2
=
( )o 0 32W 2 27J m m 27 3 3
m 2kg s s
= = = =
5.10
m 20kg=
, h 1,8m= , .
, .
. ,
.
. m
2s
= .
2
mg 10
s= .
. N
-
-
. :
... () (): ( ) ( )AE E=
U U+ = +
2
1 m0 mgh m 0 2gh 6
2 s+ = + = =
U 0=
()
-
118
. .... () ()
W = 2
B T
1m 0 W W
2 = +
2
T
1m mgh W
2= +
2
T
1W m mgh 320J
2= =
-
B
W mgh=
5.11
R 0,8m= ,
m 0,5kg= . -
36%
.
-
M 0,5kg= . -
0,2= .
;
2
mg 10
s= .
. (m) ,
. -
(m) () ().
W = 2
B T N
1m 0 W W W
2 = + +
2
1m mgR 0,36mgR 0
2= +
m 2 0,64gR 3,2
s= = ( )
. ..
p p= ( )m 0 m M V + = +
m mV 1,6
m M s= =
+
. ()
oB
, o
,
-
119
A
( )y o
F 0 N m M g= = = +
( )1 T m M g= = +
.... () () W =
( )
2
T B N
10 m M V W W W
2 + = + +
( ) ( )21
m M V m M g S2
+ = +
2V
S 0,64m2g
= =
5.12
m 0,2kg= 1
h 1,8m=
,
2h 0,8m.=
:
. ,
. ,
.
2
mg 10
s=
. .... () () -
: K W = 2
1 1
1m 0 mgh
2 =
1 1
m 2gh 6
s= =
. .... () ():
K W = 2
2 2
10 m mgh
2 =
2 2
m 2gh 4
s= =
.
2 2
1 2
1 1Q m m 2J
2 2= =
-
120
5.13
m 2000kg= -
km72
h t 20s= ,
.
F 500= , -
-
(t).
0 t 20s
2
m t 1
t s= = =
: x
F m =
F F m F m F F 2500N = = + =
F F t 2500t= = = (S.I)
t 20s>
F F 500N= = =
. F 10000W= =
5.14
m 2kg= ,
F
, ,
x :
F 8 4x= + ( )F ,xm , 0 x 4m F 32 2x= ( )F ,xm , x 4m
. x 16m= m
12s
= , -
.
. ;
( ).
. ;
.
F
.
. x 4m= , -
. ; 2
mg 10
s=
-
121
A
: B
, F
, T
N
.
( )F f x= :
. (...)
x 0m= x 16m= .
2
F T
1m 0 W W
2 = + 2
T F
1W W m
2= +
( )( )T
8 24 1W 4 24 16 4 J 144J
2 2
+ = + +
TW 64J=
yF 0 = N B m g 20N= = = TW T x mg x= = 0,2=
. ( )xF F T 8 4 N 4N 0 = = = > .
xF 0 = . xF 0< .
max
= x
F 0= F T mg 4N= = =
i. 1
0 x 4m : 1
F 8 4x 4= + = 1
x 4m=
ii. 1
x 4m : 1
F 32 2x 4= = 1x 14m=
... x 0m= 1
x 14m=
2
max F T
1m 0 W W
2 = +
( ) ( )( )2max
8 24 24 412 4 14 4 0,2 2 10 14 J
2 2 2
+ + = +
max
m 148
s=
-
122
. 2
x 2
F 32 2x= .
... x 16m= 2
x x=
2
F T
10 m W W
2 = ( ) ( ) ( )2 2 2
10 144 32 2x x 16 mg x 16
2 =
2
2
2x 28x 48 0 + =
2x 1,83m=
2x 16m>
2x 26,165m= .
. x 16m= F 0= xF F T 4N = =
x
m JouleF 4 12 48
t s s= = =
. ... x 0m= x 4m=
2
F T
1m 0 W W
2 =
( )2
8 241m 4 0,2 2 10 4
2 2
+=
m 48
s=
x 4m= F 24N=
F
W JF 24 48
t s= =
:
( ) x J
F 24 4 48 20 48t s
= = =
:
( )TWQ m
T 4 48t t s
= = =
F T W W
t t t= +
5.15
m 50kg=
h 5,2m=
, -
R 2m= . :. ()
. -
()
2
mg 10
s=
-
123
A
. , N
.
() :
( )( )1gR
0N
R
mNB
R
mF
NBF
min,
2
2
=
=
=+
=
+=
.... () ():
W =
( )2,min B T N1m 0 W W W 2
2 = + +
N
W 0= ( )
( )B
W B h 2R= (
)
(2) ( )T
1mgR mg h 2R W
2= +
TW 100J=
. mgh, :
T
A
W 100u % % 3,8%
E 2600= = =
5.16
M 1kg=
m 0,1kg= . -
,
16cm= ,
,
. -
-
. 0
, -
, -
.
2
mg 10
s=
. ...
0 p p 0 m = =
( )0
1
m=
-
124
. (), (R = ).
( )
22
B TF
R
T 0 F B T
+ ==
== +
( )2
,min
,min
MMg g 2= =
. .. () ()
( ) AE E U U= + = + 2 2
,min
1 1 0 M Mg 2
2 2+ = +
Ag 4g 5g= + =
(1), (3), 0
m 5g 20 2
m s= =
5.17
M 5kg=
0,1= . -
0
100m/ s= m 0,1kg=
0
/ 2 . :
. .
. -
.
.
t = 0,01 s.
.
,
.
. ...
0 0
, , 0 0
p p m V m m m V
2 2
= = + =
0 0 m
m MV V V 1m/s2 2
= = =
.M p p p p V 0 p 5kg1m / s 5kgm / s
= = = =
(..).
p p=
-
125
A
. M MM M
p p 5kgm / sF F F 500N
t t 0,01s
= = = =
.y
F 0 0 g= = = = T g= =
....: F
W = B N T
W W W K + + =
B N
W W K 0= = =
21Mgx V2
=
2Vx
2g= x 0,5m =
TQ W= Q gx= Q 2,5J=
5.18
B 1000N= -
-
0,03= .
F 250N=
36 km/h.
:
. .
. F
.
.
.
2
mg 10
s=
:
. W 5000J= .
FW 5.681,8J=
. W 681,8J=
5.19
65kg -
20 ,
25cm. ;
2
mg 10
s=
: B
W 3.250J=
5.20
18kg -
.
( 0,6=
0,8= ) -
.
0,5= . -
:
. .
.
20m,
;
.
0
m2s
= -
, -
;
2
mg 10
s=
:
. F 90N= . F
W 108J=
. x 0,4m=
-
126
5.21
100kg -
F 500N= ,
.
,
x 10m=
1x 6m= . -
0,05= :
.
.
.
. .
2
mg 10
s=
:
. 1
W 5000J= . T
K 2700J=
. 2
W 3000J=
5.22
()
H 50m= ,
m 0,1 g= . 1
2 -
, -
d 0,5m= ,
m
4s
= .
:
.
.
.
. .
2
mg 10
s=
:
. 47,7J
= . 46m / s
=
5.23
m 2kg= ,
-
0,1= .
.
:
. F 0
6m.
. -
.
. x 6m= .
2
mg 10
s=
:
. F
W 46J= .T
W 12J=
. 34m / s=
5.24
.
F
x, -
F 1000 50x= (S.I).
:
-
127
A
. ;
. F
;
.
, ;
:
. x 20m= . 4F
W 10 J= . F 500N=
5.25
500g
-
R 1m= . - -
m
4sec
,
2
mg 10
s=
: T
W 1J=
5.26
m 2kg=
.
.
2
mg 10
s=
60=
: N 120N, N 90N= =
5.27
m
.
1T
2
T
-
1 2T T 6mg = .
5.28
1m= m.
( )3
OK4
=
-
. 0
-
, -
,
.
2
mg 10
s=
: 0
11gm / s
4
=
-
128
5.29
m 4kg=
0,1= .
F
F 100 2N=
45= .
:
.
x 1m=
. F
-
;
.
-
.
2
mg 10
s=
:
. 5m / s = . 40 =
. 0,6 =
-
129
6.
-
. , -,
.
:
Nf
t=
1
fT
= .
.
.
.
-
( )0x A t = + (..) (..).
0 ,
-
. ,
m, -
1
x
(..-
.).
x A= , -
. -
-
-
x At= .
-
130
( -)
2
2f
= = (rad/sec). t: ( rad).
(..),
. , t, -
x .. :
( ) ( )0x A t 1= +:
x: ... , A x A +A: , ...
: . ( S.I. 1rad/s)
2f 2 / = =
f: :
0t + : .
0 : .
0 x
t 0= .
0
0 2 < (0
0= x 0= 0> )
-
( ... .)
:
( ) ( ) ( )0 0 0 A t t 2= + = +
0
A= ( ).
( ... .)
:
( ) ( ) ( )2 0 0 0 A t t 3= + = +
: 20
A= ( ).
(1) (3) :
( )2 x 4=
:
.
..., , ( ).
-
131
...
m ... H F
, : ( )F Dx 5=
(-) -
. -
.
: 2D m = , .
H = F Dx ...
m
T 2D
= (6)
.
( )x f t= , ( ) f t= ( ) f t= ,
0 0= , .
( )F f x= : F Dx=
( )F f t= : ( )2F m A t=
-
132
...
,
: 2 2 2 21 1
K m m A t2 2
= = (7)
,
: 2 2 2 21 1
U Dx m A t2 2
= = (8)
D .
-
...
, -
.
2 2 2 2 2 2
1 1E U K m A t m A t
2 2= + = + =
( )2 2 2 2 2 2 21 1 1m A t t m A D A
2 2 2+ = =
( )maxU - , 100%
( )maxK .
: i. ( )U f t= , ( )K f t= ii.
( )U f x= , ( )K f x= .
E
U K2
= =
:
.
21
E x2
= x
21
E Dx2
= x .
-
133
m -
-
, -
-
. -
-
-
3. -
,
.
,
.
:
x
xF mg mg= = =
mgF x
=
mg
D =
.
: m m
T 2 2 2mgD g
= = =
.
: 3<
m g.
-
134
... .
x1
, ..
...
A ( )0x A t = + , 1x x= 1t
2t
1 2t t< 1 .
t1 1
1x
2t 2
1x .
:
2 2 x=
....
U K E+ =
2 2 21 1 1Dx m DA
2 2 2+ = 2 2 2 2 2m x m m A+ =
2 2 2 2 2 x A+ = ( )2 2 2 2 A x= ( )2 2 2 2 2 A x A x= =
:
. (+)
, (-) -
-
.
. ,
.
, ,
.
d
: d
2
=
1
x
, .
.. 1,
, : 1 0 A= = .
-
135
x, , F.
( )0x A t = +
( )0 0 0 0
t t 2
= + = + +
( ) ( )0 0 0 0 t t = + = + +
x /2 rad.
/2 rad.
x rad.
F .
0
0 x t 0=
.
00 2 < .
t = 0
.
(1), (2) (3) ,
-
:
( )0x A t = +
( )0 0 t = +
( )0 0 t = +
t 0= , x 0= 0> 0 [0,2) .
..
F m
t. 2 2 0F m m t m At m x Dx= = = = = .
2D m= . D .
:
.
. .
2
2 2 2
2
4 m mD m D m 4 T 2
D D
= = = =
-
136
A
m -
D. D = m
T 2K
= .
.. ( )
.
-
F 0=
( ) 1B F mg Kx 1= = .
A .
( ) 1F B F mg x x= = +
1F mg x x x= =
F Kx= -
D K= .
(, )
.
.
,
E
20
1U DA
2=
2
0 0
1K m
2= .
E
. 0 0
E = = = +
.
( )22 2 2 2
0 0 0
1 1 1 1U DA m A m A m
2 2 2 2= = = = = .
2 2 2 2 2 2 20
1 1 1 m m A t DA t E t
2 2 2= = = =
2 2 2 21 1
U Dx DA t E t2 2
= = = .
: 2
2
tK U E
U E t
= + =
=
-
137
6.1
-
.
2,5cm.
, -
. 2g 10m/ s= .
, F 0=E
m xF x mg
g = = = x -
.
: 3m x
T 2 2 2 2,5 10 s g
= = =
4 2
T 2 25 10 s 2 5 10 s s10
= = = .
6.2
10cm
T s= . ( )t 0= x 10cm= .
. x At= -
.
( )0x A t = + .
. 0
x 10cm= t 0= ;
x A t= ( )t 0= -
( )x 0= . t 0= x 10cm A= = .
( )0x A t = + 0 0 0
A 1 2
= = = .
x 0,1 2t2
= +
2 2 rad / s 2rad / s
= = = . , t 0=
- +,
0 ( )0x A t = + . 0 -
.
-
138
6.3
m 2kg= A 0,5m= .
1
x 0,3m=
1 4m / s= .
. .
.
2
x 0,4m= .
: 2 2 2
2 2 2 2
x A t x A t
A t t
= =
= = 2 2
2 2
2 2 2
x t t 1 1
A + = + = 2 2 2 2 2 x + =
.
. 1
x 0,3m= , 1 4m / s= , A 0,5m= :
( )2 2 0,009 16 0,25 S.I. + =
10rad / s =
2D m 200 /m= =
. 2
x 0,4m= , A 0,5m= , 10rad / s= :
( )2 2 2 2 2 22 2
x 25 16 S.I.= =
( )2
3m / s 3m / s =
6.4
2m. -
:
. x 1m=
. x 0,5m=
.
;
.
2
2
22
O max
1Dx
U U x 12
1E U A 4DA
2
= = = =
-
139
.
2 2
2 2
O max
22
1 1DA Dx
E U U UK A x2 215
1U U U xDx
2
= = = = =
. max
1 1K U E U U U E U U
2 2= = = =
2 21 1 1 2Dx DA x A 2m
2 2 2 2 = = =
6.5
100 /m
M 1kg= . 5cm
. :
.
.
. .
2g 10m/ s=
. 5
T 2 s f z 5
= = =
. max
= 2 5cm 5 10 m= =
2
max
5 2f A 2 5 10 m/s 0,5m /s
= = =
. 2 2max
5m/s= =
6.6
m 0,2kg= -
-
20/m .
-
,
. ,
.
. /2
.
-
140
. .. (
)
. (-
) x, .
F x= . -
,
() .
.2
T 2 2 10 s 0,2 s
= = =
. 2
2 21 1 A 1U x A2 2 4 8
= = =
2 2 2O max
1 1 3 E U U U A x A
2 2 8= = = =
3U= .
6.7
-
m 2kg= . 1
120N /m=
2 80N /m= . -
-
.
x1, x
2 .
( )1 1 2 2F 0 x x 1= = . () x,
. 1
1x x+
2
2x x+ .
, ( ) ( )1 1 2 2F x x x x= + =
1 1 1 2 2 2 x x x x= + + =
( )1 2 x= + .
F .. -
:
-
141
1 2
mT 2 0,2 s
= =
+
.
-
.
6.8
10
. -
20cm.
x 10cm=
x 5cm= + .
f 5Hz=
A 10cm 0,1m= =
10cm x 0= T 1
t s4 20
= = .
x 0= x 5cm= + : x A t=
( )1
5 10 2f t 10t 2 6
= = = 1
t s60
= .
x 10cm= x 5cm= + :
1 1 1t s s s
20 60 15= + =
6.9
-
100/m
M 0,9kg= . m 100g= -
50m / s= .
:
.
. .
.
... :
( )m m = + 5m / s=
.
-
142
,
=
2 10rad / s
M m= = =
+
A 0,5m
= = 2
T 0,2s
= = .
6.10
-
( )x 0,220t S.I.= . :
.
. x 0,1m=
. 2m / s= .
. .
, :
x A= + /4 x A=
3T 3 2 3 3s s
4 4 40 40= = =
. x 0,220t= x 0,1m= : 1
20t 2 6
= =
2
620t
2
6
+ +
( ) 0= 7
20t t s6 120
= + =
. t 420t m / s= = 1
2 m/s :20t 20t2 3
= = =
1
t s60
=
-
143
6.11
= 20 t = 10s.
.
.
. 75%
.
.
, g
g =16
-
.
:
.T 0,5s, f 2Hz= =
. 40
. 5
6.12
... -
A 10cm= , T 2s= -
0
0= .
. ( )x f t= ,
( ) f t= ( ) f t= .
. 1
t s2
=
;
2 10= .
:
. ( ) ( )
( )
x 0,1 t , 0,1 t
t
= =
=
. x 0,1cm=
6.13
m 0,1kg=
...
216m/s .
d = 8cm.
.
. D
.
2cm.
: 2 10=
:
.
T s10
= . N
D 40m
=
. 1m
4 10 3s
=
6.14
2m 10 kg
=
...
:
( ) 10 2t= (S.I). :
. A, o
0
.
t 0,125s=
.
10m/s=
. -
t 2s=
: 2 10=
:
. 0A 5m, T 1s, 0= = =
. 2 m
x 5 m, 5 22 s
= =
. 1
t s2
=
. F 0=
-
144
6.15
m 0,01kg= ..-
. 0,2m. D 10N /m= , -
:
. 0,1m
.
. 1
U3
=
:
. m
30s
= . 2
m 100
s=
. 3
x A2
=
6.16
m=1,5Kg ...
-
16x= (S.I). A -
=0,12J t=0
:
.
.
. -
.
.
-
.
:
. rad
4s
=
. D 24N /m, A 0,1m= =
.
2
At 0,4 4t (S.I)
At 1,64t (S.I)
= =
= =
6.17
...,
0y . -
t 0= : x 0= -
:
.
. -
;
:
. A 2
x2
=
. 1 2 3 4
T 3T 5T 7Tt , t , t , t
8 8 8 8= = = =
6.18
1
m 9kgr= -
A 0,2m=
-
N
100m
= .
2
m 1kgr=
100m/s -
.
.
:
( )2 2
1 2
m 100 1 10m/s
m m 10= = =
+
500J=
-
145
7.
Coulomb -
:
,
. -
, .
.
-
( )
eQ N q=
N = , e
q = -
(qe = 1,6 1019 C)
-
.
Coulomb
-
, -
-
.
= 1 2c c 2
Q QF k
r
2
9
c 2
N mk 9 10
C
=
Coulomb:
. ( )
. ( )
.
. 14
10 m
.
, -
(), .
-
(
q) , .
.
F =
q
. S.I. 1 N/C.
-
146
:
. A -
.
-
, (-
), -
.
: F q =
-
,
Coulomb,
:
c 2c
c 2
Q qk QF r k
q q r
= = = (1)
-
147
Coulomb
Q (
Coulomb) q
, r Q, :
A AU W = A c
Q qU k
r
= (2)
, :
Q q 0 > , , -
q
.
Q q 0 < , ,
-
q .
:
W U= (3)
( ) W U U=
W U U=
:
-
.
-
, -
-
, -
.
-
148
-
-
V
-
, q, -
, q.
A
A
UV
q= (4)
CA
A A c
Q qk
W QrV V k
q q r
= = = (5)
S.I.
Volt: 1J
1V1C
=
(1) -
:
A
A
WV
q
= (6)
A B
V V AB
V -
q
, :
A
AB
WV
q
= (7)
:
1. -
-
-
-
.
-
:
A B ABW q V =
-
149
-
() . -
.
.
C
Q
V.
( )Q
C 9V
=
S.I. Farad: 1C
1F1V
=
...
,
.
S . -
, :
0 0
SC (10)=
0
c
c=
0
SC
= .
