Механика УМК

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A BBYY				c

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

0 t . t = 0

= 0

w

m

w

w. .

o

A B BYY

c

t

dt

M

u

u ln

.

M t

M t

0

, : = 366

: = 366 .

1 ( 1)

m1 . l

m2 , .

.

.

2 m1 2gl sin	.
	m2 cos .

2 ( 2) , 12 , -

, 10 5 .

25 30° . -

. [26 , 74º.]

4.2.1.4 . .

1 ( 1)

, , -

m = 1 1			= 2 2 = 6 /c S = 10 .
Fòð = 2 .
:				:
m = 1
1 = 2				:
	2	= 6		W	W Aâñåõ .
				k2	k1
S = 10				-
Fòð = 2				-
				.

111

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

w

w.

o

.

c

— ?

A BBYY

m 22 m 12 A Aòð .

2 2

Aòð FòðS .

:

A m ( 22 12 ) FS .

2

A 1 (36 4) 2 10 36 ( ). 2

: 36 .

2 ( 2)

m =4

,

x 6 5t t2 . t1 = 10 c?

P t2 = 5 c?

:	:
m =4	:
x 6 5t t2		A12 FsdS .
t1 = 10 c					C
t2 = 5 c				x2
		A Fs dS .
				x1
A — ?	F	ma	x	,	a	x		d 2 x	2	( 2).
	x

dt2

P — ?

Fx 2 4 8 ( ).

x2

A ( 8)dx .

x1

t2 = 10 c

x2 6 5 10 100 44 ( ).

t1 = 0

x1 6 ( ).

:

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A B BYY				c

112

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A BBYY				c

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

44

w

w.

.

o

A B BYY

c

A 8dx 8( 44 6) 400 ( ).

6

P dA d (Fx dx) d ( 48 40t 8t 2 ) 40 16t 40 ).

dt	dt	dt
: = 400 ,	P = 40 .

1 ( 2) m = 80 -

a = 1 ². l = 3 ,

30°, µ = 0,15.

: 1) , ; 2) -

; 3) . [1) A = m l (a + g sin + µ g

	cos ) = 1,72 ; 2) < P > =			A	= A		= 702 ; 3) P = F at = 1,41 .]
						a / 2
				t			max
		2 ( 2) m
	F 2ti		3tj . P(t) , -

t . [ (t) = (2t³ + 3t5 )/m.]

4.2.1.5.

.

.

1 ( 1)

m

mB

F 2 i , x , B . , -

			x
.
:				:
	mB
F		i ,		,
	x2
				: 1 x , y , z 2	x		, y		, z		.
						2		2		2
				1 1 1
				F
				-
				m 1
				2 :

113

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

w

w. .

o

2

2

A BBYY

c

A12 F dS F dS cos ,

1

1

dS cos dx , x .

2 mB

x2

dx

mB

mB

A12

dx mB

.

x2

x2

x1

x2

1

x1

A12 ,

. , -

F .

2 ( 2)

,

= 8° , . -

, , -

, .

:	:
= 8°

l= S

— ?

. 2.19

-, , -

. -

.

: — ; -

; ( 0, -

); — . -

: (1) -

(2) — .

. :

Wk2 Wn2 Wk1 Wn1 Aòð1 Aòð2

Aòð1 ;

Aòð2 ;

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A B BYY				c

114

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A BBYY				c

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

w

w.

o

.

Wk 0

, ;

A B BYY

c

1

Wn1 mgh ;

Wk2 0 , ;

Wn2 0 , h 0 ;

Aòð1 Fòð S cos FòðS .

Fòð N , N — .

(

) OY :

N1 mg cos 0 ,

N1 mg cos ,

Fòð1 mg cos .

:

N2 mg 0 .

:

Aòð1 mg cos l ,

Aòð2 mgS .

:

(0 0) (0 mgh) mg cos l mg S .

	mg , l S						h			. :
				sin
	cos		h							h	h ,
			sin						sin
		1								= 0,07.
		ctg					1
					sin

: 0,07.

1 ( 1) -

Wï 2x2 3y2 0,5z . , -

x1 = 0,5 , y1 = = 1 , z1 = 0,2 x2 = 0,1 , y2 = 0,75 ,

z2 =0,1 .

115

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

w

w. .

o

,

A BBYY

c

. [ = 1,8425 ;

F=-(4xi 6 yj 0,5k );

F1 = 6,34 , F2 = 4,545 .]

2 ( 2) m = 200 , =60°

, S = 10 t = 20 .

, ,

;

[ = 10 ; W =2,5 ; W = 7,5 .]

4.3.1

»

,

1.. , -

,

.

.

2.— , . .

. — , -

.

3.. ,

,

:

N

Fi

i 1

a . m

4. :

N ma Fi .

i1

5.. -

, .

, -

, .

, -

:

m aýò mýò , mýò 1 ( ). a

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A B BYY				c

116

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A BBYY				c

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

mýò 1 .

w

o

w. .

A B BYY

c

7. -

,

Fòð N , :

Fòð .

N

.

8. ,

.

9. p , -

:

p m .

10. N

, :

N

P p .

i 1

11., -

, ( ).

12.. -

:

Pconst .

13.-

.

14.t

, -

:

P Fi t .

				i
15. A F , -
dr ,							,
F dr :
		F				cos FdScos .
A F dr				dr

16. Wk ,

N , -

:

	N		m	2
W	A ,	W			.
k	i	k	2
	i 1

.

117

F Tran

sf

D

Y

P

or

e

B

Y

m

B

buy

r

3

0

A

to

.

here

Click

w

m

w

w. .

o

17. ,

A BBYY

c

, , , -

.

18. Wï -

, , « »,

:

Wï Aêîí .

, -

, .

19. Fêîí , , -

:

Fêîí gradWï .

20.( )

.

21.( ) -

N ,

:

N

Wìåõ (Wki Wï i ) .

i1

22..

, -

, :

Wìåõ const .

23. -

N ,

:

N

Wìåõ Aíåê i . i 1

,

( . 2.20). ,

-

. , mg , Fòð ,

N . , :

			(1)
mg	N Fòð ma .

y

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A B BYY				c

118

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A BBYY				c

	N	F
l		h1
	mg
x		h2

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A B BYY				c

. 2.20

,

. , -

.

(1) OX,OÓ :

mg sin Fòð ma ,	(2)
mg cos N 0 .	(3)
-
:
Fòð N ,
N — . (3) :
Fòð mg cos .	(4)
(4) (2) :
mg sin mg cos ma ,
g sin g cos a .	(5)

cos 0 ( . ),

2

g(tg ) a . cos

,

( min ),

( a 0 ). : g(tg min ) 0 ,

:

tg min .	(6)
(6)	-

,

min .

. ,

min ( 1 ), .			t1
l .
	at2
l	1	,	(7)
2

119

						F Tran			sf
					D
			Y	P						or	e
	B	Y								m
B									buy		r
											3
											0
A								to			.
							here
					Click
	w										m
			w							o
					w. .
								:
						A BBYY				c

a 2l ,

t12

(5) :

g sin	g cos		2l	.	(8)
1	1		t2
		1

:

tg 1		2l
			.	(9)
	gt 2	cos
1		1

, , -

.

1 2 ,

l , t2 t1 .

:

g sin 2 g cos 2	2l
		.	(10)
	t2
2

(8) (10) sin 2 sin 1 -

, :

											sin				sin	2
g(sin		cos cos				sin ) 2l							1				.
													t2		t2
	2	1			2			1
													2		1
:
			sin						sin	2
			2l			1
						t2			t2
						2		1				.					(11)

gsin( 2 1)

(9) (11) -

l , .

, . (8)(10), :

sin	cos		t2
1	1		2	,
sin 2			t12
	cos 2

:

	t2	sin	t 2	sin	2
	1	1	2				.	(12)
	t2	cos	t 2
				cos		2
1		1	2

(9) ( , (11), (12)) -

,

.

AB

					F Tran			sf
				D
		Y	P						or	e
B	Y								m
								buy		r
										3
										0
							to			.
						here
				Click
w										m
		w		w.					o
								.
					A B BYY				c

120