Πανεπιστημιακή Φυσική Τόμος Α Hugh D. Young Μέρος Α - Κεφάλαιο 4 Ορμή και Ώθηση

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( , (Hugh D. Young).

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.

(Hugh D. Young).

8, .

.

.

(, ..) mail:

[email protected]

[email protected]



4: ........................................................................... 1 : ................................................. 4

.1) .................................................................................................................. 4 .2) 2 ..... 4 .3) ................................................................. 4 ....................................................... 5 .4) ........................................................................................ 8 .5) ..................................................................................... 9 .6) ......................................................................................... 11 ............................................................................ 15

: - .................................... 31 .1) .............................................................................................. 31 .2) ................................................................................ 31 - ................................................................... 33

: ........................ 36 .1) ................................................................................................... 36 .2) ..................................................................................... 37 ................................................................ 39

: ............................................................................. 51 .1) ....................................................................................... 51 ........................................................................................................... 53 .1) ............................................................................................................. 57 .................................................................................................. 58



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8-23 . 225 , (Hugh D. Young)

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8-43 . 227 , (Hugh D. Young)

. .

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1

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10

1

20.

8-44 . 227 , (Hugh D. Young)

( 2,00kg), B ( 3,00kg) C ( 4,00kg) . . ) . x y C, ;

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s

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8-47 . 227 , (Hugh D. Young)

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kg

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8-49 . 227 , (Hugh D. Young)

, 0,200kg, 0,050m. , 0,200kg, 0,300m . .

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8-50 . 227 , (Hugh D. Young)

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8-53 . 228 , (Hugh D. Young)

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8-56 . 228 , (Hugh D. Young)

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: -

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8-27 . 226 , (Hugh D. Young)

. , 0,0450kg, , 50,0m/s .

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; ;

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0



8-29 . 226 , (Hugh D. Young)

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b) , 2t ;

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3

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8-67 . 229 , (Hugh D. Young)

120 . 10,0g 500m/s.

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8-69 . 229 , (Hugh D. Young)

2,00kg jtiatF

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s500,0 ;

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j

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0

0



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kg

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321

332211 (.1.2)

cmr

,..., 21 rr

:

i

i

i

ii

cmm

rm

mmm

rmrmrmr

...

...

321

332211 (.1.3)

, :

...

...

...

...

321

332211

321

332211

mmm

mmm

mmm

xmxmxm

dt

d

dt

dx xxxcmxcm

(.1.4)

...

...

...

...

321

332211

321

332211

mmm

mmm

mmm

ymymym

dt

d

dt

dy yyycmycm

(.1.5)

(.1.4) (.1.5) :

...

...

321

332211

mmm

mmmcm

(.1.6)

H (.1.6) :

...... 332211321

mmmmmm cm



PmmmM cm

...332211 (.1.7)

Mmmm ...321 P

.

, . , , , .

.2)

:

...

...

...

...

321

332211

321

332211

mmm

amamam

mmm

mmm

dt

d

dt

da xxxxxxxcmxcm

(.1.8)

...

...

...

...

321

332211

321

332211

mmm

amamam

mmm

mmm

dt

d

dt

da

yyyyyyycm

ycm

(.1.9)

(.1.8) (.1.9) :

...

...

321

332211

mmm

amamamacm

(.1.10)

H (.1.10) :

...... 332211321 amamamammm cm

FamamamaM cm

...332211 (.1.11)

Mmmm ...321 F

.

: cmaMFFF

(.1.12)

, F

F

.

3

, 0

F .

(.1.12) : cmaMF

(.1.13)



, :

, .

: . , . .

: .

: 0

F .

cm

. P

. .

, , , . , , . , .

: ) ) .

.

02211

mm ,



. . . . , . , ( ) .

8-31 . 226 , (Hugh D. Young)

- . :

: kgm 241098,5

: kgm 221035,7

( ): mx 91038,0

0x , 0y . :

mm

xmxmxcm

kgkg

mkgxcm 2224

922

1035,71098,5

1038,01035,7mxcm6106,4

.

: mr 61038,6 .

- !

8-32 . 226 , (Hugh D. Young)

: (1) 2,00kg (4,00m, 2,00m) , (2) 3,00kg (1,00m, -4,00m) , (3) 4,00kg, (-3,00m, 6,00m). .

x

x

y



321

32211

mmm

xmxmxmxcm

kgkgkg

mkgmkgmkgxcm

00,400,300,2

)00,3(00,400,100,300,400,2

mxcm 111,0

321

332211

mmm

ymymymycm

kgkgkg

mkgmkgmkgycm

00,400,300,2

00,600,4)00,4(00,300,200,2

mycm 778,1

8-55 . 228 , (Hugh D. Young)

m m

. a) , :

2222

1

2

1 mmMK cm

, cm cm

,

cm

.

. b) , , ;

a) :

22

2

1

2

1 mmK (1)

,

cm

cm

. , cmcm

2222 (2)

:

cm

cm

. , cmcm

2222 (3)



(2), (3) (1) :

cmcmcmcm mmK

22

12

2

1 2222

cmcm mmmmmmK

222

2

1

2

1 (4)

, .

, , . , .

,

cmMmm

0

mm

, Mmm . (4)

2222

1

2

1 mmMK cm

b) cmMP

2

2

1cmM

.

, 02

1 22 mm .

:

2

min2

1cmMK

8-61 . 228 , (Hugh D. Young)

. .

L L L

L

L L

L

L L

L

a b c d

0



.

a

y x . y.

2cos22/2

cos L

yL

y

2cos2

Ly

.

b

:

23

20

321

332211 L

m

LmL

mm

mmm

xmxmxmxcm

33

20

2

321

332211 L

m

Lmm

Lm

mmm

ymymymycm

2

a 2

a 2/L 2/L

x

y

x

y

x



L/3 .

c

42

20

21

2211 L

m

Lmm

mm

xmxmxcm

42

02

21

2211 L

m

mL

m

mm

ymymycm

90 :

22 cmcm yxd 16

2 2Ld

22

Ld

4

2Ld

d

y

x

y

x

60

60sin2

L

60cos2

L

60cos2

LL



23

60cos22

60cos2

321

332211 L

m

LLm

Lm

Lm

mmm

xmxmxmxcm

6

3

3

60sin2

060sin2

321

332211 L

m

Lmm

Lm

mmm

ymymymycm

6

3L .

8-59 . 228 , (Hugh D. Young)

, , . 80,0kg, 50,0kg 20,0kg. 5,00m/s . a) , ; b) ; (: ). c) ;

a) +x.

:

:

: sm /00,5

: sm /00,5

. :

00,20/00,50,50/00,50,800 kgsmkgsmkgmmm

smkgkgkg

smkgsmkg/34,4

205080

/00,50,50/00,580

b) - .

. :



00,200,50/00,50,800 kgkgsmkgmmm

smkgkg

smkg/67,2

0,700,80

/00,50,80

(

). :

smkgkgsmkgmmmm /67,20,700,20/00,50,50

smkgkg

smkgsmkg/24,6

0,200,50

/67,20,70/00,50,50

c) :

- .

. :

00,200,80/00,50,500 kgkgsmkgmmm

smkgkg

smkg/67,1

0,1000,50

/00,50,50

(

). :

smkgkgsmkgmmmm /67,10,1000,20/00,50,80

smkgkg

smkgsmkg/67,5

0,200,80

/67,10,100/00,50,80

8-64 . 229 , (Hugh D. Young)

70,0kg 30,0kg 5,00m. 1,00m , 1,00m . , ;

.

, 0cm .

x.

1x 1x . : 2x 2x .

:



mm

xmxmxcm

111

:

mm

xmxmxcm

222

0cm :

21 cmcm xx 2211 xmxmxmxm (1)

.

, mx 00,11

mx 50,21 . 1m

1,5m ()

2x 21 xx .

: mxx 50,122 .

(1) :

22 0,3050,10,7050,20,3000,10,70 xkgmxkgmkgmkg mx 40,02

, : mmmxx 10,250,240,012 , 2,10m

.

8-73 . 230 , (Hugh D. Young)

. , .1 .2 :

xdmMxcm

1 , ydmMycm

1

, x,y dm . . L , M A . x .

a) VM / ,

x . b) x, ax

, x .



dm : dVdm

dV : AdxdV

Mdm

x, 0cmy .

,

L

cm xdmM

x0

1

L

cm dVxM

x0

1

L

cm xdVM

x0

L

cm AdxxM

x0

L

cm xdxM

Ax

0

2

2L

M

Axcm

(1)

, ALV ALVM (2)

(2) (1) :

2

2L

AL

Axcm

2

Lxcm

x .

b) L

cm xdmM

x0

1

L

cm dVxM

x0

1

L

cm AdxaxxM

x0

1 dxxM

aAx

L

cm

0

2

3

3L

M

aAxcm (3)

dmM dVM L

AdxaxM0

L

xdxaAM0

2

2LaAM (4)

(4) (3) :

3

2

3

2

3

2

LL

LaA

aAxcm

8-74 . 230 , (Hugh D. Young)

8-73 x y



t. a, taM 2

2

1 .

.

cmx .

dx y ( ). :

222 ayx . : 22 xay (1)

: tydxdmdVdm (2)

(1) (2) :

dxxatdm 22 (3)

(3) :

dxxaxM

tx

a

a

cm

22

(4)

:

22 xau xdxdu 2

:



ax : 22 xau 0 22 aau

ax : 22 xau 0 22 aau

(4) :

02

0

0

duuMt

xcm

( .

cmy .

dy

x ( b). :

222 ayx . : 2222 yax (5)

: txdydmdVdm 2 (6)

(6) (5) :

dyyatdm 222 (7)

(7) :

a

cm dyyayM

ty

0

222 (8)

:

22 yau ydydu 2

:

ay : 22 yau 0 22 aau

0y : 22 yau 2au

(8) :

022

0

2

duuM

tx

a

cm

0

3

2 3aM

txcm

M

taxcm

1

3

2 3 (9)

: taM 2

2

1

(9) :



3

4

2

1

1

3

2

2

3 a

ta

taxcm

8-75 . 230 , (Hugh D. Young)

.

l . ( ) l . ; 2 . a) . . b) , . . 2 .

a) y ym .

:

gymFy ygFy

:

0

4/l

y dyyFW 0

4/l

ydygW 32

2glW

b) 8

lycm

:

cmMgyW cmygyW 84

lgl

W 32

2glW

.



:

.1)

2 amF

. . . . , , . x.

t m x . H x

mP . dt dm , dmm . ( 0dm ). dt , dm

. ex

. :

exfuel

x dm :

exfuel dmdm

dt : ddmm .

mdmm 0dm . , d , 0d , .

-

dt :

exdmddmmP

t

dtt . :

exdmddmmm



dmdmdmddmmdmm ex

dmdmdmd ex0

dmmd ex0 dt

dm

dt

dm ex

0

dt

dmF ex0

dt

dmF ex (.1.1)

(.1.1) ( )

. dt

dm

.

dt

dm

dt

dm ex

0 :

dt

dmma ex0

dt

dm

ma ex

(.1.2)

m . dt

dmex (

) .

:

dt

dm

dt

dm ex

0 dmmd ex0

m

dmd ex

m

m

exm

dmd

00

00 lnln mmex0

0 lnm

mex (.1.3)

: m m .

(.1.3):

0

m

0m

0



8-33 . 226 , (Hugh D. Young)

. 5000kg 2000m/s,

20,02/ sm ;

dt

dm

ma ex

ex

am

dt

dm

sm

kgsm

dt

dm

/2000

5000/0,20 2skg

dt

dm/50

50kg .

8-35 . 226 , (Hugh D. Young)

0,0500kg/s, 800m/s . a) ; b) ; c) ; ;

a) dt

dmF ex skgsmF /0500,0/800 NF 0,40

b)

c) .

180 ( ).

8-37 . 226 , (Hugh D. Young)

, ; . a) 2000m/s

c31000,1 , c ,

; b) 3000m/s;



a) 0

0 lnm

mex

0

lnm

mex

exm

m

0

ln

exem

m

0

3

3

102

10

0

c

em

m

ce

m

m 7105

0

150

0

em

m 66

0

102,7 m

m

b)

exem

m

0

3

3

102

103

0

em

m

2

3

0

em

m223,0

0

m

m

8-71 . 230 , (Hugh D. Young)

. 12.000kg 9000kg . 1000kg 600kg .

ex . (

). a) 13.000kg. , ; b) , ; . . c) ;

d) ex

8,00km/s;

a) H kgkgkgm 000.13000.1000.120

kg9600

:

kgkgkgm 400.3600.9000.13

,

0

0 lnm

mex

0

lnm

mex

m

mex

0ln kg

kgex

400.3

000.13ln ex 34,1

b) H kgkgkgm 000.13000.1000.120

kg9000



1 :

kgkgkgm 000.4000.9000.13

,

0

0 lnm

mex

0

lnm

mex

m

mex

0ln kg

kgex

000.4

000.13ln ex 18,1

c) H 2 kgm 000.10

2 kg600

2 :

kgkgkgm 400600000.1

,

0

0 lnm

mex

0

ln18,1m

mexex

m

mexex

0ln18,1

kg

kgexex

400

000.1ln18,1 ex 10,2

d) ex 10,2 10,2

ex skmex /

10,2

00,8 skmex /82,3

8-72 . 230 , (Hugh D. Young)

. , . . . :

dt

dm

dt

dm

dt

dpFext

.

kxm , k kdt

dm . mgFext , :

kdt

dmmg

k ,

2

dt

dxxg



at , a . . a) , a . b) st 500,0 .

c) mgk /00,2 , st 500,0 .

a) y .

.

:

adt

d

, at 2

2

1atx

2

dt

dxxg :

22222

2

1

2

1tatagat aag 2 3/ga

: g

.

b) 2

2

1atx 2

6

1gtx mssmx 41,0500,0/90,9

2

1 22

c) kxm mmgm 41,0/00,2 gm 82,0



.1)

.

1 2

E :

2

22

2

11212

1

2

1 mmKKE (.2.1)

. :

22110 mm 2211 mm 2

22

2

11 mm 2

22

2

112

1

2

1 mm

222

2

2

1

2

12

1

2

1 mm 2211 KmKm

2

1

1

2

m

m

K

K (.2.2)

(.2.1) (.2.2) :

12

1121 Km

mKEKKE

2

11 1

m

mKE E

mm

mK

21

21

: Emm

mK

21

12

.

2 E . 3 , , .

Rn222

.

Rn222 ( ). 2 .

( )

( ) ( ).

Bi210 , Po210 . 2 .

Po210 383.000 . 2 . ! .



. Pauli 1931 . Pauli . . 3 . 3 E . , E ( 3 ).

,

.

8-39 . 226 , (Hugh D. Young)

, J310070,1

Q J310090,1 . ;

EQ .

Emm

mKa

EmmmKa

a

aa

KE

mKm

J

kgJm

3

273

1002,0

1064,610070,1kgm 25106,3

8-41 . 226 , (Hugh D. Young)

Bi210 Po210 .

. skgm/1060,3 22

skgm/1040,4 22 . Po210

kg251050,3 . :

a) Po210 , b) .

a) skgmskgmskgmP /1069,5/1040,4/1060,3 22222222

) smkg

skgm

m

PmP /1062,1

1050,3

/1069,5 325

22



JsmkgmK 19226252 1062,4/1064,21050,32

1

2

1

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Πανεπιστημιακή Φυσική Τόμος Α Hugh D. Young Μέρος Α - Κεφάλαιο 4 Ορμή και Ώθηση