Centre of Mass Theory_E

8/12/2019 Centre of Mass Theory_E

1/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" =

OEÔY L@ F QQAOEÔY L@ FMQQ

Ovory pgyscamk systof gms msslacmton wctg ct m aortmce plcet wglso fltcle agmrmatorcsos tgo fltcle l`tgo wglko systof. Xgoe tgo systof flvos uenor slfo oxtoremk `lraos, tgoe tgcs plcet flvos ms c` tgo

oetcro fmss l` tgo systof cs aleaoetrmton mt tgcs plcet men mksl tgo oxtoremk `lrao cs mppkcon mt tgcs

plcet `lr trmeskmtclemk fltcle. ^gcs plcet cs amkkon tgo aoetor l` fmss l` tgo systof.

AOEÔY L@ FMQQ L@ M QSQÔF L@ E NCQAYOÔ ZMY^CAKOQAlescnor m systof l` E plcet fmssos f

=, f

9, f

1, ................ f

ewglso

plsctcle voatlrs `rlf lrchce L mro hcvoe iy =r

, 9r

, 1r

,...............

er

rospoatcvoky. ^goe tgo plsctcle voatlr l` tgo aoetor l` fmss A l` tgo

systof cs hcvoe iy.

afr

6e9=

ee99==

f........ff

rf........rfrf

4 afr

6

e

=c

c

e

=ccc

f

rf

afr

6F

=

e

=c

ccrf

wgoro, ccrf

cs amkkon tgo flfoet l` fmss l` tgo pmrtcako w.r.t L.

F 6

e

=c

cf cs tgo tltmk fmss l` tgo systof.

Elto2 C` tgo lrchce cs tmdoe mt tgo aoetor l` fmss tgoe

e

=c

ccrf

6:. goeao, tgo ALF cs tgo plcet milut wgcag

tgo suf l` fmss flfoets l` tgo systof cs zorl.

ZLQC^CLE L@ ALF L@ ^XL ZMY^CAKOQAoetor l` fmss l` twl pmrtcakos l` fmssos f

=men f

9sopmrmton iy m ncstmeao r kcos ce iotwooe tgo

twl pmrtcakos. ^go ncstmeao l` aoetor l` fmss `rlf mey l` tgo pmrtcako (r) cs cevorsoky prlplrtclemk

tl tgo fmss l` tgo pmrtcako (f)

c.o. r =/f

lr9

=

r

r6

=

9

f

f

lr f=r

=6 f

9r

9

lr r=

6

=9

9

ff

fr men r

96

9=

=

ff

fr

Goro, r =6 ncstmeao l` ALF `rlf f

=

men r 96 ncstmeao l` ALF `rlf f

9

@rlf tgo milvo ncsausscle, wo soo tgmt

r=

6 r9

6 =/9 c` f=

6 f9, c.o., ALF kcos fcnwmy iotwooe tgo twl pmrtcakos l` oqumk fmssos.

Qcfckmrky, r= 5 r9 c ` f= ; f 9 men r= ; r9 c ` f 9 ; f=, c.o., ALF cs eomror tl tgo pmrtcako gmvceh kmrhorfmss.


2/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 9

Oxmfpko =. ^wl pmrtcakos l` fmss = dh men 9 dh mro klamton mt x 6 : men x 6 1 f. @cen tgo plsctcle l` tgocr

aoetor l` fmss.

Qlkutcle 2

Qceao, iltg tgo pmrtcakos kcos le x-mxcs, tgo ALF wckk mksl kco le x-mxcs. Kot tgo ALF cs

klamton mt x 6 x, tgoe

r=

6 ncstmeao l` ALF `rlf tgo pmrtcako l` fmss = dh 6 x

m en r 9

6 ncstmeao l` ALF `rlf tgo pmrtcako l` fmss 9 dh 6 (1 x)

_sceh9

=

r

r6

=

9

f

flr

x1

x

6

=

9lr x 6 9 f

^gus, tgo ALF l` tgo twl pmrtcakos cs klamton mt x 6 9 f. Mes.

Oxmfpko 9. ^go plsctcle voatlr l` tgroo pmrtcakos l` fmssos f=

6 = d h, f9

6 9 dh men f1

6 1 dh mro

f)d

b

>c

(r=

, f)d

b

c

(r9

men f)d

9b

c

9(r1

rospoatcvoky. @cen tgo plsctcle voatlrl` tgocr aoetor l` fmss.

Qlkutcle 2

^go plsctcle voatlr l` ALF l` tgo tgroo pmrtcakos wckk io hcvoe iy

19=

1199==ALF

fff

fff

rrrr

Quistctutceh tgo vmkuos, wo hot

19=

)d9bc1)(9()dbc((9))db>c((=)rALF

f)dbc1(

9

= Mes.

Oxmfpko 1. @lur pmrtcakos l` fmss = dh, 9 dh, 1 dh men > dh mro pkmaon mt tgo `lur vortcaos M, I, A men Nl` m squmro l` scno = f. @cen tgo plsctcle l` aoetor l` fmss l` tgo pmrtcakos.

Qlkutcle 2

Ms su fc eh N ms tgo lr chce, NA ms x -m xc s me n NM ms y-m xc s, wo gmvo

f=

6 = dh, (x=, y

=) 6 (:, =f)

f9

6 9 dh, (x9, y

9) 6 (=f, =f)

f1

6 1 dh, (x1, y

1) 6 (=f, :)

men f>

6 > dh, (x>, y

>) 6 (:, :)

Al-lrncemtos l` tgocr ALF mro

xALF

6>19=

>>1199==

ffff

xfffxfxf

6>19=

):(>)=(1)=(9):)(=(

6=:

?6

9

=f 6 :.? f

Qcfckmrky, yALF

6>19=

>>1199==

ffff

yfyfyfyf

6>19=

):(>):(1)=(9)=)(=(

6=:

16 :.1 f

(xALF

, yALF

) 6 (:.? f, :.1 f) Mes.

^gus, plsctcle l` ALF l` tgo `lur pmrtcakos cs ms sglwe ce `churo.


3/48

ZGSQCAQ


Oxmfpko >. Alescnor m twl-pmrtcako systof wctg tgo pmrtcakos gmvceh fmssos f=men f

9. C` tgo `crst pmrtcako

cs pusgon tlwmrns tgo aoetor l` fmss tgrluhg m ncstmeao n, iy wgmt ncstmeao sglukn tgo

soalen pmrtcako io flvon sl ms tl doop tgo aoetor l` fmss mt tgo smfo plsctcle

^go y-allrncemto l` ALF cs

yALF

6

nf

nfy

6 :

Qcfckmrky, zALF

6 :

c.o., tgo allrncemtos l` ALF l` tgo rln mro

:,:,9

K

, c.o. ct k cos mt tgo aoetor l` tgo rln.

Oxmfpko ?. M rln l` koehtg K cs pkmaon mkleh tgo x-mxcs iotwooe x 6 : men x 6 K. ^go kceomr noescty (fmss/koehtg)

l` tgo rln vmrcos wctg tgo ncstmeao x `rlf tgo lrchce ms 6 Yx. Goro, Y cs m plsctcvo alestmet. @centgo plsctcle l` aoetor l` fmss l` tgcs rln.

Qlkutcle 2

Fmss l` okofoet nx sctumton mt x 6 x cs

nf 6nx 6 Yx nx^go ALF l` tgo okofoet gms allrncemtos (x, :, :).

^goro`lro, x-allrncemto l` ALF l` tgo rln wckk io

x6Kx6xx6:

nx

y

x

xALF

6

nf

nfxK

:

6

K

:

K

:

nx)Yx(

nx)Yx)(x(

6

K

:

K

:

9

nxxY

nxxY

61

K9

9

x

1

x

K

:

9

K

:

1

^go y-allrncemto l` ALF l` tgo rln cs yALF

6

nf

nfy

6 : (ms y 6 :)

Qcfckmrky, zALF

6 :

Goeao, tgo aoetor l` fmss l` tgo rln kcos mt

:,:,

1

K9Mes.

AOE^OY L@ FMQQ L@ M QOFCACYA_KMY YCEH@churo sglws tgo liboat (sofc acraukmr rceh). Iy lisorvmtcle wo ame smy tgmt tgo x-allrncemto l`

tgo aoetor l` fmss l` tgo rceh cs zorl ms tgo gmk` rceh cs syffotrcamkm ilut y-mxcs le iltg scnos l`

tgo lrchce. Leky wo mro roqucron tl `cen tgo y-allrncemto l` tgo aoetor l` fmss.

S

yaf

n y6Ysce

Yn

W


5/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" ?

^l `cen yaf

wo uso yaf

6F

= ynf ...(c)

Goro `lr nf wo alescnor me okofoetmk mra l` tgo rceh mt me mehko `rlf tgo x-ncroatcle l` mehukmrwcntg n. C` rmncus l` tgo rceh cs Y tgoe cts y allrncemto wckk io Y sce, goro nf cs hcvoe ms

nf 6

Y

F

Y n

Ql `rlf oqumtcle ---(c), wo gmvo

yaf

6F

=

YnY

F

:

(Y sce) 6Y

nsce

:

yaf

6Y9

...(cc)

AOE^OY L@ FMQQ L@ QOFCACYA_KMY NCQA@churo sglws tgo gmk` ncsa l` fmss F men rmncus Y. Goro, wo mro leky roqucron tl `cen tgo y-

allrncemto l` tgo aoetor l` fmss l` tgcs ncsa ms aoetor l` fmss wckk io klamton le cts gmk` vortcamk

ncmfotor. Goro tl `cen yaf

, wo alescnor m sfmkk okofoetmk rceh l` fmss nf l` rmncus x le tgo ncsa

(ncsa ame io alescnoron tl io fmno up suag tgce rcehs l` cearomsceh rmncc) wgcag wckk io cetohrmton

`rlf : tl Y. Goro nf cs hcvoe ms

nf 6 9Y

F9

( x)nx

S

yaf

WY

nx

x

Elw tgo y-allrncemto l` tgo okofoet cs tmdoe msx9

, ms ce provclus soatcle, wo gmvo norcvon tgmt

tgo aoetor l` fmss l` m sofc acraukmr rceh cs aleaoetrmton mtY9

Goro yaf

cs hcvo e m s yaf

6

F

=

Y

:

nf

x96

F

=

Y

:

9

9 nxx

Y

F>

yaf

61Y>

AOE^OY L@ FMQQ L@ M QLKCN GOFCQZGOYO^go gofcspgoro cs l` fmss F men rmncus Y. ^l `cen cts aoetor l` fmss (leky y-allrncemto), wo

alescnor me okofoet ncsa l` wcntg ny, fmss nf mt m ncstmeao y `rlf tgo aoetor l` tgo gofcspgoro.

^go rmncus l` tgcs okofoetmk ncsa wckk io hcvoe ms

r 6 99 yY


6/48

ZGSQCAQ


^go fmss nf l` tgcs ncsa ame io hcvoe ms

nf 6 1Y9

F1

r9 ny

6 1Y9

F1(Y 9 y9) ny

yaf

l` tgo gofcspgoro cs hcvoe ms

yaf

6F

= Y

:

ynf 6F

= Y

:

1Y9

F1(Y 9 y9) ny y

6 nyy)yY(Y9

1 Y

:

99

1

yaf

67

Y1

AOE^OY L@ FMQQ L@ M GLKKLX GOFCQZGOYOM gl kklw go fc sp goro l` fms s F me n rmnc us Y. El w wo al es cnor me okof oetmk ac ra uk mr strc p l`

mehukmr wcntg nmt me mehukmr ncstmeao `rlf tgo imso l` tgo gofcspgoro. ^gcs strcp wckk gmvo memrom.

nQ 6 9Y als Yn

Cts fmss nf cs hcvoe ms

nf 6 9Y9

F

9Y als Yn

Goro y-allrncemto l` tgcs strcp l` fmss nf ame io tmdoe ms Y sce. Elw wo ame litmce tgo aoetorl` fmss l` tgo systof ms.

yaf

6F

=

9

:

sceYnf 6F

=

9

:

9

9 nalsY9

Y9

FY sce

6 Y

9

:

nalssce yaf

69

Y


7/48

ZGSQCAQ


AOE^OY L@ FMQQ L@ M QLKCN ALEOM sl kcn al eo gms fm ss F, gochgt G men imso rm ncus Y. Li vclusk y tg o aoet or l` fm ss l` tgcs aleo

wckk kco slfowgoro le cts mxcs, mt m gochgt koss tgme G/9. ^l klamto tgo aoetor l` fmss wo alescnor

me okofoetmk ncsa l` wcntg ny men rmncus r, mt m ncstmeao y `rlf tgo mpox l` tgo aleo. Kot tgo fmss

l` tgcs ncsa io nf, wgcag ame io hcvoe ms

nf 6GY

F19

r9 ny

goro yaf

ame io hcvoe ms

yaf

6F

= G

:

nfy

6F

=yny

G

Yy

GY

F1G

:

9

9

6 1G

1 G

:

1 nyy 6>

G1

Oxmfpko 0. @cen lut tgo aoetor l` fmss l` me cslsaokos trcmehko l` imso koehtg m men mktctuno i.Mssufo tgmt tgo

fmss l` tgo trcmehko cs uec`lrfky ncstrciuton lvor cts mrom.

Qlkutcle 2

l klamto tgo aoetor l` fmss l` tgo trcmehko, wo tmdo m strcp l` wcntg nx mt m ncstmeao x `rlf tgo vortox

l` tgo trcmehko. Koehtg l` tgcs strcp ame io ovmkumton iy scfckmr trcmehkos ms

6 x. (m/i)

F ms s l ` t go s tr cp c s n f 6mi

F9nx

Ncstmeao l` aoetor l` fmss `rlf tgo vortox l` tgo trcmehko cs

xAF

6F

=nfx

6 i

:

9

9

nxi

x9

61

9i

Zrlaoonceh ce tgo scfckmr fmeeor, wo ame `cen tgo ALF l` aortmce rchcn ilncos. Aoetor l`

fmss l` slf o wokk delwe rchcn ilncos mro hcvoe ioklw 2

=. Aoetor l` fmss l` m uec`lrf roatmehukmr, squmro lr acraukmr pkmto kcos mt cts aoetor. Mxcs l` syffo-

try pkmeo l` syffotry.


8/48

ZGSQCAQ


9. @lr m kmfcemr typo (9-ncfoesclemk) ilny wctg uec`lrf eohkchciko tgcadeoss tgo `lrfukmo `lr `cenceh

tgo plsctcle l` aoetor l` fmss mro ms `lkklws 2

....tMtM

....rtMrtM

....ff

....rfrfr

9=

99==

9=

99==ALF

( f 6 Mt )

lr....MM

....rMrMr

9=

99==ALF

Goro, M stmens `lr tgo mrom,

1. C` slfo fmss l` mrom cs roflvon `rlf m rchcn ilny, tgoe tgo plsctcle l` aoetor l` fmss l` tgo

rofmceceh plrtcle cs litmceon `rlf tgo `lkklwceh `lrfukmo2

(c)9=

99==ALF

ff

rfrfr

lr9=

99==ALF

MM

rMrMr

(cc)9=

99==ALF

ff

xfxfx

lr

9=

99==ALF

MM

xMxMx

9=

99==ALF

ff

yfyfy

lr9=

99==ALF

MM

yMyMy

men9=

99==ALF

ff

zfzfz

lr9=

99==ALF

MM

zMzMz

Goro, f=, M

=, =r

, x=, y

=men z

=mro tgo vmkuos `lr tgo wglko fmss wgcko f

9, MM

9, 9r

, 9x

, y9

men z9

mro tgo vmkuos

`lr tgo fmss wgcag gms iooe roflvon. Kot us soo twl oxmfpkos ce supplrt l` tgo milvo tgolry.

Oxmfpko 8. @cen tgo plsctcle l` aoetor l` fmss l` tgo uec`lrf kmfcem sglwe ce `churo.

WmL

S

Qlkutcle 2 Goro,

M=

6 mrom l` alfpkoto acrako 6m9

M9

6 mrom l` sfmkk acrako 6 9

9

m

6

>

m9

(x=, y

=) 6 allrncemtos l` aoetor l` fmss l` kmrho acrako 6 (:, :)

men (x9

, y9

) 6 allrncemtos l` aoetor l` fmss l` sfmkk acrako 6

:,

9

m

_ sc eh xALF

69=

99==

MM

xMxM

wo hot xALF

6

>

mm

9

m

>

m

99

9

6

>

1

7

=

m 6 0

m

men yALF

6 : ms y=

men y9

iltg mro zorl.

^goro`lro, allrncemtos l` ALF l` tgo kmfcem sglwe ce `churo mro

:,

0

mMes.


9/48

ZGSQCAQ


AOE^OY L@ FMQQ L@ QLFO ALFFLE QSQ^OFQ M systof l` twl plcet fmssos f

=r

=6 f

9r

9

^go aoetor l` fmss kcos aklsor tl tgo gomvcor fmss.

Yoatmehukmr pkmto (Iy syffotry)

xa6

9

iy

a6

9

K

M trcmehukmr pkmto (Iy qumkctmtcvo mrhufoet)

mt tgo aoetrlcn 2 ya6

1

g

M sofc-acraukmr rceh

ya6

Y9

xa6 :

M sofc-acraukmr ncsa

ya6

1Y>

xa6 :

M gofcspgorcamk sgokk

ya 6 9

Y

xa 6 :

M slkcn gofcspgoro

ya6

7

Y1x

a6 :

M acraukmr aleo (slkcn)

ya6

>

g


10/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" =:

M acraukmr aleo (glkklw)

ya6

1

g

FL^CLE L@ AOE^OY L@ FMQQ MEN ALEQOYPM^CLE L@ FLFOE^_F 2Poklacty l` aoetor l` fmss l` systof

afv

6F

nt

rnf..............

nt

rnf

nt

rnf

nt

rnf ee

11

99

==

6F

vf..........vfvfvf ee1199==

Goro euformtlr l` tgo rchgt gmen scno torf cs tgo tltmk flfoetuf l` tgo systof c.o., suffmtcle

l` flfoetuf l` tgo cencvcnumk alfpleoet (pmrtcako) l` tgo systof

Goeao voklacty l` aoetor l` fmss l` tgo systof cs tgo rmtcl l` flfoetuf l` tgo systof tl tgo fmss l` tgo

systof.

QystZ 6 F afv

Maaokormtcle l` aoetor l` fmss l` systof

afm

6

F

nt

vnf..............

nt

vnf

nt

vnf

nt

vnf ee

11

99

==

6F

mf..........mfmfmf ee1199==

6F

systofle`lraoEot6

F

@lraooremkcetEot@lraoOxtoremkEot 6

F

@lraoOxtoremkEot

( matcle men romatcle iltg l` me cetoremk `lrao fust io wctgce tgo systof. Poatlr suffmtcle wckk

ameaok mkk cetoremk `lraos men goeao eot cetoremk `lrao le systof cs zorl)

oxt@

6 F afm

wgorooxt@

cs tgo suf l` tgo 'oxtoremk' `lraos matceh le tgo systof. ^go cetoremk `lraos wgcag tgo

pmrtcakos oxort le leo meltgor pkmy mislkutoky el rlko ce tgo fltcle l` tgo aoetor l` fmss.

C` el oxtoremk `lrao cs matceh le m systof l` pmrtcakos, tgo maaokormtcle l` aoetor l` fmss l` tgo

systof wckk io zorl. C` ma

6 :, c t cfpkcos tgmt va

fust io m alestmet men c` vaf

cs m alestmet, ct

cfpkcos tgmt tgo tltmk flfoetuf l` tgo systof fust rofmce alestmet. Ct komns tl tgo prceacpmk l`

alesorvmtcle l` f lfoetuf ce misoeao l` oxtoremk `lraos.

C` :@oxt

tgoe afv

6 alestmet

C` rosuktmet oxtoremk `lrao cs zorl le tgo systof, tgoe tgo eot flfoetuf l` tgo systof

fust rofmce alestmet.

Fltcle l` ALF ce m flvceh systof l` pmrtcakos2(=) ALF mt rost 2

C` @ox t

6 : men Paf

6 :, tgoe ALF rofmces mt rost. Cencvcnumk alfpleoets l` tgo systof fmy

flvo men gmvo ele-zorl flfoetuf nuo tl f utumk `lraos (cetoremk), iut tgo eot flfoetuf l`

tgo systof rofmces zorl.

(c) Mkk tgo pmrt ca kos l` tgo s ys tof mro mt ros t.

(cc) Zmrtcakos mro flvceh suag tgmt tgocr eot flfoetuf cs zorl.

oxmfpko2


11/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" ==

(ccc) M ilfi mt rost sunnoeky oxpklnos cetl vmrclus sfmkkor rmhfoets, mkk flvceh ce nc òroet ncroatcles

tgoe, sceao tgo oxpklscvo `lraos mro cetoremk & tgoro cs el oxtoremk `lrao le tgo systof `lr oxpkl-

scle tgoro`lro, tgo ALF l` tgo ilfi wckk rofmce mt tgo lrchcemk plsctcle men tgo `rmhfoet `ky suag

tgmt tgocr eot flfoetuf rofmces zorl.

(cv) ^wl foe stmenceh le m `rcatclekoss pkmt`lrf, pusg omag ltgor, tgoe mksl tgocr eot flfoetuf

rofmces zorl ioamuso tgo pusg `lraos mro cetoremk `lr tgo twl foe systof.

(v) M ilmt `klmtceh ce m kmdo, mksl gms eot flfoetuf zorl c tgo polpko le ct agmehos tgocr plsctcle,

ioamuso tgo `rcatcle `lrao roqucron tl flvo tgo polpko cs cetoremk l` tgo ilmt systof.

(vc) Liboats cectcmkky mt rost, c` flvceh uenor futumk `lraos (okoatrlstmtca lr hrmvctmtcle)mksl gmvo eot

flfoetuf zorl.

(vcc) M kchgt sprceh l` sprceh alestmet d dopt alfprosson iotwooe twl iklads l` fmssos f=me n

f9le m sflltg glrczletmk sur`mao. X goe rokomson, tgo iklads maqucro voklactcos ce lppl-

scto ncroatcles, suag tgmt tgo eot flfoetuf cs zorl.

(vccc) Ce m `me, mkk pmrtcakos mro flvceh iut ALF cs mt rost

(9) A LF flvceh w ctg uec`lrf voklacty 2

C` @oxt

6 : , t go e Paf

rofmces alestmet tgoro`lro, eot flfoetuf l` tgo systof mksl rofmces

alesorvon. Cencvcnumk alfpleoets l` tgo systof fmy gmvo vmrcmiko voklacty men flf oetuf nuo tl

futumk `lraos (cetoremk), iut tgo eot flfoetuf l` tgo systof rofmces alestmet men ALF aletce-

uos tl flvo wctg tgo cectcmk voklacty.

(c) Mkk tgo pmrtcakos l` tgo systof mro flvceh wctg smfo voklacty.

o.h.2 M amr flvceh wctg uec`lrf spoon le m strmchgt rlmn, gms cts ALF flvceh wctg m

alestmet voklacty.

(cc) Cetoremk oxpklscles / iromdceh nlos elt agmeho tgo fltcle l` ALF men eot flfoetufrof mces ale sorv on. M ilfi flvceh ce m strmchgt kceo sunnoeky oxpklnos cetl vmrclus sfmkkor

`rmhfoets, mkk flvceh ce nc`òroet ncroatcles tgoe, sceao tgo oxpklscvo `lraos mro cetoremk & tgoro cs

el oxtoremk `lrao le tgo systof `lr oxpklscle tgoro`lro, tgo ALF l` tgo ilfi wckk aletceuo tgo

lrchcemk fltcle men tgo `rmhfoet `ky suag tgmt tgocr eot flfoetuf rofmces alesorvon.

(ccc) Fme bufpceh `rlf amrt lr iuhhy mksl oxort cetoremk `lraos tgoro`lro eot flfoetuf l` tgo

systof men goeao, Fltcle l` ALF rofmces alesorvon.

(cv) ^wl flvceh iklads aleeoaton iy m kchgt sprceh le m sflltg glrczletmk sur`mao. C` tgo

matceh `lraos cs leky nuo tl sprceh tgoe ALF wckk rofmce ce cts fltcle men flfoetuf wckk

rofmce alesorvon.

(v) Zmrtcakos alkkcnceh ce misoeao l` oxtoremk cfpukscvo `lraos mksl gmvo tgocr flfoetuf ale-sorvon.

(1) A LF flvceh w ctg maaokormtcle 2

C` me oxtoremk `lrao cs prosoet tgoe ALF aletceuos cts lrchcemk

fltcle ms c` tgo oxtoremk `lrao cs matceh le ct, crrospoatcvo l`

cetoremk lraos.

Oxmfpko2

Zrlboatcko fltcle 2Me mxo tgrlwe ce mcr mt me mehko wctg tgog lr cz le tm k w ck k p or `l rf m a lf pk ca mt on f lt cl e l `

rltmt c le ms w ok k ms pmrmilk c a fltc le uenor tgo o`òa t l`

hrmvctmtcle


12/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" =9

Galf

6h9

sceu 99

Yalf

6h

9sceu9 ^ 6

h

sceu9

Oxmfpko2

Acraukmr Fltcle 2M rln gcehon mt me oen, rltmtos, tgme ctsALF por`lrfs acraukmr fltcle. ^go aoetrcpotmk `lrao (@

a)

roqucron ce tgo acraukmr fltcle cs mssufon tl io matceh le

tgo ALF.

ALF9

a Yf@

fh

fh

fh

@a

@a@a

fh

@a

Yalf

Oxmfpko 7. Mprlboatcko cs `cron mt m spoon l` =:: f/smt me mehko l` 18 milvo tgo glrczletmk. Mt tgo gchgost plcet,

tgo prlboatcko iromds cetl twl pmrts l` fmss rmtcl = 2 1, tgo kchgtor pcoao alfceh tl rost. @cen tgo

ncstmeao `rlf tgo kmueagceh plcet tl tgo plcet wgoro tgo gomvcor pcoao kmens.

Qlkutcle 2Cetoremk `lrao nl elt o``oat tgo fltcle l` tgo aoetor l` fmss, tgo aoetor l` fmss gcts tgo

hrluen mt tgo plsctcle wgoro tgo lrchcemk prlboatcko wlukn gmvo kmenon. ^go rmeho l` tgo

lrchcemk prlboatcko cs,

xALF

6h

alssceu9 9 6 f

=:

?

>

?

1=:9 >

6 30: f

^go aoetor l` fmss wckk gct tgo hrluen mt tgcs plsctcle. Ms tgo sfmkkor iklad alfos tl rost

m`tor iromdceh, ct `mkks nlwe vortcamkky men gcts tgo hrluen mt gmk` l` tgo rmeho, c.o., mt x 6

>7: f. C` tgo gomvcor iklad gcts tgo hrluen mt x9, tgoe

xALF

69=

99==

ff

xfxf

30: 6)f1f(

)x)(f1()>7:)(f( 9

x9

6 ==9: f Mes.

Flfoetuf Alesorvmtcle 2^go tltmk kceomr flfoetuf l` m systof l` pmrtcakos cs oqumk tl tgo prlnuat l` tgo tltmk fmss l` tgo

systof men tgo voklacty l` cts aoetor l` fmss. Z

6 F afv

oxt@

6nt

nZ

C oxt@

6 :

nt

nZ

6 : 4 Z

6 alestmet

Xgoe tgo voatlr suf l` tgo oxtoremk `lraos matceh le m systof cs zorl, tgo tltmk kceomr flfoetuf l`

tgo systof rofmces alestmet.

=Z

+9

Z

+1

Z

+ ...............+e

Z

6 alestmet.


13/48

ZGSQCAQ


Oxmfpko 3. M sgokk cs cron `rlf m ameele wctg m spoon l` =:: f/s mt me mehko 0: wctg tgo glrczletmk (plsctcvo x-

ncroatcle). Mt tgo gchgost plcet l` cts trmboatlry, tgo sgokk oxpklnos cetl twl oqumk `rmhfoets. Leo l`

tgo `rmhfoets flvos mkleh tgo eohmtcvo x-ncroatcle wctg m spoon l` ?: f/s. Xgmt cs tgo spoon l` tgo

ltgor `rmhfoet mt tgo tcfo l` oxpklscle.

Qlkutcle 2

Ms wo de lw ce miso ea o l` ox tore mk `l ra o tgo fl tc le l` ao etor l` fm ss l` m ilny ro fmces

uem``oaton. ^gus, goro tgo aoetor l` fmss l` tgo twl `rmhfoets wckk aletceuo tl `lkklw tgo

lrchcemk prlboatcko pmtg. ^go voklacty l` tgo sgokk mt tgo gchgost plcet l` trmboatlry cs

vF

6 uals 6 =:: als0: 6 ?: f/s.Kot v

=io tgo spoon l` tgo `rmhfoet wgcag flvos mkleh tgo eohmtcvo x-ncroatcle men tgo

ltgor `rmhfoet gms spoon v9,. wgcag fust io mkleh plsctcvo x-ncroatcle. Elw `rlf flfoe-

tuf alesorvmtcle, wo gmvo

fv 69

fv

=+

9

fv

9 lr 9v 6 v

9 v

=

lr v9

6 9v + v=

6 (9 ?:) + ?: 6 =?: f/s

Oxmfpko =:. M fme l` fmss f cs stmenceh le m pkmt lrf l` fmss F dopt le sflltg cao. C` tgo fme stmrts flvceh

le tgo pkmt`lrf wctg m spoon v rokmtcvo tl tgo pkmt`lrf, wctg wgmt voklacty rokmtcvo tl tgo cao nlos tgo

pkmt lrf roalck

Qlkutcle 2

^gcs amso cs scfckmr tl tgo provclus oxmfpko, oxaopt elw tgo amr cs flvceh io`lro bufp. Goro mksl

el oxtoremk `lrao cs matceh le tgo systof ce glrczletmk ncroatcle, goeao flfoetuf rof mces ale-

sorvon ce tgcs ncroatcle. M`tor bufp amr mttmces m voklacty v9

ce tgo smfo ncroatcle, wgcag cs koss

tgme v=, nuo tl imadwmrn pusg l` tgo agckn `lr bufpceh. M`tor bufp agckn mttmces m voklacty u + v

9ce

tgo ncroatcle l` f ltcle l` amr, wctg rospoat tl hrluen.

f

F

///////////////////////////////////////////////////////////////////

v=

uv9 F

f

///////////////////////////////////////////////////////////

Ma alrnceh tl fl fo etuf ales orvmt cle

(F + f)v=6 Fv

9+ f (u + v

9)

Poklacty l` amr m`tor bufp cs

v9

6fF

fuv)fF( =

Poklacty l` agckn m`tor bufp cs u + v 9 6 fF

u)F(v)fF( =

Oxmfpko =1. Omag l` tgo iklads sglwe ce `churo gms fmss = dh. ^go romr iklad flvos wctg m spoon l`

9 f/s tlwmrns tgo `rlet iklad dopt mt rost. ^go sprceh mttmagon tl tgo `rlet iklad cs kchgt men

gms m sprceh alestmet ?: E/f. @cen tgo fmxcfuf alfprosscle l` tgo sprceh. Mssufo, le m

`rcatcle koss sur`mao

/////////////////////////////////////////////////////////////////

=dh=dhd6?:E/f

Qlkutcle 2

Fmxcfuf alfprosscle wckk tmdo pkmao wgoe tgo iklads flvo wctg oqumk voklacty. Ms el eotoxtoremk glrczletmk `lrao mats le tgo systof l` tgo twl iklads, tgo tltmk kceomr flfoetuf wckk

rofmce alestmet. C` P cs tgo alffle spoon mt fmxcfuf alfprosscle, wo gmvo,

(= dh) (9 f/s) 6 (= dh)P + (= dh)P

lr, P 6 = f/s.

Cectcmk dceotca oeorhy 69

=(= dh) (9 f/s)9 6 9 B.

@cemk dceotca oeorhy

69

=(= dh) (=f/s)9 +

9

=(= dh) (= f/s)9 6 = B

^go dceotca oeorhy klst cs stlron ms tgo okmstca oeorhy ce tgo sprceh.

Goeao,9

=(?: E/f) x9 6 9B =B 6 = B

lr, x 6 :.9 f.

Oxmfpko =>. @churo sglws twl iklads l` fmssos ? dh men 9 dh pkmaon le m `rcatclekoss sur`mao men ale-

eoaton wctg m sprceh. Me oxtoremk dcad hcvos m voklacty => f/s tl tgo gomvcor iklad tlwmrns tgo

kchgtor leo. Nonuao (m) voklacty hmceon iy tgo aoetor l` f mss men (i) tgo sopmrmto voklactcos l`

tgo twl iklads wctg rospoat tl aoetor l` fmss bust m`tor tgo dcad.

//////////////////////////////////////////////////////////////

?dh 9dh


15/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" =?

Qlkutcle 2

(m) Poklac ty l` a oetor l` fmss c s

vaf

69?

:9=>?

6 =: f/s

(i) Nuo tl dcad le ? dh iklad, ct stmrts flvceh wctg m voklacty => f/s cffoncmtoky, iut

nuo tl ceortcm 9 dh iklad rofmces mt rost, mt tgmt flfoet. ^gus, voklacty l` ? dh

iklad wctg rospoat tl tgo aoetor l` fmss cs v=

6 => =: 6 > f/s men tgo voklacty

l` 9 dh iklad w.r.t. tl aoetor l` fmss cs v 9 6 : =: 6 =: f/s

Oxmfpko =?. M kchgt sprceh l` sprceh alestmet d cs dopt alfprosson iotwooe twl iklads l` fmssos f men F

le m sflltg glrczletmk sur`mao. Xgoe rokomson, tgo iklads maqucro voklactcos ce lpplscto nc-

roatcles.

^go sprceh klsos aletmat wctg tgo iklads wgoe ct maqucros emturmk koehtg. C` tgo sprceh wms

cectcmkky alfprosson tgrluhg m ncstmeao x, `cen tgo `cemk spoons l` tgo twl iklads.

Qlkutcle 2

Alescnor tgo twl iklads pkus tgo sprceh tl io tgo systof. El oxtoremk `lrao mats le tgcs

systof ce glrczletmk ncroatcle. Goeao, tgo kceomr flfoetuf wckk rofmce alestmet. Qup-

plso, tgo iklad l` fmss F flvos wctg m spoon v=men tgo ltgor iklad wctg m spoon

vm`tor

klsceh aletmat wctg tgo sprceh. @rlf alesorvmtcle l` kceomr flfoetuf ce glrczletmk ncroa-tcle wo gmvo

Fv= fv

96 : lr v

=6

F

fv

9, ....(c)

Cectcmkky, tgo oeorhy l` tgo systof 69

=dx 9

@cemkky,tgo oeorhy l` tgo systof 69

=fv

99 +

9

=Fv

=9

Ms tg oro cs el `r ca tcle , fo ag meca mk oeorhy wck k ro fm ce ales or von.

^goro`lro,9= fv

99 +

9= Fv

=9 6

9= dx 9 ...(cc)

Qlkvceh Oqs. (c) men (cc), wo hot

lr, v9

6 x)fF(f

dF 9/=

men v=

6

9/=

)fF(F

df

x Mes.

Oxmfpko =0. M iklad l` fmss f cs aleeoaton tl meltgor iklad l` fmss F iy m fmsskoss sprceh l` sprceh

alestmet d. ^go iklads mro dopt le m sflltg glrczletmk pkmeo men mro mt rost. ^go sprceh cs

uestrotagon wgoe m alestmet `lrao @ stmrts matceh le tgo iklad l` fmss F tl pukk ct. @cen tgo

fmxcfuf oxtoescle l` tgo sprceh.

Qlkutcle 2

Xo s lkvo tgo sctumtcle ce tgo ro`oroeao `rmfo l` aoetor l` fmss. Ms leky @ cs tgo oxtoremk

`lrao matceh le tgo systof, nuo tl tgcs `lrao, tgo maaokormtcle l` tgo aoetor l` fmss cs @/

(F + f). ^gus wctg rospoat tl aoetor l` fmss tgoro cs m Zsounl `lrao le tgo twl fmssos

ce lpplscto ncroatcle, tgo `roo ilny ncmhrmf l` f men F wctg rospoat tl aoetor l` fmss

(tmdceh aoetor l` fmss mt rost) cs sglwe ce `churo.


16/48

ZGSQCAQ


^mdceh aoetor l` fmss mt rost, c` f flvos fmxcfuf iy m ncstmeao x=men F flvos fmxcfuf

iy m ncstmeao x9, tgoe tgo wlrd nleo iy oxtoremk `lraos (ceakunceh Zsounl `lrao) wckk io

X 6

Ff

f@

. x=

+

Ff

F@@ . x

9 6

Ff

f@

. (x=

+ x9)

^gcs wlrd cs stlron ce tgo `lrf l` pltoetcmk oeorhy l` tgo sprceh ms

_ 69

=d( x

=+ x

9)9

^gus le oqumtceh wo hot tgo fmxcfuf oxtoescle ce tgo sprceh, ms m`tor tgcs cestmet tgo

sprceh stmrts aletrmatceh.

9

=d( x

=+ x

9)9 6

Ff

f@

. (x

=+ x

9)

xfmx

6 x=

+ x9

6)Ff(d

f@9

C F Z _ K Q OCfpukso l` m `lrao @

matceh le m ilny `lr tgo tcfo cetorvmk t 6 t

=tl t 6 t

9cs no`ceon ms 2-

6 9

=

t

tnt@

6 nt@

6 ntvn

f

nt 6 vnf

6 f( =9 v-v

) 6 Z

6 agmeho ce flfoetuf nuo tl `lrao @

Mksl, sYo

6 9

=

t

t sYo nt@

6 Z

(cfpukso - flfoetuf tgolrof)

Elto2 Cfpukso mppkcon tl me liboat ce m hcvoe tcfo cetorvmk ame mksl io

amkaukmton `rlf tgo mrom uenor `lrao tcfo (@-t) hrmpg ce tgo smfo

tcfo cetorvmk.

Cestmetmeolus Cfpukso 2

^goro mro fmey amsos wgoe m `lrao mats `lr suag m sglrt tcfo tgmt tgo o``oat cs cestmetmeolus,

o.h., m imt strcdceh m imkk. Ce suag amsos, mktgluhg tgo fmhectuno l` tgo `lrao men tgo tcfo `lr

wgcag ct mats fmy omag io uedelwe iut tgo vmkuo l` tgocr prlnuat (c.o., cfpukso) ame io delwe iy

fomsurceh tgo cectcmk men `cemk flfoetm. ^gus, wo ame wrcto.

c` ZZZnt@

Cfplrtmet Zlcets 2

(=) Ct cs m voatlr qumetcty.

(9) Ncfoescles 6 TFK^=V

(1) Q uect 6 dh f/s(>) Ncroatcle cs mkleh agmeho ce flfoetuf.

(?) Fmhectuno cs oqumk tl mrom uenor tgo @-t. hrmpg.

(0)

6 nt@

6 m@

nt 6 t@mv

(8) Ct cs elt m prlporty l` m pmrtcako, iut ct cs m fomsuro l` tgo nohroo tl wgcag me oxtoremk `lrao

agmehos tgo flfoetuf l` tgo pmrtcako.


17/48

ZGSQCAQ


Oxmfpko =8. ^go gorl l` m stuet `ckf `cros ?: h iukkots `rlf m fmagceo hue, omag mt m spoon l` =.: df/s. C`

go `cros 9: iukkots ce > soalens, wgmt mvormho `lrao nlos go oxort mhmcest tgo fmagceo hue

nurceh tgcs porcln.

Qlkutcle 2

^go flfoetuf l` omag iukkot

6 (:.:?: dh) (=::: f/s) 6 ?: dh-f/s.

^go hue gms iooe cfpmrton tgcs fuag mfluet l` flfoetuf iy omag iukkot `cron. ^gus, tgo rmto

l` agmeho l` flfoetuf l` tgo hue

6s>

9:)s/fdh?:( 6 9?: E.

Ce lrnor tl glkn tgo hue, tgo gorl fust oxort m `lrao l` 9?: E mhmcest tgo hue.

Cf pukscvo `lr ao 2M ` lrao, l` rokmtcvoky gchgor fmhectuno men matceh `lr rokmtcvoky s glrtor tcfo, cs amkkon cfpukscvo `lrao.

Me cfpukscvo `lrao ame agmeho tgo flf oetuf l` m ilny ce m `cecto fmhectuno ce m vory sglrt tcfo

cetorvmk. Cfpukscvo `lraocs m rokmtcvo torf. ^goro cs el akomr iluenmry iotwooe me cfpukscvo men Ele-

Cfpukscvo `lrao.Elto2 _sumkky alkkcnceh `lraos mro cfpukscvo ce emturo.

Qceao, tgo mppkcamtcle tcfo cs vory sfmkk, goeao, vory kcttko fltcle l` tgo pmrtcako tmdos pkmao.

Cfplrtmet plcets 2

=. Hrmvctmtclemk `lrao men sprceh `lrao mro mkwmys ele-Cfpukscvo.

9. Elrfmk, toescle men `rcatcle mro amso nopoenoet.

1. Me cfpukscvo `lrao ame leky io imkmeaon iy meltgor cfpukscvo `lrao.

=. Cfpukscvo E lrfmk 2Ce amso l` alkkcscle, elrfmk `lraos mt tgo sur`mao l` alkkcscle mro mkwmys cfpukscvo

oh. Ec6 Cfpukscvo4 E

h6 Ele-cfpukscvo

E=

E9/////////////////////////////

Iltg elrfmks mro Cfpukscvo

//////////////////////////////////////////////////////////////////

E9E 1

E=

E=

E=, E

16 Cfpukscvo4 E

96 ele-cfpukscvo


18/48

ZGSQCAQ


Iltg elrfmks mro Cfpukscvo

9. Cfpukscvo @rcatcle 2C` tgo elrfmk iotwooe tgo twl liboats cs cfpukscvo, tgoe tgo `rcatcle iotwooe tgo

twl wckk mksl io cfpukscvo.

@rcatcle mt iltg sur`maos cs cfpukscvo

@rcatcle nuo tl E9cs ele-cfpukscvo men nuo tl E

1men E

=mro cfpukscvo

1. Cfpukscvo oescles 2 Xgoe m strceh bords, oqumk men lpplscto toescle mat sunnoeky mt omag

oen. Alesoquoetky oqumk men lpplscto cfpuksos mat le tgo ilncos mttmagon wctg tgo strceh ce tgo

ncroatcle l` tgo strceh. ^goro mro twl amsos tl io alescnoron.

(m) L eo oen l` tgo strceh cs ` cxon 2

^go cfpukso wgcag mats mt tgo `cxon oen l` tgo strceh ameelt agmeho tgo flfoetuf l` tgo

`cxon liboat tgoro. ^go liboat mttmagon tl tgo `roo oen glwovor wckk uenorhl m agmeho ce

flfoetuf ce tgo ncroatcle l` tgo strceh. ^go flfoetuf rofmces ueagmehon ce m ncroatcle

porpoencaukmr tl tgo strceh wgoro el cfpukscvo `lraos mat.

(i) Iltg oens l` tgo strceh mttmagon tl flvmiko liboats 2

Ce tgcs amso oqumk men lpplscto cfpuksos mat le

tgo tw l lib oa ts , prlnua c eh oqumk men lppls c to

agmehos ce flfoetuf. ^go tltmk flfoetuf l` tgo

systof tgoro`lro rofmces alestmet, mktgluhg tgo

flfoetuf l` omag cencvcnumk liboat cs agmehon ce

tgo ncroatcle l` tgo strceh. Zorpoencaukmr tl tgo strceh

glwovor, el cfpukso mats men tgo flfoetuf l` omag

pmrtcako ce tgcs ncroatcle cs ueagmehon.

///////////////////

Mkk elrfmk m ro cfpuksc vo iut toescle^ cs cfpukscvo leky `lr tgo imkk M

^^ cs Cfpukscvo

^ cs ele-cfpukscvo

^ cs ele-cfpukscvo

M

I

A

@lr tgcs oxmfpko2

Ce amso l` rln, ^oescle cs mkwmys cfpukscvo men ce amso l` sprceh, ^oescle cs mkwmys ele-cfpukscvo.


19/48

ZGSQCAQ


Oxmfpko =7. M iklad l` fmss f men m pme l` oqumk fmss mro aleeoaton iy m strceh

hlceh lvor m sflltg kchgt pukkoy. Cectcmkky tgo systof cs mt rost wgoe m

pmrtcako l` fmss f `mkks le tgo pme men stcads tl ct. C` tgo pmrtcako strcdos

tgo pme wctg m spoon v, `cen tgo spoon wctg wgcag tgo systof flvos bust

m`tor tgo alkkcscle.

f

f

v

f

//////////

Qlkutcle 2

Kot tgo roqucron spoon cs P.

@urtgor, kot B=6 cfpukso iotwooe pmrtcako men pme

men B9

6 cfpukso cfpmrton tl tgo iklad men tgo pme iy tgo strceh

_sceh, cfpukso 6 agmeho ce flfoetuf

@lr pmrtcako B=

6 fv fP ....(c)

@lr pme B=

B9

6 fP .....(cc)

@lr iklad B9

6 fP .....(ccc)

Q lk vc eh , t go so t gr oo o qu mt cl e, wo h ot P 61

vMes.

Mktoremtcvo Qlkutcle 2

Mppk yceh ales orvmtcl e l` kceomr fl fo etuf mk leh tgo strc eh4

fv 6 1fP

wo hot, P 61

vMes.

Oxmfpko =3. ^wl cnoetcamk iklad M men I, aleeoaton iy m fmsskoss strceh mro pkmaon le m `rcatclekoss glrczletmk

pkmeo.M iukkot gmvceh smfo fmss, flvceh wctg spoon u strcdos iklad I `rlf iogcen ms sglwe. C` tgo

iukkot hots ofionnon cetl tgo iklad I tgoe `cen 2

M I

f uA

ff

/////////////////////////////////////////////////////

(m) ^go voklacty l` M,I,A m`tor alkkcscle.

(i) Cfpukso le M nuo tl toescle ce tgo strceh

(a) Cfpukso le A nuo tl elrfmk `lrao l` alkkcscle.

(n) Cfpukso le I nuo tl elrfmk `lrao l` alkkcscle.

Qlkutcle 2

(m) Iy Alesorvmtcle l` kceomr flfoetuf v 61

u

(i)1

funt^

(a)

u1

ufntE 61

fu9

(n) nt)^E( 6 1fu

nt^Ent

1

fu9ntE


20/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 9:

ALKKCQCLE LY CFZMAÂlkkcscle cs me ovoet ce wgcag me cfpukscvo `lrao mats iotwooe twl lr flro ilncos `lr m sglrt tcfo,

wgcag rosukts ce agmeho l` tgocr voklactcos.

Elto 2 (m) Ce m alkkcscle, pmrtcakos fmy lr fmy elt alfo ce pgyscamk aletmat.

(i) ^go nurmtcle l` alkkcscle,t cs eohkchciko ms alfpmron tl tgo usumk tcfo cetorvmks l` lisorvmtclel` fltcle.

(a) Ce m alkkcscle tgo o`òat l` oxtoremk ele cfpukscvo `lraos suag ms hrmvcty mro elt tmdoe cetl m

maaluet ms nuo tl sf mkk nurmtcle l` alkkcscle (t) mvormho cfpukscvo `lrao rosplesciko `lr alkkc-scle cs fuag kmrhor tgme oxtoremk `lraos matceh le tgo systof.

^go alkkcscle cs ce`mat m roncstrciutcle l` tltmk flfoetuf l` tgo pmrtcakos. ^gus, kmw l`

alesorvmtcle l` kceomr flfoetuf cs cencspoesmiko ce nomkceh wctg tgo pgoelfoele l` alkkcscle

iotwooe pmrtcakos.

Kceo l` Cfpmat^go kceo pmssceh tgrluhg tgo alffle elrfmk tl tgo sur`maos ce aletmat nurceh cfpmat cs amkkon kceo l`

cfpmat. ^go `lrao nurceh alkkcscle mats mkleh tgcs kceo le iltg tgo ilncos.

Ncroatcle l` Kceo l` cfpmat ame io notorfceon iy2(m) Holfotry l` alkkcnceh liboats kcdo spgoros, ncsas, wonho ota.

(i) Ncroatcle l` agmeho l` flfoetuf.

C` leo pmrtcako cs stmtclemry io`lro tgo alkkcscle tgoe tgo kceo l` cfpmat wckk io mkleh cts fltcle m`tor

alkkcscle.

Akmssc` camtcle l` alkkcscles(m) Le tgo imscs l` kceo l` cfpmat

(c) Gomn-le alkkcscle 2 C` tgo voklactcos l` tgo alkkcnceh pmrtcakos mro mkleh tgo smfo kceo io`lro

men m`tor tgo alkkcscle.

(cc) Likcquo alkkcscle 2 C` tgo voklactcos l` tgo alkkcnceh pmrtcakos mro mkleh nc`òroet kceos io`lro

men m`tor tgo alkkcscle.

(i) Le tgo imscs l` oeorhy 2

(c) Okmstca alkkcscle 2Ce me okmstca alkkcscle, tg o alkkcnceh pmrtcakos rohmce tgocr sgmpo men sczo

alfpkotoky m`tor alkkcscle. c.o., el `rmatcle l` foagmecamk oeorhy rofmces stlron ms no`lr-

fmtcle pltoetcmk oeorhy ce tgo ilncos. ^gus, dceotca oeorhy l` systof m`tor alkkcscle cs oqumk

tl dceotca oeorhy l` systof io`lro alkkcscle. ^gus ce mnnctcle tl tgo kceomr f lfoetuf, dc-

eotca oeorhy mksl rofmces alesorvon io`lro men m`tor alkkcscle.

(cc) Ceokmstca alkkcscle 2Ce me ceokmstca alkkcscle, tgo alkkcnceh pmrtcakos nl elt rohmce tgocr sgmpo

men sczo alfpkotoky m`tor alkkcscle. Qlfo `rmatcle l` foagmecamk oeorhy cs rotmceon iy tgo

alkkcnceh pmrtcakos ce tgo `lrf l` no`lrfmtcle pltoetcmk oeorhy. ^gus, tgo dceotca oeorhy l`

tgo pmrtcakos m`tor alkkcscle cs elt oqumk tl tgmt l` io`lro alkkcscle. Glwovor, ce tgo misoeaol` oxtoremk `lraos, kmw l` alesorvmtcle l` kceomr flfoetuf stckk glkns hlln.

(ccc) Zoròatky ceokmstca 2 C` voklacty l` sopmrmtcle mkleh tgo kceo l` cfpmat bust m`tor alkkcscle io-

alfos zorl tgoe tgo alkkcscle cs poròatky ceokmstca. Alkkcscle cs smcn tl io poròatky

ceokmstca c` iltg tgo pmrtcakos stcad tlhotgor m`tor alkkcscle men flvo wctg smfo voklacty,

Elto 2 Matumkky alkkcscle iotwooe mkk romk liboats mro eoctgor poròatky okmstca elr poròatky ceokmstca, cts ceokms-

tca ce emturo.


21/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 9=

Oxmfpkos l` kceo l` cfpmat men alkk cscles imson le kceo l` cfpmat(c) ^wl imkks M men I mro mpprlmagceh omag ltgor suag tgmt tgocr aoetors mro flvceh mkleh kceo AN.

Gomn le Alkkcscle

(cc) ^wl imkks M men I mro mpprlmagceh omag ltgor suag tgmt tgocr aoetor mro flvceh mkleh nltton kceos ms sglwe

ce `churo.

Likcquo Alkkcscle

(ccc) Imkk cs `mkkceh le m stmtclemry wonho.

Likcquo Alkkcscle

ALO@@CACOE^ L@ YOQ^C^_^CLE o)^go a lo`` c a c oet l` ros t c tut c le c s no` c eon ms tgo rmt c l l` tgo c fpuk s os l` ro`lrfmtc le men

no`lrfmtcle l` octgor ilny.

o 6eno`lrfmtcll`puksoCf

ero`lrfmtcll`puksoCf6

nt@

nt@

n

r

6cfpmatl`kceomklehmpprlmagl`Poklacty

cfpmatl`kceomklehsopormtclel`Poklacty

^go flst hoeormk oxprosscle `lr alo``cacoet l` rostctutcle cs

o 6cfpmatl`kceomklehaletmatl`plcetl`mpprlmagl`voklacty

cfpmatl`kceomklehaletmatl`plcetsl`sopmrmtclel`voklacty


22/48

ZGSQCAQ


Oxmfpko `lr amkaukmtcle l` o

^wl sflltg imkks M men I mpprlmagceh omag ltgor suag tgmt tgocr aoetors mro flvceh mkleh kceo AN ce

misoeao l` oxtoremk cfpukscvo `lrao. ^go voklactcos l` M men I bust io`lro alkkcscle io u=

men u9

rospoatcvoky.

^go voklactcos l` M men I bust m`tor alkkcscle io v=

men v9

rospoatcvoky.

Kceo l` cfpmat

u = u 9

Bust Io`lro alkkcscle

A NM I

Kceo l` cfpmat

v= v9

Bust M`tor alkkcscle

A NM I

EN

f=

EN

f9

u=

u9

EY

f=EY

f9

vv

f= f9

v=

v9

No`lrfmtcle Yo`lrfmtcle

u 5 u= 9 v ; v= 9

@oxt

6 : flfoetuf cs alesorvon `lr tgo systof.

f=u

=+ f

9u

96 (f

=+ f

9)v 6 f

=v

=+ f

9v

9

v 69=

99==

ff

ufuf

69=

99==

ff

vfvf

.......(=)

Cfpukso l` No`lrfmtcle 2

BN

6 agmeho ce flfoetuf l` mey leo ilny nurceh no`lrfmtcle.

6 f9(v u

9) `lr f

9

6 f= (v + u=) `lr f=

Cfpukso l` Yo`lrfmtcle 2

BY

6 agmeho ce flfoetuf l` mey leo ilny nurceh Yo`lrfmtcle.

6 f9(v

9 v) `lr f

9

6 f=(v v

=) `lr f

=

o 6)(

)(

N

Y

BeNo`lrfmtcll`Cfpukso

BeYo`lrfmtcll`Cfpukso

69=

=9

uu

vv

6

cfpmatl`kceomklehmpprlmagl`Poklacty

cfpmatl`kceomklehsopmrmtclel`Poklacty

Elto 2 o cs cenopoenoet l` sgmpo men fmss l` liboat iut nopoens le tgo fmtorcmk.

^go alo``cacoet l` rostctutcle cs alestmet `lr m pmcr l` f mtorcmks.

(m) o 6 = Cfpukso l` Yo`lrfmtcle 6 Cfpukso l` No`lrfmtcle Poklacty l` sopmrmtcle 6 Poklacty l` mpprlmag Dceotca oeorhy l` pmrtcakos m`tor alkkcscle fmy io oqumk tl tgmt l` io`lro alkkcscle. Alkkcscle cs okmstca.

(i) o 6 : Cfpukso l` Yo`lrfmtcle 6 : Poklacty l` sopmrmtcle 6 : Dceotca oeorhy l` pmrtcakos m`tor alkkcscle cs elt oqumk tl tgmt l` io`lro alkkcscle. Alkkcscle cs por`oatky ceokmstca .

(a) : ; o ; = Cfpukso l` Yo`lrfmtcle ; Cfpukso l` No`lrfmtcle Poklacty l` sopmrmtcle ; Poklacty l` mpprlmag

Dceotca oeorhy l` pmrtcakos m`tor alkkcscle cs elt oqumk tl tgmt l` io`lro alkkcscle. Alkkcscle cs Ceokmstca.


23/48

ZGSQCAQ


Elto 2 Ce amso l` aletmat alkkcscles o cs mkwmys koss tgme uecty.

: o =

Cfplrtmet Zlcet 2

Ce amso l` okmstca alkkcscle, c` rluhg sur`mao cs prosoet tgoe

d`; d

c (ioamuso `rcatcle cs cfpukscvo)

Xgoro, d cs Dceotca Oeorhy.

/////////////////////////////////////

Yluhg

M pmrtcako I flvceh mkleh tgo nltton kceo alkkcnos wctg m rln mksl ce stmto l` fltcle ms sglwe ce tgo `churo.

^go pmrtcako I alfos ce aletmat wctg plcet A le tgo rln.

^l wrcto nlwe tgo oxprosscle `lr alo``cacoet l` rostctutcle o, wo `crst nrmw tgo kceo l` cfpmat. ^goe wo roslkvo

tgo alfpleoets l` voklactcos l` plcets l` aletmat l` iltg tgo ilncos mkleh kceo l` cfpmat bust io`lro men bust

m`tor alkkcscle.

^goe o 6x9x=

x=x9

uu

vv

Alkk cscle ce leo ncfoescle Gomn le)f= f9

u= u9

(m)Io`lro Alkkcscle

f= f9

v= v9

(i)M`tor Alkkcscle

u=

5 u9

v95 v

=

o 69=

=9

uu

vv

(u= u

9)o 6 (v

9 v

=)

Iy flfoetuf alesorvmtcle,

f=

u=

+ f9

u9

6 f=

v=

+ f9

v9

v96 v

=+ o(u

= u

9)

men v=6

9=

9=999==

ff

)uu(ofufuf

v96

9=

9==99==

ff

)uu(ofufuf

Qpoacmk Amso 2

(=) o 6 :

v=

6 v9

`lr por`oatky ceokmstca alkkcscle, iltg tgo ilncos, flvo wctg smfo vok. m`tor alkkcscle.


24/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 9>

(9) o 6 =

men f=6 f

96 f,

wo hot v=6 u

9men v

96 u

=

c.o., wgoe twl pmrtcakos l` oqumk fmss alkkcno okmstcamkky men tgo alkkcscle cs gomn le, tgoy ox-

agmeho tgocr voklactcos., o.h.

f f9f/s

Io`lro Alkkcscle

v 6:=

(1) f=55 f

9

f=+ f

9 f=men :ff

=

9

v=

6 u=

El agmeho

men v96 u=+ o(u= u9)

Oxmfpko 9:. ^wl cnoetcamk imkks mro mpprlmagceh tlwmrns omag ltgor le m strmchgt kceo wctg voklacty 9 f/s

men > f/s rospoatcvoky. @cen tgo `cemk voklactcos, m`tor okmstca alkkcscle iotwooe tgof.

ff >f/s9f/s

Qlkutcle 2

^go twl voklactcos wckk io oxagmehon men tgo `cemk fltcle cs rovorso l` cectcmk fltcle `lr iltg.

f f>f/s 9f/s

Oxmfpko 9=. ^groo imkks M, I men A l` smfo fmss f mro pkmaon le m `rcatclekoss glrczletmk pkmeo ce m

strmchgt kceo ms sglwe. Imkk M cs flvon wctg voklacty u tlwmrns tgo fcnnko imkk I. C` mkk tgo

alkkcscles mro okmstca tgoe, `cen tgo `cemk voklactcos l` mkk tgo imkks.

M I Au

f f f

//////////////////////////////////////////

Qlkutcle 2

M alkk cnos okms tcamkky wctg I men alf os tl rost iut I stmrts flvceh wctg voklacty u

M I Au

f f f

//////////////////////////////////////////

M`tor m wgcko I alkkcnos okmstcamkky wctg A men alfos tl rost iut A stmrts flvceh wctg voklacty u

M I A u

f f f

//////////////////////////////////////////

@cemk voklactcosP

M6 :4

PI

6 : men PA

6 u Mes.


25/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 9?

Oxmfpko 99. @lur cnoetcamk imkks M, I, A men N mro pkmaon ce m kceo le m `rcatclekoss glrczletmk sur`mao. M men

N mro flvon wctg smfo spoon u tlwmrns tgo fcnnko ms sglwe. Mssufceh okmstca alkkcscles,

`cen tgo ` cemk voklactcos.

M I Au

N

/////////////////////////////////////////////////////

u

Qlkutcle 2

M men N alkk cnos okmstca mkky wctg I men A ros poat cvoky men alfo tl ros t iut I men A stmrts

flvceh wctg voklacty u tlwmrns omag ltgor ms sglwe

M I Au

N

/////////////////////////////////////////////////////

u

I men A alkkcnos okmstcamkky men oxagmeho tgocr voklactcos tl flvo ce lpplscto ncroatcles

M I Au

N

/////////////////////////////////////////////////////

u

Elw, I men A alkkcnos okmstcamkky wctg M men N rospoatcvoky men alfo tl rost iut M men N stmrts

flvceh wctg voklacty u mwmy `rlf omag ltgor ms sglwe

M I Au

N

/////////////////////////////////////////////////////

u

@cemk voklactcos PM

6 u ( ) 4 PI

6 :4 PA

6 : men PN

6 u ( ) Mes.

Oxmfpko 91. ^wl pmrtcakos l` fmss f men 9f flvceh ce lpplscto ncroatcles le m `rcatclekoss sur`mao alkkcno

okmstcamkky wctg voklacty v men 9v rospoatcvoky. @cen tgocr voklactcos m`tor alkkcscle, mksl `cen tgo

`rmatcle l` dceotca oeorhy klst iy tgo alkkcnceh pmrtcakos.

f 9f9v v

Qlkutcle 2

Kot tgo `cemk voklactcos l` f men 9f io v=

men v9rospoatcvoky ms sglwe ce tgo `churo2

f 9fv9

Iy alesorvmtcle l` flfoetuf2

f(9v) + 9f(v) 6 f(v=) + 9f (v

9)

lr : 6 fv=+ 9fv

9

lr v=+ 9v

96 : .........(=)

men sceao tgo alkkcscle cs okmstca2

v9 v

=6 9v (v)

lr v9 v

=6 1v .........(9)

Qlkvceh tgo milvo twl oqumtcles, wo hot,

v96 v men v =6 9v Mes.

c.o., tgo fmss 9f rotures wctg voklacty v wgcko tgo fmss f rotures wctg voklacty 9v ce tgo

ncroatcle sglwe ce `churo2

f 9fv

^go alkkcscle wms okmstca tgoro`lro, el dceotca oeorhy cs klst, DO klss 6 DOc- DO

`

lr,

9999 v)f9(

9

=)v9(f

9

=)v)(f9(

9

=)v9(f

9

=6 :


26/48

ZGSQCAQ


Oxmfpko 9>. Le m `rcatclekoss sur`mao, m imkk l` fmss f flvceh mt m spoon v fmdos m gomn le alkkcscle wctg

me cnoetcamk imkk mt rost. ^go dceotca oeorhy l` tgo imkks m`tor tgo alkkcscle cs 1/>tg l` tgo lrchcemk.

@cen tgo alo``cacoet l` rostctutcle.

Qlkutcle 2

Ms wo gmvo sooe ce tgo mi lvo nc sa us sc le, tgmt ueno r tgo hc voe al enctcl es 2

Iy usceh alesorvmtcle l` kceomr flfoetuf men oqumtcle l` o, wo hot,

v9

o='v=

men v

9

o='v 9

Hcvoe tgmt c` D>

1D

lr9

=fv

=9 +

9

=fv

99 6

>

1

9fv9

=

Quistctutceh tgo vmkuo, wo hot

9

9

o=

+

9

9

o=

6

>

1

lr o 69

=Mes.

Oxmfpko 9?. M iklad l` fmss 9 dh cs pusgon tlwmrns m vory gomvy liboat flvceh wctg 9 f/s aklsor tl tgo

iklad (ms sglwe). Mssufceh okmstca alkkcscle men `rcatclekoss sur`maos, `cen tgo `cemk voklactcos

l` tgo iklads.

9d h

vorygomvyliboat

=:f/s

9f/s

/////////////////////////////////////////////////////////////////////

Qlkutcle 2

Kot v=men v

9io tgo `cemk voklactcos l` 9dh iklad men gomvy liboat rospoatcvoky tgoe,

v=

6 u=+ = (u

= u

9) 6 9u

= u

96 => f/s

v96 9f/s

Oxmfpko 90. M imkk cs flvceh wctg voklacty 9 f/s tlwmrns m gomvy wmkk flvceh tlwmrns tgo imkk wctg spoon

=f/s ms sglwe ce `ch. Mssufceh alkkcscle tl io okmstca, `cen tgo voklacty l` tgo imkk cffoncmtoky

m`tor tgo alkkcscle.


27/48

ZGSQCAQ


Qlkutcle 2

^go spoon l` wmkk wckk elt agmeho m`tor tgo alkkcscle. Ql, kot v io tgo voklacty l` tgo imkk

m`tor alkkcscle ce tgo ncroatcle sglwe ce `churo. Qceao alkkcscle cs okmstca (o 6 =),

sopmrmtcle spoon 6 mpprlmag spoon

lr v = 6 9 + =

lr v 6 > f/s Mes.

Oxmfpko 98. ^wl imkks l` fmssos 9 dh men > dh mro flvon tlwmrns omag ltgor wctg voklactcos > f/s men

9 f/s rospoatcvoky le m `rcatclekoss sur`mao. M`tor alkkcnceh tgo 9 dh imkk rotures imad wctg

voklacty 9f/s.

9dh >dh>f/s 9f/s

//////////////////////////////////////////// ////////////////////////////////////////

9dh9f/s

>dh v9

Bust i o`l ro alkkc sc le Bust m tor alkkc sc le

^goe `cen2

(m) voklacty l` > dh imkk m`tor alkkcscle

(i) alo` cacoet l` rostctutcle o.

(a) Cfpukso l` no`lrfmtcle BN

.

(n) Fmxcfuf pltoetcmk oeorhy l` no`lrfmtcle.

(o) Cfpukso l` ro`lrfmtcle BY

.

Qlkutcle 2

(m) Iy flfoetuf alesorvmtcle,

9(>) >(9) 6 9(9) + >(v9) v

96 = f/s

(i) o 6mpprlmagl`voklacty

sopmrmtclel`voklacty6

0

1

)9(>

)9(=

6 :.?

(a) Mt fmxcfuf no`lrfon stmto, iy alesorvmtcle l` flfoetuf, alffle voklacty cs v 6 :.

BN

6 f=(v u

=) 6 f

9(v u

9) 6 9(: >) 6 7 E -s

6 >(: 9) 6 7 E - s

lr 6 >(: 9) 6 7 E - s

(n) Zltoetcmk oeorhy mt fmxcfuf no`lrfon stmto _ 6 klss ce dceotca oeorhy nurceh no`lrfmtcle.

lr _ 6

999

9== uf

9

=uf

9

=

9

=(f

=+ f

9)v9

6

99 )9(>

9

=)>(9

9

=

9

=(9 + >) (:)9

lr _ 6 9> Bluko

(o) BY

6 f=(v

= v) 6 f

9(v v

9) 6 9 (9 :) 6 > E-s

lr 6 >(: =) 6 > E-s

lr o 6N

Y

B

B

BY

6 oBN

6 (:.?) (7) 6 > E-s


28/48

ZGSQCAQ


Al kk c sc le ce t wl nc fo esc le l ik cq uo)=. M pmcr l` oqumk men lpplscto cfpuksos mat mkleh alffle elrfmk ncroatcle. Goeao, kceomr flfoe-

tuf l` cencvcnumk pmrtcakos nl agmeho mkleh alffle elrfmk ncroatcle. C` fmss l` tgo alkkcnceh

pmrtcakos rofmce alestmet nurceh alkkcscle, tgoe wo ame smy tgmt kceomr voklacty l` tgo cencvcnumk

pmrtcakos agmeho nurceh alkkcscle ce tgcs ncroatcle.

9. El alfpleoet l` cfpukso mat mkleh alffle tmehoet ncroatcle. Goeao, kceomr flfoetuf lr kceomr

voklacty l` cencvcnumk pmrtcakos (c` f mss cs alestmet) r ofmce ueagmehon mkleh tgcs ncroatcle.

1. Eot cfpukso le iltg tgo pmrtcakos cs zorl nurceh alkkcscle. Goeao, eot flfoetuf l` iltg tgopmrtcakos rofmce alesorvon io`lro men m`tor alkkcscle ce mey ncroatcle.

>. No`cectcle l` alo``cacoet l` rostctutcle ame io mppkcon mkleh alffle elrfmk ncroatcle, c.o., mkleh

alffle elrfmk ncroatcle wo ame mppky

Yokmtcvo spoon l` sopmrmtcle 6 o (rokmtcvo spoon l` mpprlmag)

Oxmfpko 97. M imkk l` fmss f gcts m `kllr wctg m spoon v:fmdceh me mehko l` ceacnoeao m wctg tgo elrfmk.

^go alo``cacoet l` rostctutcle cs o. @cen tgo spoon l` tgo ro`koaton imkk men tgo mehko l` ro`koatcle

l` tgo imkk.

Qlkutcle 2

^go alfpleoet l` voklacty v: mkleh alffle tmehoetcmk ncroatcle v: s ce wckk rofmce ue-agmehon. Kot v io tgo alfpleoet mkleh alffle elrfmk ncroatcle m`tor alkkcscle. Mppkyceh,

Yokmtcvo spoon l` sopmrmtcle 6 o (Yokmtcvo spoon l` mpprlmag)

mkleh alffle elrfmk ncroatcle, wo hot

v 6 ov:a ls

v sce:

v'

v (6 ov als ):

^gus, m`tor alkkcscle alfpleoets l` voklacty v mro v:s ce men ov

:a ls

9:

9: )alsov()scev('v Mes.

men tme 6

alsov

scev

:

:

lr tme 6o

tme Mes.

Elto 2 @lr okmstca alkkcscle, o 6 =

v 6 v:

men 6

Oxmfpko 93. M imkk l` fmss f fmdos me okmstca alkkcscle wctg meltgor cnoetcamk imkk mt rost. Qglw tgmt c` tgo

alkkcscle cs likcquo, tgo ilncos hl mt rchgt mehkos tl omag ltgor m`tor alkkcscle.


29/48

ZGSQCAQ


Qlkutcle 2

Ce gomn le okmstca alkkcscle iotwooe twl pmrtcakos, tgoy oxagmeho tgocr voklactcos. Ce tgcs

amso, tgo alfpleoet l` imkk = mkleh alffle elrfmk ncroatcle, v als

ioalfos zorl m`tor alkkcscle, wgcko tgmt l` 9 ioalfos v als . Xgcko tgo alfpleoetsmkleh alffle tmehoet ncroatcle l` iltg tgo pmrtcakos rofmce ueagmehon. ^gus, tgo alf-

pleoets mkleh alffle tmehoet men alffle elrf mk ncroatcle l` iltg tgo imkks ce tmiukmr

`lrf mro hcvoe ioklw.

Imkk

Io`lro alkkcscle M tor alkkcscle Io`lro alkkcscle M tor alkkcscle

= v sce v sce v als 9 v als

Alfpleoet mkleh alffle

tmehoet ncroatcle

Alfpleoet mkleh alffle elrfmk

ncroatcle

@rlf tgo milvo tmiko men `churo, wo soo tgmt iltg tgo imkks flvo mt rchgt mehko m`tor

alkkcscle wctg voklactcos v sce men v als .

Elto 2 Xgoe twl cnoetcamk ilncos gmvo me likcquo okmstca alkkcscle, wctg leo ilny mt rost io`lro alkkcscle, tgoe

tgo twl ilncos wckk hl ce ncroatcles.

Oxmfpko 1:. ^wl spgoros mro flvceh tlwmrns omag ltgor. Iltg

gmvo smfo rmncus iut tgocr fmssos mro 9 dh men >

dh. C` tgo voklactcos mro > f/s men 9 f/s rospoa-

tcvoky men alo``cacoet l` rostctutcle cs o 6 =/1, `cen. Y

I

M

9dh

>dh

9f/s

>f/s

Y

(m) ^go alffle voklacty mkleh tgo kceo l` cfpmat.

(i) @cemk voklactcos mkleh kceo l` cfpmat.

(a) Cfpuksol` no`lrfmtcle.

(n) cfpuksol` ro`lrfmtcle.

(o) Fmxcfuf pltoetcmk oeorhy l` no`lrfmtcle.( ) Klss ce dceotca oeorhy nuo tl alkkcscle.

Qlkutcle 2

A

I

Y

Y

M

Y

Kceo l` fltcle

Kceo l` cfpmat

Kceo l` fltcle9dh

>dh9f/s

>f/s


30/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 1:

Ce MIA sce 6MI

IA6

Y9

Y6

9

=lr6 1:

(m) Iy alesorvmtcle l` flfoetuf mkleh kceo l` cfpmat.

1:

1:>f/s

>sce1:

I >dh

>als1:

9als1:

Bust Io`lro Alkkcscle Mkleh KLC

9f/s

9dh

9sce1:

9sce1:

>sce1:

v

Fmxcfuf No`lrfon Qtmto

9(> als 1:) >(9als 1:) 6 (9 + >)v

lr v 6 : (alffle voklacty mkleh KL)

(i)

9sce1:

>sce1:

I

v=

v9

M9dh

>dh

Bust M`tor Alkkcscle Mkleh KLC

Kot v=men v

9io tgo cemk voklacty l` M men I rospoatcvoky tgoe, iy alesorvmtcle l` flfoetuf mkleh kceo

l` cfpmat,

9(>als 1:) >(9als 1:) 6 9(v=) + > ( v

9)

lr : 6 v=+ 9v

9......... (=)

Iy alo``cacoet l` rostctutcle,

KLmklehmpprlmagl`voklacty

KLmklehsopmrmtclel`voklactyo

lr

1

=6

1:als91:als>

vv =9

lr v

9 v

=6 1 ........ (9)

`rlf tgo milvo twl oqumtcles,

v=6 s/f

1

9men v

96 s/f

1

=.

(a) BN

6 f=(v u

=) 6 9(: > als 1:) 6 1> E-s

(n) BY

6 oBN

61

=(> 1 ) 6 1

>E-s

(o) Fmxcfuf pltoetcmk oeorhy l` no`lrfmtcle cs oqumk tl klss ce dceotca oeorhy nurceh no`lrfmtcle uptl

fmxcfuf no`lrfon stmto,

_ 69= f=(u= als)9 + 9

= f9(u9 als)9 9= (f= + f9)v

9

69

=9(> als 1:)9 +

9

=>(9als 1:)9

9

=(9 + >) (:)9 lr _ 6 =7 Bluko.

( ) Klss ce dceotca oeorhy,

DO 69

=f

=(u

=als)9 +

9

=f

9(u

9als)9

999

9== vf

9

=vf

9

=

69

=9(> als 1:)9 +

9

=>(9 als 1:)9

99

1

=>

9

=

1

99

9

=

DO 6 =0 Bluko


31/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 1=

Oxmfpko 1=. ^wl plcet pmrtcakos M men I mro pkmaon ce kceo le m `rcatclekoss glrczletmk pkmeo. C` pmrtcako M

(fmss = dh) cs flvon wctg voklacty =: f/s tlwmrns stmtclemry pmrtcako I (fmss 9 dh) men m`tor

alkkcscle tgo twl flvo mt me mehko l` >? wctg tgo cectcmk ncroatcle l` fltcle, tgoe `cen 2

=dh

M I

=:f/s 9dh

(m) Poklactcos l` M men I bust m`tor alkkcscle.

(i) Alo` cacoet l` rostctutcle.

Qlkutcle 2^go vory `crst stop tl slkvo suag prlikofs cs tl `cen tgo kceo l` cfpmat wgcag cs mkleh tgo ncroatcle l`

`lrao mppkcon iy M le I, rosuktceh tgo stmtclemry I tl flvo. ^gus, iy wmtagceh tgo ncroatcle l`

fltcle l` I, kceo l` cfpmat ame io notorfceon. Ce tgcs amso kceo l` cfpmat cs mkleh tgo ncroatcle l`

fltcle l` I. c.o. >? wctg tgo cectcmk ncroatcle l` fltcle l` M.

vM

>?

>? vI

Kceo l` cfpmat

y

x

(m) Iy alesorvmtcle l` flfoetuf, mkleh x ncroatcle2 fM uM6 fMvM als >? + fIvI als >?

lr =(=:) 6 =(vM

als >?) + 9(vI

(als >?)

lr vM

+ 9vI

6 9=: ............ (=)

mkleh y ncroatcle

: 6 fM

vM

sce >? + fI

vI

sce >?

lr : 6 =(vM

sce >?) 9(vI

sce >?)

lr vM

6 9vI

............ (9)

slkvceh tgo twl oqumtcles,

vM 6 9

=:

f/s men vI6 9

?

f/s.

(i)cfpoatl`kceomklehmpprlmagl`voklatcy

cfpoatl`kceomklehsopmrmtclel`voklactyo

uM

>?u als>?M

KL

M

vM

3:

v als 3:M

KL

IvI

M

Io`lro alkkcscle M`tor alkkcscle

lr o 6>?alsu

3:alsvv

M

MI6

9

=:

:9

?

69

=Mes.


32/48

ZGSQCAQ


Oxmfpko 19. M sflltg spgoro l` fmss f cs flvceh le m glrczletmk pkmeo wctg m voklacty bc1 wgoe ct

alkkcnos wctg m vortcamk wmkk wgcag cs pmrmkkok tl tgo voatlr b . C` tgo alo``cacoet l` rostctutcle

iotwooe tgo spgoro men tgo wmkk cs9

=, `cen

(m) tgo voklacty l` tgo spgoro m`tor cfpmat,

(i) tgo klss ce dceotca oeorhy amuson iy tgo cfpmat.(a) tgo cfpukso B

tgmt mats le tgo spgoro.

Qlkutcle 2

Kot v

io tgo voklacty l` tgo spgoro m`tor cfpmat.

^l `cen v

wo fust sopmrmto tgo voklacty alfpleoets pmrmkkok men porpoencaukmr tl tgo

wmkk.

_sceh tgo kmw l` rostctutcle tgo alfpleoet l` voklacty pmrmk-

kok tl tgo wmkk rofmces ueagmehon wgcko alfp leoet porpoe-

ncaukmr tl tgo wmkk ioalfos o tcfos ce lpplscto ncroatcle.

^gus, bc

9

1v

(m) ^goro`lro, tgo voklacty l` tgo spgoro m`tor cfpmat cs 6 bc9

1 Mes.

(i) ^ go kls s ce D. O. 69

=fv 9

9

=fu 9 6

9

=f( 19 + =9)

9

=f f7

98=

9

1 99

Mes.

(a) B

6 )u(f)v(fZZZ c`

6 f

bc

9

1 f bc1 6 cf

9

3 Mes.

Oxmfpko 11. ^wl sflltg spgoros, M men I, gmvceh oqumk rmncc, kco le m glrczletmk tmiko. M cs l` fmss f men I

cs l` fmss 1f. ^go spgoros mro prlboaton tlwmrns omag ltgor wctg voklacty voatlr b9c? men

bc9 rospoatcvoky men wgoe tgoy alkkcno tgo kceo blceceh tgocr aoetors cs pmrmkkok tl tgo voatlr c .

C` tgo alo``cacoet l` rostctutcle iotwooe M men I cs1

=, `cen tgo voklactcos m`tor cfpmat men tgo klss ce

dceotca oeorhy amuson iy tgo alkkcscle. @cen mksl tgo fmhectuno l` tgo cfpuksos tgmt mat mt tgo

cestmet l` cfpmat.

Qlkutcle 2

^go kceo l` aoetors mt cfpmat, cs pmrmkkok tl tgo voatlr c , tgo voklacty alfpleoets l` M men

I porpoencaukmr tl c mro ueagmehon iy tgo cfpmat.

9

uf 1f

v

=

y

x

Mppk yceh ales orvmtcle l` kceomr fl fo etuf men tgo kmw l` ro st ctutcle, wo gmvo

ce x ncroatcle ?f + (1f) (9) 6 fu + 1 fv .... (c)

men 1

=

(? 9) 6 v u .....(cc)


33/48

ZGSQCAQ


Qlkvceh tgoso oqumtcles, wo gmvo u 6 9 men v 6 1

^go voklactcos l` M men I m`tor cfpmat mro tgoro`lro,

b9c9 me n bc1 rospoatcvoky Mes.

Io`lro cfpmat tgo dceotca oeorhy l` M cs

9

=f( ?9 + 99) 6

9

93f

men l` I cs9

=(1f)(99 + =9) 6

9

=?f

M`to r cf pmat tgo dc eotc a oeor hy l` M cs

9

=f( 99 + 99) 6 >f

men l` I cs9

=(1f) (19 + =9) 6 =? f

^goro`lro, tgo klss ce D.O. mt cfpmat cs

9

93f +

9

=?f >f =?f 6 1f Mes.

^l `cen vmkuo l` B, wo alescnor tgo agmeho ce flfoetuf mkleh c `lr leo spgoro leky..

@lr spgoro I B 6 1f ( 1 9)

lr B 6 1f Mes.

Oxmfpko 1>. M sfmkk stook imkk M cs suspoenon iy me ceoxtoesciko tgromn l` koehtg

I

M

L 6 =.? `rlf L. Meltgor cnoetcamk imkk cs tgrlwe vortcamkky nlwewmrns

suag tgmt cts sur`mao rofmces bust ce aletmat wctg tgromn nurceh

nlwewmrn fltcle men alkkcnos okmstcamkky wctg tgo suspoenon imkk. C`

tgo suspoenon imkk bust alfpkotos vortcamk acrako m`tor alkkcscle,

amkaukmto tgo voklacty l` tgo `mkkceh imkk bust io`lro alkkcscle.

(h 6 =: fs9)

Qlkutcle 2

Poklacty l` imkk M bust m`tor alkkcscle cs hk?

Kot rmncus l` omag imkk io r men tgo blceceh aoetors l` tgo twl imkks fmdos me mehko wctgtgo vortcamk mt tgo cestmet l` alkkcscle, tgoe

sc e 6r9

r6

9

=lr 6 1:

I M

v als1::

v sce1::v:

Kot voklacty l` imkk I (bust io`lro alkkcscle) io v:. ^gcs voklacty ame io roslkvon cetl twl

alfpleoets, (c) v:als 1:, mkleh tgo kceo blceceh tgo aoetor l` tgo twl imkks men (cc) v

:s ce

1: elrfmk tl tgcs kceo. Gomn -le alkkcscle tmdos pkmao nuo tl v:

als 1: men tgo alfpl-

eoet v:sce 1: l` voklacty l` imkk I rofmces ueagmehon.

Qceao, imkk M cs suspoenon iy me ceoxtoesciko strceh, tgoro`lro, bust m`tor alkkcscle, ct ame

flvo mkleh glrczletmk ncroatcle leky. Goeao, m vortcamkky upwmrn cfpukso cs oxorton iy

tgromn le tgo imkk M. ^gcs fomes tgmt nurceh alkkcscle twl cfpuksos mat le imkk M scfuktm-

eolusky. Leo cs cfpukscvo cetormatcle B iotwooe tgo imkks men tgo ltgor cs cfpukscvo roma-

tcle B l` tgo tgromn.

Poklacty v=

l` imkk I mkleh kceo l` alkkcscle cs hcvoe iy

B fv:

als 1: 6 fv=


34/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 1>

lr v=

6f

B v

:als 1: ............ ( c)

Glrczletmk voklacty v9

l` imkk M cs hcvoe iy B sce 1: 6 fv9

lr v9

6f9

B........(cc)

fv=

fv als 1::

fv sce 1::

fv sce 1::B

B1:

'B'

M

fv9

I

sfmkk sczo

v9v sce1:9

v=

Qceao, tgo imkks alkkcno okmstcamkky, tgoro`lro, alo``cacoet l` rostctutcle cs o 6 =.

Goeao, o 6 =:1:alsv

)v(1:scev

:

=9

............ (ccc)

Qlkvceh Oqs. (c), (cc), men (ccc), B 6 =.0 fv:

als 1:

v=

6 :.0 v:

als 1: men v9

6 :.7 v:

als 1:

Qceao, imkk M bust alfpkotos vortcamk acrako, tgoro`lro v9

6 h?

:.7v

:als 1: 6

h? lr v

:6 =9.? fs= Mes.

PMYCMIKO FMQQ QSQ^OF 2C` m fmss cs mnnon lr oboaton `rlf m systof, mt rmto dh/s men rokmtcvo voklacty rokv

(w.r.t. tgo systof),

tgoe tgo `lrao oxorton iy tgcs fmss le tgo systof gms fmhectuno rokv

.

^grust @lrao ( t@

)

nt

nfv@ rokt

Qupplso mt slfo flfoet t 6 t fmss l` m ilny cs f men cts voklacty cs v . M`tor slfo tcfo mt t 6

t + nt cts fmss ioalfos (f nf) men voklacty ioalfos vnv

. ^go fmss nf cs oboaton wctg

rokmtcvo voklacty rv

. Mislkuto voklacty l` f mss nf cs tgoro`lro (v

+ rv

). C` el oxtoremk `lraos mro

matceh le tgo systof, tgo kceomr flfoetuf l` tgo systof wckk rofmce alesorvon, lr

`c ZZ

lr fv

6 (f nf) (v

+ nv

) + nf (v

+ rv

)

lr fv

6 fv

+ fnv

(n f) v

(n f) (n v

) + (nf)v

+ rv

nf

^go torf (nf) (nv

) cs tll sfmkk men ame io eohkoaton.

fn v

6 rv

nf


35/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" 1?

lr

nt

vnf

6

nt

nfvr

Goro,

nt

vnf

6 tgrust `lrao t@

men nt

nf6 rmto mt wgcag fmss cs oboatceh

lr

nt

nfv@ rt

Zrlikofs rokmton tl vmrcmiko fmss ame io slkvon ce `lkklwceh `lur stops

=. Fmdo m kcst l` mkk tgo `lraos matceh le tgo fmce fmss men mppky tgof le ct.

9. Mppky me mnnctclemk tgrust `lrao t@

le tgo fmss, tgo fmhectuno l` wgcag cs

nt

nfv r

men ncroa-

tcle cs hcvoe iy tgo ncroatcle l` rv

ce amso tgo fmss cs cearomsceh men ltgorwcso tgo ncroatcle l`

rv c` ct cs noaromsceh.

1. @cen eot `lrao le tgo fmss men mppky

nt

vnf@eot

(f 6 fmss mt tgo pmrtcaukmr cestmet)

>. Cetohrmto ct wctg prlpor kcfcts tl `cen voklacty mt mey tcfo t.

Elto 2 Zrlikofs l` leo-ncfoesclemk fltcle (wgcag mro flstky msdon ce BOO) ame io slkvon ce omscor

fmeeor bust iy msscheceh plsctcvo men eohmtcvo sches tl mkk voatlr qumetctcos. Goro mro `ow ox-

mfpko ce supplrt l` tgo milvo tgolry.

Oxmfpko 1?. M ` kmt amr l` fmss f:

stmrts flvceh tl tgo rchgt nuo tl m

alestmet glrczletmk `lrao @. Qmen spckks le tgo `kmt amr `rlf

m stmtclemry glppor. ^go rmto l` klmnceh cs alestmet men

o q u m k t l d h / s . @ c e n t g o t c f o n o p o e n o e a o l ` t g ovoklacty men tgo maaokormtcle l` tgo `kmt amr ce tgo prlaoss

l` klmnceh. ^go `rcatcle cs eohkchciky sfmkk.@

f:

Qlkutcle 2 Cectcmk voklacty l` tgo `kmt amr cs zorl. Kot v io cts voklacty mt tcfo t men f cts fmss mt tgmt

cestmet. ^goe

Mt t 6 :, v 6 : men f 6 f:

mt t 6 t, v 6 v men f 6 f:

+ tGor o, v

r6 v (imadwmrns)

nt

nf6

@t6 v

r ntnf 6v (imadwmrns)


36/48

ZGSQCAQ


Eot `lrao le tgo `kmt amr mt tcfo t cs @eo t

6 @ @t

lr fnt

nv6 @ v ....(c)

lr (f:

+t)nt

nv6@ v

lr v

: v@

nv

6 t

: : tf

nt

=

Te (@ v)V:

v 6 =

Te (f:

+t) V:

t

e

v@

@6e

:

:

f

tf

v@

@

6

:

:

f

tf lr v 6

tf

@t

: Mes.

@rlf Oq. (c),nt

nv6 maaokormtcle l` `kmt amr mt tcfo t

lr 6f

v@

m 6

tf

tf

t@@

:

:lr m 6 9

:

:

)tf(

@f

Mes.

Oxmfpko 10. M amrt klmnon wctg smen flvos mkleh m glrczletmk `kllr nuo tl m alestmet `lrao @ alceacnceh ce

ncroatcle wctg tgo amrts voklacty voatlr. Ce tgo prlaoss smen spckks tgrluhg m glko ce tgo ilttlf

wctg m alestmet rmtodh/s. @cen tgo maaokormtcle men voklacty l` tgo amrt mt tgo flfoet t, c` mttgo cectcmk flfoet t 6 : tgo amrt wctg klmnon smen gmn tgo fmss f

:men cts voklacty wms oqumk

tl zorl. @rcatcle cs tl io eohkoaton.

Qlkutcle 2 Ce tgcs prlikof tgo smen spckks tgrluhg m glko ce tgo ilttlf l` tgo amrt. Goeao, tgo rokmtcvo

voklacty l` tgo smen vrwckk io zorl ioamuso ct wckk maqucro tgo smfo voklacty ms tgmt l` tgo

amrt mt tgo flfoet.

vr6 :

v

@

v

f

^ gus , @t6 :

nt

nfv@ms rt

men tgo eot `lrao wckk io @ leky.

@eo t

6 @

lr f

nt

nv6 @ ....(c)

Iut goro f 6 f:

t

(f:

t)nt

nv6 @

lr

v

: nv 6 t

: : tf

nt@


37/48

ZGSQCAQ


v 6 t

:: )tf(e

@

lr v 6 @

ke

tf

f

:

:Mes.

@rlf oq. (c), maaokormtcle l` tgo amrt

m 6ntnv

6f

@

lr m 6tf

@

: Mes.

Yl adot prl pu ksc le 2Kot f

:io tgo fmss l` tgo rladot mt tcfo t 6 :. f cts fmss mt mey tcfo t men v cts voklacty mt tgmt

flfoet. Cectcmkky, kot us supplso tgmt tgo voklacty l` tgo rladot cs u.

@urtgor, kot

nt

nfio tgo fmss l` tgo hms oboaton por uect tcfo men v

rtgo oxgmust voklacty l` tgo

hmsos wctg rospoat tl rladot. _sumkky

nt

nfmen v

rmro dopt alestmet tgrluhglut tgo blureoy l`

tgo rladot. Elw, kot us wrcto `ow oqumtcles wgcag ame io uson ce tgo prlikofs l` rladot prlpuk-

scle. Mt tcfo t 6 t,

=. ^grust lrao le tgo rladot @t6 v

r

nt

nf(upwmrns)

9. X ochgt l` tgo rladot X 6 fh (nlwewmrns)

1. Eot lrao le tgo rladot @eot

6 @t X (upwmrns)

lr @eo t

6 vr

nt

nff h

>. Eot maaokormtcle l` tgo rladot m 6f@

lrnt

nv6

f

vr

nt

nfh

lr nv 6f

vr nf h nt

lr v

u rvnv

f

f: f

nfh

t

:nt

^gus, v 6 u ht + vr

e

f

f:

...(c)


38/48

ZGSQCAQ


Elto 2 =. @t6 v

r

nt

nfcs upwmrns, ms v

r cs nlwewmrns men

nt

nfcs eohmtcvo.

9. C` hrmvcty cs chelron men cectcmk voklacty l` tgo rladot u 6 :, Oq. (c) ronuaos tl v 6 vrk e

f

f:.

Oxmfpko 18. M rladot, wctg me cectcmk fmss l` =::: dh, cs kmueagon vortcamkky upwmrns `rlf rost uenor hrmvcty.

^go rladot iures `uok mt tgo rmto l` =: dh por soalen. ^go iuret fmttor cs oboaton vortcamkky

nlwewmrns wctg m spoon l` 9::: fs= rokmtcvo tl tgo rladot. C` iureceh stlps m`tor leo fceuto.

@cen tgo fmxcfuf voklacty l` tgo rladot. (^mdo h ms mt =: fs9)

Qlkutcle 2

_sceh tgo voklacty oqumtcle

v 6 u ht + vrk e

f

f:

Goro u 6 :, t 6 0:s, h 6 =: f/s 9, vr6 9::: f/s, f

:6 =::: dh

men f 6 =::: =: 0: 6 >:: dh

Xo hot v 6 : 0:: + 9::: ke

>::

=:::

lr v 6 9::: ke 9.? 0::

^go fmxcfuf voklacty l` tgo rladot cs 9::(=: ke 9.? 1) 6 =919.0 fs= Mes.

KCEOMY FLFOE^_F ALEQOYPM^CLE CE ZYOQOEAO L@ OW^OYEMK @LYAO.oxt@

6nt

Zn

oxt@

nt 6 Zn

Zn

6 oxt@

)cfpukscvo

nt

C` oxt@

)cfpukscvo

6 :

Zn

6 :

lr Z

cs alestmet

Elto2 Flfoetuf cs alesorvon c` tgo oxtoremk `lrao prosoet cs ele-cfpukscvo. oh. hrmvctmtcle lr sprceh `lrao


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"fmecsgdufmrpgyscas.ce" >:

Qlkutcle 2 fv als (f (u als )) 6 ntE

fv sce fu sce 6 ntE

men o 6

alsu

alsv v als 6 ou als

lr fv sce fu sce 6 (fv als + fu als)

lr v 6sce

uTsce a ls (o + =)V Mes.

Z r l i k o f = . ^groo pmrtcakos l` fmssos :.? dh, =.: dh men =.? dh mro pkmaon mt tgo tgroo alreors l` m rchgt

mehkon trcmehko l` scnos 1.: af, >.: af men ?.: af ms sglwe ce `churo. Klamto tgo aoetor l`

fmss l` tgo systof.

Qlkutcle 2 =.? dh

:.? dh=.: dh(M) (I)

(A)

x

y

1af?af

>af

tmdceh x men y mxos ms sglwe.

allrncemtos l` ilny M 6 (:,:)

allrncemtos l` ilny I 6 (>,:)

allrncemtos l` ilny A 6 (:,1)

x - allrncemto l` a.f. 6AIM

AAIIMM

fff

rFxfxf

6?.=:.=?.:

:?.=>:.=:?.:

61

>

dh

af6 af 6 =.11af

scfckmrky y - allrncemto l` a.f. 6?.=:.=?.:

1?.=::.=:?.:

61

?.>6 =.? af

Ql, aortor l` fmss cs =.11 af rchgt men =.? af milvo pmrtcako M.


41/48

ZGSQCAQ

"fmecsgdufmrpgyscas.ce" >=

Z r l i k o f 9 . ^go kceomr fmss noescty l` m strmchgt rln l` koehtg K vmrcos ms 6 M + Ix wgoro x cs tgoncstmeao `rlf tgo ko`t oen. Klamto tgo aoetor l` fmss `rlf ko`t oen.

Qlkutcle 2

Kot tmdo m strcp l` wcntg jnxj mt ncstmeao x `rlf leo oen.

nf 6 fmss l` jnxj strcp 6 ` nx

nf 6 (M+Ix)nx ......(=)

Iy no`cectcle Walf

6

K

L

K

L

nf

xnf

`rlf oq (=)Walf

6

K

L

K

L

nx)IxM(

nx)IxM(x

6

K

L

K

L

9

nx)IxM(

nx)IxMx(

6

9

IKMK

1

IK

9

MK

9

19

6

)IKM9(1

IK9MK1 9

Mes.

Z r l i k o f 1 . Gmk` l` tgo roatmehukmr pkmto sglwe ce `churo cs fmno l` m fmtorcmk l` noescty=men tgo ltgor

gmk` l` noescty9 ^go koehtg l` tgo pkmto cs K. Klamto tgo aoetor l` fmss l` tgo pkmto.

A= A9

=

L

K

Qlkutcle 2

Yopkmaceh gmk` l` roatmehukmr pkmto iy m plcet fmss mt cts aoetor l` fmss ms sglwe ce `churo.

F=6

9

K=i F

96

9

K9i

Walf

69=

>K

9>K

=

ff

1ff

6

i.9

K

.i.9

K

.

>

K1.i.

9

K.

>

Ki.

9

K.

9=

9=

6)(>

)1(

9=

9=

K `rlf plcet L.


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ZGSQCAQ

"fmecsgdufmrpgyscas.ce" >9

Z r l i k o f > . Ce m ilmt l` fmss > F men koehtgle m `rcatclekoss wmtor sur`mao. ^wl foe M (fmss 6 F) men

I (fmss 9F) mro stmenceh le tgo twl lpplscto oens. Elw M trmvoks m ncstmeao/> rokmtcvo tl

ilmt tlwmrns cts aoetor men I flvos m ncstmeao 1/> rokmtcvo tl ilmt men foot M. @cen tgo

ncstmeao trmvokkon iy tgo ilmt le wmtor tckk M men I foot.

Qlkutcle 2 Kot x cs ncstmeao trmvokkon iy ilmt.

Cectcmk plsctcle l` aoetor l` fmss

6IMIlmt

IIMMIlmtIlmt

FFF

WFWFWF

6F9FF>

.F9:.FF>9

6

F8

F> 6

8

>

@cemk plsctcle l` aoetor l` fmss

6

F8

xF9xFxF>>>9

6F8

Fx8F99

F>

F

6F8

xF8>F==

68

x8>F==

sceao tgoro cs el glrczletmk `lrao, plsctcle l` aoetor l` fmss rofmces ueagmehon.

aoetor l` fmss cectcmkky 6 aoetor l` fmss `cemkky

8

>6

8

x8>

==

>6 x8>

== x 697

?

Z r l i k o f ? . M iklad M (fmss 6 >F) cs pkmaon le tgo tlp l` m wonho I l` imso koehtg k (fmss 6 9: F) ms

sglwe ce `churo. Xgoe tgo systof cs rokomson `rlf rost. @cen tgo ncstmeao flvon iy tgo wonho

I tckk tgo iklad M romagos mt klwost oen l` wonho. Mssufo mkk sur`maos mro `rcatclekoss.

Qlkutcle 2 Cectcmk plsctcle l` aoetor l` fmss

6II

MMII

FFFWFW

6

F9>F>.F9:.WI 6

0W? I

@cemk plsctcle l` aoetor l` fmss

6F9>

Fx>F9:)xW( I 6

0

x)xW(? I

sceao tgoro cs el glrczletmk `lrao le systof

aoetor l` fmss cectcmkky 6 aoetor l` fmss `cemkky.

?WI

+6 ?WI

+ ?x + x

6 0x

0x


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ZGSQCAQ


Z r l i k o f 0 . Me cslkmton pmrtcako l` fmss f cs flvceh ce m glrczletmk xy pkmeo, mkleh x-mxcs. Mt m aortmce

gochgt milvo hrluen, ct sunnoeky oxpklnos cetl twl `rmhfoets l` fmssos f/> men 1f/>. Me

cestmet kmtor, tgo sfmkkor `rmhfoet cs mt y 6 + =? af. @cen tgo plsctcle l` gomvcor `rmhfoet mt

tgcs cestmet.

Qlkutcle 2 Ms pmrtcako cs flvceh mkleh x-mxcs, sl, y-allrncemto l` ALF cs zorl.

SF

F 6>FS

>

F+

>F1S

>

F1 : F 6 =?

>

F+

>F1S

>

F1

af?>

S F1

Z r l i k o f 8 . M sgokk cs `cron `rlf m ameele wctg m spoon l` =:: f/s mt me mehko 1: wctg tgo vortcamk

(y-ncroatcle). Mt tgo gchgost plcet l` cts trmboatlry, tgo sgokk oxpklnos cetl twl `rmhfoets l`

fmssos ce tgo rmtcl = 2 9. ^go kchgtor `rmhfoet flvos vortcamkky upwmrns wctg me cectcmk spoon l`

9:: f/s. Xgmt cs tgo spoon l` tgo gomvcor `rmhfoet mt tgo tcfo l` oxpklscle.

Qlkutcle 2 Poklacty mt gchgost plcet 6 u als 6 =:: als 0: 6 ?:f/s.

tmdceh, x men y mxos ms sglwe ce `churo,

Poklacty l` f 6 ?:c f / soa

f oxpklnos cetl1

fmen

1

f9, voklacty l`

1

f6 9::b f/soa

Kot, voklacty l`1

f96

P

Mppkyceh kmw l` alesorvmtcle l` flfoetuf

f ?: c 6 1

f 9:: b +

1

f9 P ?: c 1

9::b 6

1

9 P

sl

P 6 8? c =:: b

spoon 6 |

P| 6 99 )=::()8?( 6 9? =03 6 =9? f/soa Mes.

Z r l i k o f 7 . M sgokk mt rost mt lrchce oxpklnos cetl tgroo `rmhfoets l` fmssos = dh, 9 dh men f dh. ^go

`rmhfoets l` fmssos = dh men 9 dh `ky l`` wctg spoons =9 f/s mkleh x-mxcs men 7 f/s mkleh y-

mxcs rospoatcvoky. C` f dh `kcos l`` wctg spoon >: f/s tgoe `cen tgo tltmk fmss l` tgo sgokk.

Qlkutcle 2 Ms cectcmk voklacty 6 :, Cectcmk flfoetuf 6 (= + 9 + f) : 6 :

@cemkky, kot voklacty l` F 6

P. X o delw |

P| 6 >: f/s.

Cectcmk flfoetuf 6 `cemk flfoetuf .

: 6 = =9 c + 9 7 b + f

P

P 6 f

)b=0c=9(

|

P| 6 9

99

f

)=0()=9( 6

f

=99

)=0()=9( 6 >: {hcvoe}

f 6 >:

)=0()=9( 99 6 :.? dh

^ltmk fmss 6 = + 9 + :.? 6 1.? dh


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ZGSQCAQ

"fmecsgdufmrpgyscas.ce" >>

Z r l i k o f 3 . M iklad flvceh glrczletmkky le m sflltg sur`mao wctg m spoon l` 9: f/s iursts cetl twl oqumk

pmrts aletceuceh ce tgo smfo ncroatcle. C` leo l` tgo pmrts flvos mt 1: f/s, wctg wgmt spoon

nlos tgo soalen pmrt flvo men wgmt cs tgo `rmatclemk agmeho ce tgo dceotca oeorhy l` tgo

systof.

Qlkutcle 2

Mppkyceh flfoetuf alesorvmtcle 4

f 9: 69

fP +

9

f 1: 9: 6

9

P+ =?

Ql, P 6 =: f/s

cectcmk dceotca oeorhy 69

=f (9:)9 6 9:: f

`cemk dceotca oeorhy 69

=.

9

f. (=:)9 +

9

=

9

f(1:)9 6 9? f + 99? f 6 9?: f

`rmatclemk agmeho ce dceotca oeorhy 6O.Dcectcmk

)OD.cectcmk()OD.`cemk( 6

f9::

f9::f9?: 6

>

=

Z r l i k o f = : . M iklad mt rost oxpklnos cetl tgroo oqumk pmrts. ^wl pmrts stmrts flvceh mkleh W men S mxos

rospoatcvoky wctg oqumk spoons l` =: f/s. @cen tgo cectcmk voklacty l` tgo tgcrn pmrt.

Qlkutcle 2

Kot tltmk fmss 6 1 f, cectcmk kceomr flfoetuf 6 1f :

Kot voklacty l` tgcrn pmrt 6

P

_sceh alesorvmtcle l` kceomr flfoetuf 2

f =: c + f =: b + f P 6 :

Ql,

P 6 ( =: c =: b ) f/soa.

|

P| 6 99 )=:()=:( 6 =: 9 , fmdceh mehko =1?l ioklw x-mxcs

Z r l i k o f == . ^wl porsles M men I, omag l` fmss f mro stmenceh mt tgo twl oens l` rmck-rlmn amr l`

fmss F. ^go porsle M bufps tl tgo ko`t wctg m glrczletmk spoon u wctg rospoat tl tgo amr.

^gorom`tor, tgo porsle I bufps tl tgo rchgt, mhmce wctg tgo smfo glrczletmk spoon u wctg

rospoat tl tgo amr. @cen tgo voklacty l` tgo amr m`tor iltg tgo porsles gmvo bufpon l``.

Qlkutcle 2 Xgoe porsle M bufps, kot amr magcovo voklacty P ce `lrwmrn ncroatcle. Ql, voklacty l` M w.r.t.

hrluen 6 (u v) ce imadwmrn ncroatcle.


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ZGSQCAQ


Z r l i k of = 1. M iklad l` fmss =.9 dh flvceh mt m spoon l` 9: af/s alkkcnos gomn-le wctg m scfckmr iklad dopt

mt rost. ^go alo``cacoet l` rostctutcle cs 1/?. @cen tgo klss l` tgo dceotca oeorhy nurceh tgo

alkkcscle

Qlkutcle 2 Io`lro alkkcscle 2

M`tor alkkcscle 2 kot ` cemk voklacty l` M men I io u men P rospoatcvoky

Mppkyceh flfoetuf alesorvmtcle 2

=.9 :.9 6 =.9 u + =.9 P

:.9> 6 =.9 (u + P) .......(=)

alo``cacoet l` rostctutcle

o 69.:

uP 6 ?1 .........(9)

mppkyceh flfoetuf alesorvmtcle 2

=.9> + =.9 P 6 :.9u u +P - :.9 f/s ..........(9)

Qlkvceh oqumtcle (=) men (9) wo gmvo ,

P 69?

>f/soa men u 6

9?

=f/soa

Klss l` dceotca oeorhy 6 (cectcmk dceotca oeorhy) (`cemk dceotca oeorhy)

69

= =.9 (:.9)9

99

9?

=)9.=(

9

=

9?

>9.=

9

=

6 :.0

09?

=

09?

=0:>.:

6 :.0 {:.:> :.:9?0 :.::=0}

6 :.0 {:.:=97} 6 :.::807 B 6 8.8 =:1 B

Z r l i k o f = > . ^go spgoro l` fmss f=trmvoks wctg me cectcmk voklacty u

=ncroaton ms sglwe men strcdos tgo

stmtclemry spgoro l` f mss f9gomn le. @lr m hcvoe alo``cacoet l` rostctutcle o, wgmt alenc-

tcle le tgo fmss rmtcl9

=f

foesuros tgmt tgo `cemk voklacty l` f

9 cs hromtor tgme u

=8

=u

P6

9= ff

)o=(,f

6=

9

f

f=

)o=(

=

9

f

f=

)o=(

6

=u

P5 = {hcvoe}

o

=

f

f

9

= Mes.

Z r l i k o f = ? . @cen tgo fmss l` tgo rladot ms m `ueatcle l` tcfo, c` ct flvos wctg m alestmet maaokormtcle

m, ce misoeao l` oxtoremk `lraos. ^go hms osampos wctg m alestmet voklacty u rokmtcvo tl tgo

rladot men cts cectcmk fmss wms f:.

Qlkutcle 2 _sceh, @eot

6 Prok

nt

nf

@eot

6 unt

nf.......(=)

@eot

6 fm ......(9)

Qlkvceh oqumtcle (=) men (9)

fm 6 unt

nf

f

flf

nf6

t

l u

mnte

lf

f6

u

mt

lf

f6 u/mto u

mt

off :

Mes.

Z r l i k o f = 0 . C` tgo agmce cs klworon mt m alestmet spoon v 6 =.9 f/s, notorfceo tgo elrfmk romatcle

oxorton le tgo `kllr ms m `ueatcle l` tcfo. ^go agmce gms m fmss l` 7: dh men m tltmk koehtg

l` 0 f.

0f

v6=.9 f/s

Qlkutcle 2 m`tor tcfo t

Kceomr fmss noescty l` agmce 6 0

7:6 1

>:dh/f 6


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ZGSQCAQ

Xochgt l` plrtcle l` agmce kyceh le hrluen 6 fh 61

>: =.9 t

1

>: =.9 t 6 =0 t, E

^grust ` lrao 6 @t

6 P nt

nf

6 =.9 v 6 =.9v

6 =.9 1

>:

=.9

6 =.9 >: :.> 6 =3.9E

Elrfmk romatcle oxorton le `kllr

6 X + @t

6 (=0t + =3.9) E vortcamkky nlwewmrn

Z r l i k o f = 8 . M imkk cs mpprlmagceh tl hrluen wctg spoon u. C` tgo alo``cacoet l` rostctutcle cs o tgoe `cen lut2

f

u

/////////////////////////

(m) tgo voklacty bust m`tor alkkcscle.

(i) tgo cfpukso oxorton iy tgo elrfmk nuo tl hrluen le tgo imkk.

Qlkutcle 2

o 6

mpprlmgl`voklacty

sopmrmtclel`voklacty6

u

v

(m) voklacty m`tor alkkcscle 6 P 6 ou ........(=)

(i) Cfpukso oxorton iy tgo elrfmk nuo tl hrluen le tgo imkk 6 agmeho ce flfoetuf l` imkk.

6 {`cemk flfoetuf} {cectcmk flfoetuf}

6 {f v} { fu}

6 fv + fu

6 f {u + ou}

6 fu {= + o} Mes.

Centre of Mass Theory_E

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