3U]\NáDG5DPDÄWUyMSU]HJXERZD´

:\]QDF]\üUHDNFMHZUDPLHRSRGDQ\PVFKHPDFLHVWDW\F]Q\P

q

l

l

l

5R]ZL]DQLH

8ZDOQLDP\XNáDG]ZL ]yZZSURZDG]DMFRGSRZLDGDMFHLPUHDNFMH

q

1

HB

HA

A

B

V

V

A

B

3U]HGVWDZLRQ\ XNáDG VLá PXVL VSHáQLDü ZDUXQNL UyZQRZDJL D LORü QLHZLDGRP\FK

VNáDGRZ\FK UHDNFML Z\QRVL =DWHP WH UyZQDQLD UyZQRZDJL QLH Z\VWDUF] GR Z\]QDF]HQLD

ZV]\VWNLFK QLHZLDGRP\FK ']L NL ZVSyOQHM OLQLL G]LDáDQLD UHDNFML +A i HB PR*OLZH MHVW

REOLF]HQLH ZDUWRFL VNáDGRZ\FK SLRQRZ\FK UHDNFML 9A i VB 3RQLHZD* Z SXQNFLH $

SU]HFLQDMVL OLQLHG]LDáDQLDWU]HFK]F]WHUHFKQLHZLDGRP\FKUHDNFMLZ\NRU]\VWDP\UyZQDQLH

równowagi

M

∑

= 0 do obliczenia reakcji V

iA

B (wybór tego bieguna eliminuje z równania i

SR]RVWDáH QLHZLDGRPH 3R ]DVWSLHQLX REFL*HQLD FLJáHJR MHJR Z\SDGNRZ : R ZDUWRFL

równej TO SU]\áR*RQ Z URGNX RGFLQND REFL*HQLD UyZQDQLH WR GOD QDV]HJR ]DGDQLD

SU]\MPXMHSRVWDü9B 2 l - W l/2 LVWGVB= ql/4.

Obliczenie VA PR*HP\ SU]HSURZDG]Lü DQDORJLF]QLH Z\NRU]\VWXMF UyZQDQLH UyZQRZDJL

M

∑

= 0 3U]\MPXMHRQRSRVWDü9

iB

A 2 l - W l/2 VNGVA= -ql/4 (znak minus oznacza, i

*HUHDNFMDWDPD]ZURWSU]HFLZQ\GR]DáR*RQHJRQDU\VXQNX

Trzecie równanie równowagi (np.

P

∑ = 0 SR]ZDOD MHG\QLH XVWDOLü ]DOH*QRü PL G]\

ix

i

reakcjami HA i HB.

3R SRG]LHOHQLX XNáDGX Z SU]HJXELH U\VXQHN SRQL*HM RWU]\PXMHP\ GZD XNáDG\ VLá ]

nowymi niewiadomymi V1 i H1 RGG]LDá\ZDQLDPLZW\PSRáF]HQLX=\VNXMHP\GRGDWNRZR

UyZQDQLDUyZQRZDJL NWyUH PXV] E\ü VSHáQLRQH ]DUyZQR SU]H] XNáDG VLá 9A , HA ,V1 , H1 i

:OHZDVWURQDSRG]LDáXLSU]H]XNáDG9B ,HB V1 i H1 SUDZDVWURQDSRG]LDáX

V

1

1

H1

H

V

1

1

W

HA

H

A

B

B

ql/4

ql/4

0DMFQDFHOXZ\]QDF]HQLHUHDNFML+A i HBZ\NRU]\VWDP\W\ONRWH]UyZQDUyZQRZDJLZ

NWyU\FKQLHZ\VWSLQRZHQLHZLDGRPH91 i H13XQNWHPJG]LHSU]HFLQDMVL OLQLHG]LDáDQLD

W\FK UHDNFML MHVW SU]HJXE SU]\MPLMP\ ZL F MDNR ELHJXQ REOLF]DQLD PRPHQWyZ WHQ SXQNW

=DWHPUyZQDQLHUyZQRZDJLGODF] FLSUDZHMSU]\MPLHSRVWDü

M prawa

0

∑

= ⇒ V

i1

B l - HB l VNGHB=ql/4, i

DGODF] FLOHZHM

M lewa

0

∑

= ⇒ ql/4 l + H

i1

A l + Wl/2 VNGHA= - 3/4q.

i

5R]ZL]DQLHSU]HGVWDZLDVL QDVW SXMFR

q

C

ql/4

3/4ql

ql/4

ql/4

'RVSUDZG]HQLDSRSUDZQRFLREOLF]HPR*HE\ü X*\WH QLHZ\NRU]\VWDQH ZF]HQLHMUyZQDQLH

UyZQRZDJLGODFDáHJRXNáDGX3R]RVWDá\UyZQDQLD

P

∑ = 0RηQLHPR*HE\üSURVWRSDGáD

iη

i

do linii AB) lub

M

∑

= 0 SXQNW&QLHPR*HOH*HüQDOLQLL$%

iC

i

6SUDZG]LP\F]\REOLF]RQHZDUWRFLUHDNFMLVSHáQLDMUyZQDQLHUyZQRZDJLQS

M

∑

= 0 .

iC

i

M

∑ = V

iC

A 2 l - HA l- HB l+Wl/2 = ql/4 2 l – 3/4ql l - ql/4 l +ql l/2= 0.

i

5R]ZL]DQLHWHJR]DGDQLDPR*HSU]HELHJDüQDZLHOHVSRVREyZ,VWRWQHMHVWVSRVWU]H*HQLH*H

F] ü SUDZD XNáDGX QLH MHVW REFL*RQD 1D W F] ü G]LDáDM ZL F W\ONR GZLH UHDNFMH Z

SU]HJXEDFK % L D WR R]QDF]D *H PXV] RQH PLHü ZVSyOQ\ NLHUXQHN ZDUXQHN UyZQRZDJL

GZyFKVLá

2

R1

q

1

1

d = √ 2l

B

B

A

HA

V

R

A

RB

B

x

:\NRU]\VWXMF W LQIRUPDFM R NLHUXQNX UHDNFML Z SRGSRU]H % X]\VNXMHP\ XNáDG VLá SR]ZDODMF\ REOLF]\ü ZDUWRFL UHDNFML ] WU]HFK ZDUXQNyZ UyZQRZDJL FDáHJR XNáDGX

=DSLVXMFNROHMQRUyZQDQLD

M

∑

= 0 ,

M

∑

= 0 ,

P

∑ = 0

iA

iB

ix

i

i

i

uzyskujemy R √

√

B 2 l – ql l/2 = 0,

-VA 2 l –ql l/2 = 0, HA + ql - RB 2/2 = 0

LREOLF]DP\ZDUWRFLRB = √2/4 ql, VA = - ql/4, HA = - 3/4ql.

àDWZR VSUDZG]Lü *H REOLF]RQD UHDNFMD 5B MHVW Z\SDGNRZ REOLF]RQ\FK ZF]HQLHM

VNáDGRZ\FK9B i HB .

=DGDQLHGRSU]HP\OHQLD

-DN Z\NRU]\VWDü UR]ZL]DQLH SRSU]HGQLHJR ]DGDQLD GR Z\]QDF]HQLD UHDNFML Z QDVW SXMFHM

ramie?

q

l

l

l

l

3