Przykład 1.4.

3U]\NáDG:\]QDF]DQLHSU]HPLHV]F]HZXNáDG]LHVWDW\F]QLH
wyznaczalnym
:\]QDF]\ü SU]HPLHV]F]HQLH VZRERGQHJR Z
]áD NUDWRZQLF\ 3U]HNURMH SU
WyZ LFK GáXJRFL
RUD] PRGXá <RXQJD RSLVDQH V QD U\VXQNX 'R REOLF]H SU]\Mü QDVW
SXMFH ]DOH*QRFL
A1=A, A2=2A, E1=E2=E.
2EOLF]HQLDSU]HSURZDG]LüGODQDVW
SXMF\FKGDQ\FKOLF]ERZ\FK
P = 1000N,
A = 0.0001 m2 ,
l = 1m ,
E = 2.1 1011Pa.
l
2l
l
A2 , E
A1 , E
P
5R]ZL]DQLH
:SURZDG]DP\ XNáDG ZVSyáU]
GQ\FK R]QDF]HQLD VLá Z SU
WDFK L QXPHU\ Z
]áyZ MDN QD
U\VXQNXSRQL*HM
3
2
y
.2
.1
S2
S1
x
1
P
=DXZD*P\*HMDNZSU]\NáDG]LHSRSU]HGQLP]DGDQLHMHVWVWDW\F]QLHZ\]QDF]DOQH0R*QDMH
]DWHPUR]ZL]DüZQDVW
SXMF\VSRVyE
1.
2.
3.
4.
=UyZQDUyZQRZDJLZ
]áDZ\]QDF]DP\VLá\ZHZQ
WU]QH61
i S2.
=SUDZD+RRFNH
DREOLF]DP\ZDUWRFLZ\GáX*H
=UyZQDJHRPHWU\F]Q\FKZ\]QDF]DP\VNáDGRZHZHNWRUDSU]HPLHV]F]HQLDZ
]áD
=WZLHUG]HQLD3LWDJRUDVDZ\]QDF]DP\GáXJRüZHNWRUDSU]HPLHV]F]HQLDZ
]áD
'áXJRFLSU
WyZZ\QRV]
l1 =
O
, l2 =
O
.
.W\.1L.2RNUHORQHVQDVW
SXMFR
FRV ¡
FRV ¡
=
=
O
O
O
O
=
=
VLQ ¡
VLQ ¡
=
O
O
=
=
O
O
=
,
.
2EOLF]P\VLá\ZSU
WDFK]UyZQDUyZQRZDJLZ
]áDVZRERGQHJR
∑ 3 = ⇒ −6
L[
∑3 = ⇒ 6
L\
+ 6
+ 6
=
−3 =
2EOLF]RQHZDUWRFLZ\QRV]
6
6
=
=
,
3
.
3
:\GáX*HQLD SU
WyZ Z\ZRáDQH G]LDáDMF\PL Z QLFK VLáDPL F]\OL U
ównania fizyczne
PDM
SRVWDü
∆O =
6 ( $
∆O =
,
6
(
O
$
.
:\GáX*HQLDSU
WyZZ\QRV]
∆O =
∆O =
6
O
($
6
=
O
( $
3O
($
=
3O
($
.
:]DGDQLXW\ONRZ
]HáQUMHVWVZRERGQ\PR*HVL
SU]HPLHV]F]Dü]DWHPSU]HPLHV]F]HQLD
FDáHJR XNáDGX RSLVDQH V SU]H] SU]HPLHV]F]HQLH WHJR Z
]áD / LQDF]HM SU]H] MHJR GZLH
QLH]DOH*QH VNáDGRZH X L Y =DáR*RQH NLHUXQNL L ]ZURW\ SU]HGVWDZLD SRQL*V]\ U\VXQHN /LQL
SU]HU\ZDQQDU\VXQNX]D]QDF]RQRSU
W\ZXNáDG]LHRGNV]WDáFRQ\P
2
3
1
u
/
v
u
.2
.1
.1
v
/
Zapiszemy teraz równania geometryczne
.2
u
/
v
F]\OL
Z\GáX*HQLD
SU
WyZ
Z\UD*RQH
SU]H]
SU]HPLHV]F]HQLDMHJRNRFyZ
3RXZ]JO
GQLHQLX*HZ\GáX*HQLDSU
WyZZ\QLNDMW\ONR]SU]HPLHV]F]HQLDZ
]áDUyZQDQLD
JHRPHWU\F]QHSU]\MPXMSRVWDü
∆O = Y FRV ¡ + X VLQ ¡ ∆O = Y FRV ¡ − X VLQ ¡ 2
FRSRSRGVWDZLHQLXZDUWRFLIXQNFMLWU\JRQRPHWU\F]Q\FKNWyZ.i
∆O = Y
+X
∆O = Y
−X
sprowadza je do postaci:
3RV]XNLZDQHVNáDGRZHSU]HPLHV]F]HQLDZ
]áDRNUHORQHVQDVW
SXMFR
X
=
Y
=
∆O −
∆O +
∆O ∆O .
3R SRGVWDZLHQLX Z\GáX*H Z\UD]LP\ SU]HPLHV]F]HQLH SLRQRZH L SR]LRPH Z
]áD SU]H]
ZDUWRüREFL*HQLDLV]W\ZQRFLSU
WyZ
  3O
3O

= 
−
= ($
  ($
  3O
3O
Y =
 +  ($ = ($


X
&DáNRZLWHSU]HPLHV]F]HQLH ¤
¤
=
(
)
=
+ ()
X
3O
($
+ Y wynosi
= 3O
($
.
3RGVWDZLDMFGDQHOLF]ERZHX]\VNXMHP\
X
⋅
1 P
= 1
⋅
Y
P
⋅ P ⋅
1 P
= ⋅
1
P
⋅ P LFDáNRZLWHSU]HPLHV]F]HQLH/
= ⋅ − P
= ⋅ − P
⋅ 10-5 m §PP
=DGDQLH PR*QD UR]ZL]Dü EH]SRUHGQLR EH] Z\NRU]\VW\ZDQLD VWDW\F]QHM Z\]QDF]DOQRFL
XNáDGX
1.
=
UyZQD
JHRPHWU\F]Q\FK
Z\UD*DP\
Z\GáX*HQLD
SRSU]H]
VNáDGRZH
ZHNWRUD
SU]HPLHV]F]HQLDZ
]áD
2.
= SUDZD +RRFNH
D Z\UD*DP\ VLá\ ]D SRPRF Z\GáX*H D SRGVWDZLDMF Z\QLNL S ]D
SRPRFVNáDGRZ\FKZHNWRUDSU]HPLHV]F]HQLDZ
]áD
3.
8]\VNDQH
Z
S
Z\UD*HQLD
SRGVWDZLDP\
GR
ZDUXQNyZ
UyZQRZDJL
Z
]áD
RWU]\PXMF XNáDG GZX UyZQD Z]JO
GHP GZX VNáDGRZ\FK ZHNWRUD SU]HPLHV]F]HQLD
Z
]áD
4.
5.
5R]ZL]XMFXNáDGUyZQDX]\VNDQ\ZSPDP\UR]ZL]DQLH]DGDQLD
=WZLHUG]HQLD3LWDJRUDVDZ\]QDF]DP\GáXJRüZHNWRUDSU]HPLHV]F]HQLDZ
]áD
3