3U]\NáDG8NáDGEHONRZ\]áR*RQ\

3ROHFHQLHZ\]QDF]\üUHDNFMHSRGSRURZHRUD]RGG]LDá\ZDQLDZSRáF]HQLDFKSU]HJXERZ\FK

L Z WHOHVNRSLH GOD SRQL*V]HJR XNáDGX REFL*RQHJR REFL*HQLHP FLJá\P  R VWDá\P QDW *HQLX

q

1

= q i q

2

= 2q

VLáVNXSLRQ

P = 2 ql oraz momentem skupionym M = ql

2

.

Przyjmujemy oznaczenia dla podpór: A, B, C i D

 RUD] GOD SRáF]H SU]HJXERZHJR

1 i

teleskopowego 2.

8ZDOQLDP\XNáDGRGZL ]yZ]DVW SXMFSRGSRU\UHDNFMDPL2]QDF]RQDOLWHU

A podpora jest

WXOHM D ZL F Z SXQNFLH

A

 G]LDáDM QDVW SXMFH UHDNFMH UHDNFMD SLRQRZD

R

A

 SURVWRSDGáD GR

NLHUXQNX PR*OLZHJR SU]HVXZX RUD] PRPHQW

M

A

. W miejscu podpory przegubowej

przesuwnej B

 G]LDáD UHDNFMD SLRQRZD

R

B

SURVWRSDGáD GR NLHUXQNX PR*OLZHJR SU]HVXZX

podobnie w punkcie C

G]LDáD UHDNFMD SLRQRZD

R

C

 3UDZD SRGSRUD R]QDF]RQD OLWHU

D, jest

SRGSRU SU]HJXERZ QLHSU]HVXZQ ]DWHP Z SXQNFLH

D

 G]LDáDM GZLH QLH]DOH*QH RG VLHELH

VNáDGRZH UHDNFML SLRQRZD

V

D

i pozioma H

D

 2EFL*HQLH FLJáH

q

1

=q

 ]DVW SXMHP\

Z\SDGNRZ

W

1

]D

q

2

=2q dwiema wypadkowymi W

2

i W

3

.

l

2 l

2 l

2 l

2 l

l



= 45

o

A

B

D

q

1

= q

q

2

= 2q

M = ql

2

P = 2 ql

cos

.

=

5

3

sin

.



=

5

4

cos



=

2

1

sin



=

2

1

C

2

1

.

q

1

= q

q

2

= 2q

M = ql

2

P = 2 ql

.

cos

.

=

5

3

sin

.



=

5

4

cos



=

2

1

sin



=

2

1

l

2 l

2 l

2 l

2 l

l



= 45

o

l

2 l

2 l

2 l

2 l

l

W

1

W

2

W

3

R

C

R

B



= 45

o

A

B

D

M = ql

2

P = 2 ql

.

C

2

1

M

A

R

A

V

D

H

D

x

y

2

:\SDGNRZHREFL*HQLDFLJáHJRZ\QRV]

W

1

= q

1

Â

l = q

Â

l = ql

W

2

= q

2

Â

l = 2q

Â

l = 4ql

W

3

= q

2

Â

l = 2q

Â

l = 4ql

']LHOLP\ ]áR*RQ\ XNáDG EHONRZ\ QD EHONL SRMHG\QF]H SRGXNáDG\ WZRU]F VFKHPDW SUDF\

UR]SDWU\ZDQHJR XNáDGX : SRáF]HQLX SU]HJXERZ\P

1

 G]LDáDM GZLH QLH]DOH*QH VNáDGRZH

RGG]LDá\ZDQLDSLRQRZD

V

1

i pozioma H

1

3RGREQLHZSRáF]HQLXSU]HJXERZ\P

C

Z\VW SXM

GZLHQLH]DOH*QHVNáDGRZHRGG]LDá\ZDQLDSLRQRZD

V

C

i pozioma H

C

1DWRPLDVWZSRáF]HQLX

teleskopowym 2

Z\VW SXMHRGG]LDá\ZDQLH

R

2

RNLHUXQNXSURVWRSDGá\PGRRNáDGHNWHOHVNRSX

oraz moment M

2

.

-DNRSLHUZV]H]DSLV]HP\UyZQDQLHU]XWyZVLáQDRSR]LRPGOD

III

F] FLXNáDGXZFHOX

REOLF]HQLDRGG]LDá\ZDQLDSR]LRPHJR

H

1

w przegubie 1.

∑

=

i

III

ix

P

0

:

2 ql

Â

2

1

í

H

1

= 0

⇒

H

1

= ql

3RZ\]QDF]HQLXZDUWRFLRGG]LDá\ZDQLD

H

1

]DSLV]HP\UyZQDQLHU]XWyZVLáQDRSR]LRPGOD

( I

F] FLXNáDGX]NWyUHJRREOLF]\P\ZDUWRüRGG]LDá\ZDQLD

R

2

w teleskopie 2.

l

2 l

2 l

2 l

2 l

l

90

o

R

C

R

B

B

W

1

W

2

C

2

.

V

C

H

C

M

2

R

2

IV

A

M = ql

2

P = 2 ql

1

M

A

R

A

V

1

H

1

III

D

C

V

D

H

D

W

3

V

C

H

C

II

I

.

2

1

V

1

H

1

90

o

M

2

R

2

I

I I

.

2

1

V

1

H

1

= ql

90

o

M

2

R

2

2 l

I

3

0

=

∑

i

I

ix

P

: H

1

í

R

2

ÂFRV

o

í.

) = 0

⇒

R

2

=

4

5

ql

= UyZQDQLD U]XWyZ VLá QD R SLRQRZ GOD 

I

 F] FL XNáDGX PR*HP\ MX* Z\]QDF]\ü

RGG]LDá\ZDQLHSLRQRZH

V

1

w przegubie 1.

0

=

∑

i

I

iy

P

: V

1

+ R

2

ÂFRV.

= 0

⇒

V

1

=

í

4

3

ql

Ostatnim równaniem zapisanym dla ( I

F] FLXNáDGXE G]LHUyZQDQLHPRPHQWyZZ]JO GHP

punktu 2

 ] NWyUHJR REOLF]\P\ ZDUWRü RGG]LDá\ZDQLD

M

2

w teleskopie 2. W równaniu tym

QLHZ\VWSLRGG]LDá\ZDQLD

H

1

i R

2

SRQLHZD*OLQLHG]LDáDQLDW\FKVLáSU]HFKRG]SU]H]SXQNW

2.

∑

=

i

I

i2

M

0 : M

2

í

V

1

Â

l = 0

⇒

M

2

=

í

2

3

ql

2

5HDNFM SLRQRZ

R

A

na podporze A

PR*HP\Z\]QDF]\ü]UyZQDQLDU]XWyZVLáQDRSLRQRZ

dla ( III

F] FLXNáDGX

∑

=

i

III

iy

P

0

:

R

A

í

2 ql

Â

2

1

í

V

1

= 0

⇒

R

A

=

4

1

ql

W celu wyznaczenia momentu podporowego M

A

w tulei A zapiszemy równanie momentów

Z]JO GHPSXQNWX

A dla ( III

F] FLXNáDGX

∑

=

i

III

iA

M

0

:

í

M

A

í

2 ql

Â

2

1

Â

l

í

M

í

V

1

Â

l = 0

⇒

M

A

=

4

1

ql

2

5HDNFM  SR]LRP QD SRGSRU]H

D

REOLF]\P\ ] UyZQDQLD U]XWyZ VLá QD R SR]LRP GOD FDáHJR

XNáDGX

0

=

∑

i

ix

P

: 2 ql

Â

2

1

+ H

D

= 0

⇒

H

D

=

í

ql

= UyZQDQLD PRPHQWyZ Z]JO GHP SXQNWX

C dla ( II

  F] FL XNáDGX Z\]QDF]\P\ UHDNFM

SLRQRZ

V

D

na podporze D

 : UyZQDQLX W\P QLH Z\VWSL RGG]LDá\ZDQLD

V

C

i H

C

oraz

reakcja H

D

JG\*OLQLHG]LDáDQLDW\FKVLáSU]HFKRG]SU]H]SXQNW

C.

A

M = ql

2

P = 2 ql

1

M

A

R

A

V

1

=

í

4

3

ql

H

1

= ql

III

2 l

l

II

D

C

V

D

H

D

=

í

ql

W

3

= 4 ql

V

C

H

C

2 l

4

0

=

∑

i

II

iC

M

: V

D

Â

l

í

W

3

Â

l = 0

⇒

V

D

= 2ql

.RU]\VWDMF ] UyZQDQLD U]XWyZ VLá QD R SLRQRZ GOD 

II

  F] FL XNáDGX Z\]QDF]\P\

RGG]LDá\ZDQLHSLRQRZH

V

C

w przegubie C

QDWRPLDVW]UyZQDQLDU]XWyZVLáQDRSR]LRPGOD

( II

F] FLXNáDGXZ\]QDF]\P\RGG]LDá\ZDQLHSR]LRPH

H

C

.

∑

=

i

II

iy

P

0

:

V

C

+ V

D

í

W

3

= 0

⇒

V

C

= 2ql

0

=

∑

i

II

ix

P

: H

C

+ H

D

= 0

⇒

H

C

= ql

$E\ Z\]QDF]\ü UHDNFM 

R

B

 ]DSLV]HP\ UyZQDQLH PRPHQWyZ Z]JO GHP SXQNWX

C dla ( IV )

F] FLXNáDGX

∑

=

i

IV

iC

M

0 : R

2

ÂFRV.

Â

l

í

M

2

í

R

B

Â

l + W

1

Â

2

5

l + W

2

Â

l = 0

⇒

R

B

=

8

41

ql

-DNR RVWDWQL QLHZLDGRP Z\]QDF]DP\ UHDNFM 

R

C

 ] UyZQDQLD U]XWyZ VLá QD R SLRQRZ

zapisanego dla ( IV

F] FLXNáDGX

∑

=

i

IV

iy

P

0 : R

C

+ R

B

í

R

2

ÂFRV.

í

W

1

í

W

2

= 0

⇒

R

C

=

8

21

ql

: UR]ZL]DQLX ]DGDQLD QLH Z\NRU]\VWDOLP\ UyZQDQLD U]XWyZ VLá QD R SR]LRP GOD 

IV )

F] FL XNáDGX =DSLV]HP\ WR UyZQDQLH Z FHOX VSUDZG]HQLD SRSUDZQRFL Z\NRQDQ\FK

REOLF]H

∑

=

i

IV

ix

P

0

:

R

2

ÂFRV

o

í.

)

í

H

C

=

4

5

ql

Â

5

4

í

ql = ql

í

ql = 0

5yZQDQLHVSHáQLRQHMHVWWR*VDPRFLRZR

90

o

R

C

R

B

B

W

1

= ql

W

2

= 4 ql

C

2

.

V

C

= 2ql

H

C

= ql

M

2

=

í

2

3

ql

2

R

2

=

4

5

ql

IV

2 l

l

l

2 l

2 l

2 l

2 l

l

W

1

= ql

W = 4 ql + 4 ql = 8 ql

R

C

=

8

21

ql

R

B

=

8

41

ql



= 45

o

A

B

D

M = ql

2

P = 2 ql

.

C

2

1

M

A

=

4

1

ql

2

R

A

=

4

1

ql

V

D

= 2ql

H

D

=

í

ql

5

'ODVSUDZG]HQLDZ\QLNyZVNRU]\VWDP\UyZQLH*]UyZQDQLD]DSLVDQHJRGODFDáHJRXNáDGXQD

SU]\NáDG ] UyZQDQLD PRPHQWyZ Z]JO GHP SXQNWX

D

 : UyZQDQLX W\P PR*HP\ ]DVWSLü

Z\SDGNRZH REFL*HQLD FLJáHJR

W

2

i W

3

 Z\SDGNRZ

W

 E GF LFK VXP NWyUHM OLQLD

G]LDáDQLDSU]HFKRG]LSU]H]SXQNW

C.

∑

=

L

0

iD

M

:

í

M

A

í

R

A

Â

l + 2 ql

Â

2

1

Â

l

í

ql

2

+ W

1

Â

2

9

l + W

Â

l

í

R

B

Â

l

í

R

C

Â

l =

=

í

4

1

ql

2

í

4

1

ql

Â

l + 9 ql

2

í

ql

2

+ ql

Â

2

9

l +

8 ql

Â

l

í

8

41

ql

Â

l

í

8

21

ql

Â

l =

= ql

2

(

í

4

1

í

2

5

+ 9

í

2

9

+ 16

í

2

41

í

4

21

)

= 0

3RZ\*V]HUyZQDQLHWH*VSHáQLRQHMHVWWR*VDPRFLRZR

=H Z]JO GX QD Z\VW SRZDQLH SRGSRU\ SU]HJXERZHM

C

 Z PLHMVFX SRáF]HQLD SU]HJXERZHJR

F] FL

II ) i ( IV

PR*HP\UyZQLH*SU]\MüLQQ\VFKHPDWSUDF\XNáDGX

:DUWRü UHDNFML SRGSRURZ\FK RUD] RGG]LDá\ZD Z SU]HJXELH

1 i teleskopie 2

 QLH ]DOH* RG

Z\ERUX SU]\M WHJR GR REOLF]H VFKHPDWX SUDF\ XNáDGX EHONRZHJR MHGQHJR ] GZX

SUH]HQWRZDQ\FK1DWRPLDVWZDUWRFLRGG]LDá\ZDSLRQRZ\FKZSU]HJXELH

C

E GUy*QH

V

C



V

C

*

  : SLHUZV]\P VFKHPDFLH SU]\M W\P GR UR]ZL]DQLD ]DGDQLD V WR RGG]LDá\ZDQLD

Z\VW SXMFH SR SUDZHM VWURQLH SRGSRU\

C

 : SRZ\*V]\P VFKHPDFLH RGG]LDá\ZDQLD WH

Z\VW SXM SR OHZHM VWURQLH SRGSRU\

C. Gdyby podpora C

 E\áD SRGSRU SU]HJXERZ

QLHSU]HVXZQWRPRJá\E\WH*Uy*QLüVL RGG]LDá\ZDQLDSR]LRPHZSXQNFLH

C.

I

.

2

1

V

1

H

1

90

o

M

2

R

2

I

l

2 l

2 l

2 l

2 l

l

90

o

R

B

B

W

1

W

2

C

2

.

V

C

*

H

C

*

M

2

R

2

IV

A

M = ql

2

P = 2 ql

1

M

A

R

A

V

1

H

1

III

C

V

D

H

D

W

3

V

C

*

H

C

*

II

R

C

D