T.Chhay



                                   ers‘ultg;¬kMlaMgpÁÜb¦énRbBn§½kMlaMgkñúgbøg;
                                             Resultants of Coplanar Force System



     1> kMlaMgpÁÜbénkMlaMgBIrRbsBVKña (Resultant of two concurrent forces)
        RbsinebIkMlaMgBIrRbsBVKña)anmMuEkgenaHeKGacCMnYskMlaMgTaMgBIrenaHedaykMlaMgpÁÜbmYy Edlman
tMélesIμ³
            R = FX2 + FY2

           nig tan θ    X   =
                                  FY
                                  FX
                                       ¬emIlrUbTI1¦




           k> viFIRbelLÚRkam
           kñúgkrNIEdlmMupÁúMrvagkMlaMgBIrminesμI 90 enaHkMlaMgpÁÜbkMNt;tamviFIRbelLÚRkammantMélesμI³
                                                         o



            R 2 = F12 + F22 − 2 F1 F2 cos φ


ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                         1
T.Chhay


           eday φ = 180 − α ⇒ cosφ = − cosα
           dUcenH R = F + F + 2F F cosα
                        2
                                  1
                                   2
                                               2
                                                2
                                                    1 2


           nig sin θ = F sin φ ¬emIlrUbTI2¦
                          R
                              1



           dUcKña φ = 180 − α ⇒ sin φ = sin α
           dUcenH sin θ = F sin α
                             R
                                       1



                                  F1 sin α
           ⇒ θ = arcsin
                                      R
           Edl α CamMupÁúMrvagkMlaMgTaMgBIr F nig F .     1   2




         cMNaMfa RbBn§½kMlaMgRbsBVkñúgbøg; ExSskmμrbs;kMlaMgpÁÜb EtgmancMnuccab;Rtg;kEnøgRbsBVKñaén
RbBn§½kMlaMgenaH. dUcenH TItaMgrbs;kMlaMgpÁÜbEtgEtRtUv)andwg ehIyGVIEdlcaM)ac;RtUvkaredaHRsayKW
GaMgtg;sIueténkMlaMg Tis nigTisedArbs;va.
]TahrN_³ kMNt;GaMgtg;sIuet Tis nigTisedA énkMlaMgpÁÜbrvagkMlaMgBIr F nig F RbsBVKñakñúgbøg;Rtg;cMnuc
                                                                          1   2


0 dUcbgðajkñúgrUbTI3. GaMgtg;sIuetrbs;kMlaMgTaMgBIrmantMél F = 100 N nig F = 140 N . kMlaMg F
                                                                      1           2               1


pÁúMCamYy GkS½ X )anmMu 30 enAEpñkxagelI ÉkMlaMg F pÁúMCamYyGkS½ X )anmMu 45 enAEpñkxageRkam.
                                           o
                                                                  2
                                                                                  o




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                        2
T.Chhay




dMeNaHRsay³
kMNt;mMupÁúMrvagkMlaMgTaMgBIr F nig F
                                   1      2


α = α 1 + α 2 = 30 o + 45 o = 75 o
kMNt;mMu φ
φ = 180 o − α = 180 o − 75 o = 105 o
kMNt;GaMgtg;sIueténkMlaMgpÁÜb
R 2 = F12 + F22 − 2 F1 F2 cos φ = 100 2 + 140 2 − 2 × 100 × 140 × cos105 o

⇒ R = 192 N
mMuR)ab;Tis rbs;kMlaMgpÁÜbCamYykMlaMg F       1

  R    F
     = 2
sin φ sin θ
                     F2 sin φ          140 × sin 105 o
⇒ θ = arcsin                  = arcsin                 = 44.8 o
                        R                   192
dUcenH mMuR)ab;Tisrbs;kMlaMgpÁÜbCamYyGkS½ X
θ x = θ − α 1 = 44.8 o − 30 o = 14.8 o
mMuenaHmanTisRsbtamRTnicnaLika. dUcenHTisrbs;kMlaMgpÁÜb cuHeRkameTAsþaM.


ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                              3
T.Chhay


          x> viFIbgÁúMkMlaMg
          kñúgkarkMNt;kMlaMgpÁÜb énRbBn§½kMlaMgRbsBV eRkABIviFIRbelLÚRkam eKenAmanviFImYyeTotehAfaviFI
bgÁúMkMlaMg (method of components). viFIenHCaviFImYyEdlmanlkçN³TUeTAkñúgkarkMNt;kMlaMgpÁÜb.
          viFIbgÁúMkMlaMgCaviFIEdleRbIRbBn§½kUGredaenEkg X − Y . xageRkamCaviFIsaRsþkñúgkarGnuvtþviFIenH³
          - KNnaplbUknBVn<énbgÁúMkMlaMgTaMgGs;tamGkS½ X (∑ F = R )                  X               X


          - KNnaplbUknBVn<énbgÁúMkMlaMgTaMgGs;tamGkS½ Y (∑ F = R )              Y               Y


          - KNnaGaMgtg;sIuetkMlaMgpÁÜbtamrUbmnþ R = R + R               2
                                                                        X
                                                                            2
                                                                            Y

                                                                                    ∑F
          - KNnamMuR)ab;TisrvagkMlaMgpÁÜbCamYyGkS½ X tamrUbmnþ tan θ = R = ∑ FR
                                                                                                X
                                                                                                                Y   Y

                                                                                                                X   X

]TahrN_³ kMNt;GaMgtg;sIuet Tis nigmMuR)ab;TisCamYYyGkS½ X rbs;kMlaMgpÁÜb R dUcbgðajkñúgrUbTI4.




dMeNaHRsay³
    1> bgÁúMkMlaMgpÁÜbtamGkS½ X / R esμInwgplbUknBVnþénbgÁúMkMlaMg F nig F tamGkS½ X
                                                  X                                         1               2

                                                  12
      R X = ∑ FX = F1 X + F2 X = 350                 − 300 × cos 60 o
                                                  13
      R X = +173.1N →             ¬tMél R viC¢man mann½yfaTisedArbs;kMlaMgeTAsþaM¦
                                          X



     2> bgÁúMkMlaMgpÁÜbtamGkS½ Y / R esμInwgplbUknBVnþénbgÁúMkMlaMg F nig F tamGkS½ Y
                                              Y                                         1               2




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                                         4
T.Chhay


                                           5
      RY = ∑ FY = F1Y + F2Y = −350           + 300 × sin 60 o
                                          13
      RY = +125.2 N ↑ ¬tMél R viC¢man mann½yfaTisedArbs;kMlaMgeTAelI¦
                                      Y


     3> kMlaMgpÁÜbén R nig R KW R dUcbgðajkñúgrUbTI5
                                  X   Y


      R = R X + RY
            2    2



      ⇒ R = 173.12 + 125.2 2 = 214 N
     TisedArbs;kMlaMgpÁÜb eTAelInigxagsþaM
     4> mMuR)ab;Tisrbs;kMlaMgpÁÜbeFobnwgGkS½ X
                    RY          125.2
     θ = tan −1        = tan −1       = 35.9 o
                    RX          173.1




   cMNaM³ sMrab;RbBn§½kMlaMgeRcInelIsBIBIr RbsBVKña b¤minRbsBVKña viFIbgÁúMkMlaMgCaviFIgayRsYl kñúgkarkMNt;;
kMlaMgpÁÜbénRbBn§½kMlaMgenaH.
     2> m:Um:g;énkMlaMg (Moment of forces)




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                           5
T.Chhay


           kMlaMgEdlmanGMeBIeTAelIGgÁFatumYyEdlenAes¶óm GgÁFatu enaHnwgmanclnatamBIrebobxusKña
mü:ag vaGacRtUvrMkil (translation) eTAelIb¤eTAeRkam eTAsþaMb¤eTAeqVg nigmü:ageTot vaGacRtUvvil
(rotation) CMuvijExS b¤GkS½NamYy. dUckñúgrUbTI6 (a) eRkamGMeBIénkMlaMg P EdlGnuvtþmkelIGgÁFaturwg

ÉksN§an Rtg;cMnuckNþal enaHGgÁFatunwgrMkileTAmux EdlRKb;cMnucTaMgGs; rbs;vamanKMlatesμI²Kña.
RbsinebIkMlaMgdEdl manGMeBImkelIGgÁFatuRtg;cMnucepSgeRkABIcMnuckNþal enaHGgÁFatunwgmanclnapSM
Kñarvag rMkilnigbgVilCamYyKña ¬rUbTI6(b)¦. EtRbsinebIeK eKcab;GgÁFatuenaH Rtg;cMnuc A eGayCab;edIm,I
kMueGayvarMkileTAmuxenaH vanwgekItmanEtkarbgVilEtmYyKt; ¬rUbTI6(c)¦.




GMeBIénkMlaMgEdleFVIeGayGgÁFatumYyGacvil)anRtUv)anKeGayeQμaHfam:Um:g;.
m:Um:g;RtUv)ankMNt;tamrUbmnþxageRkam³
                       M = ± Fd
           Edl     - m:Um:g;énkMlaMg ( N .m)
                       M

                F - kMlaMg ( N )

                d - cMgayEkgrvagkMlaMgnwgGkS½ b¤cMnuc ¬édXñas;¦ (m)

       eRkamGMeBIénkMlaMg GgÁFatumYymanclnavilRsbTisénRTnicnaLika RtUv)aneKsnμt;eGaym:Um:g;man
tMélGviC¢manpÞúymkvij vanwgmansBaØabUkenAeBlNaEdlGgÁFatuvilRcasTisénRTnicnaLika ¬rUbTI7¦.

]TahrN_³ kMlaMgRbsBVkñúgbøg;bImanGMeBImkelIGgÁFatumYyRtg;cMnuc A dUcbgðajkñúgrUbTI8.
k> kMNt;m:Umg;énkMlaMgnImYy²eFobcMnuc O.
x> kMNt;plbUkBiCKNiténm:Um:g;énkMlaMgTaMgbIeFobcMnuc O nigkMNt;TisénrgVil.
ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                        6
T.Chhay




dMeNaHRsay³
k> kMNt;m:Umg;énkMlaMgnImYy²eFobcMnuc O
       edaycMnuc O sßitenAelIExSskmμénkMlaMg F dUcenHm:U:m:g;énkMlaMg F esμIsUnü.
                                                     1                    1


       eRBaHkMlaMg F min)aneFVIeGayGgÁFatuvileFobcMnuc O eT.
                                  1


       m:Um:g;énkMlaMg F RtUv)anKNnatamrUbmnþ
                                      2


           M 2 = F2 d = +75 × 5 = +375 N .m
           m:Um:g;enHmansBaØaviC¢manedaysarTisénm:Um:g;RcasTisénRTnicnaLika
           m:Um:g;énkMlaMg F RtUv)anKNnatamrUbmnþ
                                      2


           M 3 = F3 d = −60 × 5 × sin 30 o = −150 N .m
       edaysarvavilRsbRTnicnaLika dUcenHm:Um:g;RtUvmansBaØadk.
x> kMNt;plbUkBicKNiténm:Um:g;énkMlaMgTaMgbIeFobcMnuc O nigkMNt;TisénrgVil
           ∑ M = 0 + 375 − 150 = +225 N .m
           m:Um:g;mansBaØabUk dUcenHrgVilmanTisRcasRTnicnaLika.
     3> eKalkarN_rbs;m:Um:g;-RTwsþIbTv:arIjú:g (Principle of moments-Varignon’s theorem)
         m:Um:g;pÁÜb b¤m:Um:g;énkMlaMgpÁÜbénRbBn§½kMlaMgmYy eFobcMnucmYymantMélesμInwgplbUkBiCKNitén
m:Um:g;rbs;bgÁúMkMlaMgnImYy²énRbBn§½enaH eFobnwgcMnucenaH.
]TahrN_³ kMNt;m:Um:g;énkMlaMg F = 200 N eFobcMnuc O EdlsßitenAelIbøg; X − Y ¬rUbTI9¦.
edayeRbInUvbec©keTsxageRkamcUrKNna³
k> edayeRbIédXñas;EdlEkgeTAnwgExSskmμénkMlaMg
x> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc M
K> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc N

ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                        7
T.Chhay




dMeNaHRsay³
kMNt;m:Um:g;énkMlaMg F = 200 N eFobcMnuc O
k> edayeRbIédXñas;EdlEkgeTAnwgExSskmμénkMlaMg d
          KNnaRbEvgédXñas; d
              a = 2 × tan 30 o = 1.15m

              b = 3 − a = 3 − 1.15 = 1.85m

              d = b × cos 30 o = 1.85 × cos 30 o = 1.6m
           m:Um:g;énkMlaMg F RtUv)anKNnatamrUbmnþ
              M O = Fd = −200 × 1.6 = −320 N .m
x> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc M
       kMlaMg F = 200 N bMEbk)anbgÁúMkMlaMgBIrKW
            FX = 200 × cos 30 o = 173.2 N

            FY = 200 × sin 30 o = 100 N
           plbUkBiCKNiténm:Um:g;énbgÁúMkMlaMg F Rtg;cMnuc O KW³
           ∑ M O = − FX × 3 + FY × 2 = −173.2 × 3 + 100 × 2 = −320 N .m
       sBaØadk bgðajfam:Um:g;manTisRsbRTnicnaLika
K> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc N
ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                           8
T.Chhay


      tameKalkarN_énviFIrMkil eK)anbgÁúMkMlaMg F Rtg;cMnuc M mantMélesμIbgÁúMkMlaMg F Rtg;cMnuc N .
      edaycMnuc O sßitelIExSskmμénbgÁúMkMlaMg F dUcenHm:Um:g;énkMlaMg F esμIsUnü. dUcenHplbUkBiC
                                                            Y              Y


KNiténm:Um:g;énkMlaMg F esμI³
           ∑ M O = − FX b = −173.2 × 1.85 = −320 N .m

     4> kMlaMgpÁÜbénRbBn§½kMlaMgRsb (Resultants of parallel force systems)
        RbBn§½kMlaMgRsb CaRbBn§½kMlaMgEdlExSskmμénkMlaMgnImYy² kñúgRbBn§½manTisRsbKña.
ExSskmμénkM -laMgpÁÜbénRbBn§½kMlaMgRsb manTisRsbnwgRbBn§½kMlaMgenaH. GaMgtg;sIuet TisedA
énkMlaMgpÁÜbRtUv)ankMNt; edayplbUkBiCKNiténRbBn§kMlaMgenaH. ¬rUbTI10¦
            R = ∑ FY = F1 + F2 + ... + Fn
           cMENkÉTItaMgénExSskmμrbs;kMlaMgpÁÜbRtUv)anedaHRsayedayeRbIeKalkarN_m:Um:g; RTwsþIbTv:arIj:úg.
           ∑ M O = R.x = F1 .x1 + F2 .x 2 + ... + Fn .x n
                  F .x + F2 .x 2 + ... + Fn .x n
           ⇒x= 1 1
                              R
                 ∑M
           b¤x=
                  R




   ]TahrN_³ kMNt;kMlaMgpÁÜbénRbBn§½kMlaMgRsb dUcbgðajkñúgrUbTI11 EdlmanGMeBIelIFñwmedk AB mYy.
kMlaMgTaMgGs;suT§EtbBaÄr. edayecalTMgn;pÞal;rbs;Fñwm.




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                     9
T.Chhay




dMeNaHRsay³
GaMgtg;sIuetkMlaMgpÁÜbénRbBn§½kMlaMgRsbesμInwgplbUlBiCKNiténkMlaMgbBaÄr
edaysnμt;TisedAeLIgelI mansBaØaviC¢man
R = ∑ FY = −25 − 15 + 10 − 20 = −50 N ↓
TItaMgrbs;kMlaMgpÁÜb RtUv)ankMNt;edayplbUkBiCKNiténm:Um:g;énkMlaMgTaMgGs;eFobcMnuc A
∑ M A = −15 × 3 + 10 × 9 − 20 × 14 = −235 N .m
       ∑ M A − 235
⇒x=          =         = 4.7m
         R       − 50
       dUcenHkMlaMgpÁÜnénRbBn§½kMlaMgRsb mantMél 50 N manTisbBaÄr TisedAcuHeRkam
nigmanTItaMgcMgay 4.7m BI cMnuc A .
     5> kMlaMgpÁÜbénbnÞúkBRgay (Resultants of distributed load)




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                        10
T.Chhay




        bnÞúkBRgay KWCakMlaMgEdlmanGMeBIelIGgÁFatu edayBRgayelIÉktaépÞ b¤rayelIÉktaRbEvg.
        bnÞúkBRgayRtUv)anEckCaBIrKW³ bnÞúkBRgayesμI (uniformly distributed load) CabnÞúkEdlBRgay
edayGaMgtg;sIuetesμI²KñaelIÉktaRbEvgnig bnÞúkBRgayminesμI (nonuniformly distributed load)
¬rUbTI12¦. xñatrbs;bnÞúkBRgayRtUv)anKitCa N m .
        bnÞúkpÁÜbénbnÞúkBRgay CaGaMgetRkalénGaMgtg;sIuettam RbEvgEdlvaBRgayelI b¤CaRkLaépÞén
bnÞúkBRgayenaH¬rUbTI13¦. bnÞúkpÁÜbenaHmanGMeBIkat;tamTIRbCMuTMgn; énragFrNImaRtrbs;bnÞúk BRgay
enaH.
            R = ∫ q ( x )dx
                      b

                      a
                  b

                  ∫ q(x ).xdx
            x=    a
                          R




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                              11
T.Chhay


]TahrN_³ kMNt;GaMgtg;sIuet nigTItaMgrbs;kMlaMgpÁÜb R énRbBn§½kMlaMgRsbdUcbgðajkñúgrUbTI14.
RbBn§½kMlaMgmanGMeBIelIFñwmedk AB . kMlaMgTaMgGs;manTisbBaÄr ehIyTMgn;pÞal;rbs;FñwmRtUv)anecal.




dMeNaHRsay³
kMlaMgpÁÜb W énbnÞúkBRgayesμI w = 2 N / m EdlrayelIRbEvg 14m esμInwg
           W = 2 × 14 = 28 N
ehIykMlaMgpÁÜbenHRtUvCMnYsedaykMlaMgcMcMnucEdlmanGMeBIkat;tamTIRbCMuTMgn;
b¤esμInwgBak;kNþalRbEvgEdlva )anBRgayelI ¬sMrab;bnÞúkBRgayragctuekaNEkg¦.
kMlaMgpÁÜbénRbBn§½kMlaMgRsb
R = ∑ Fy = −3 − 8 − 28 = −39 N

plbUkm:Um:g;énRbBn§½kMlaMgRsbeFobcMnuc A
∑ M A = −3 × 5 − 8 × (5 + 7) − 28 × (5 + 7 + 4 + 7) = −755 N
TItaMgrbs;kMlaMgpÁÜbeFobcMnuc     A
      ∑ M A − 755
x=         =      = 19.4m
        R    − 39


     6> kMlaMgbgVil (Couple)
        kalNakMlaMgBIrminenAelITMrEtmYy manExSskmμRsbKña GaMgtg;sIuetesμIKña b:uEnþmanTisedApÞúyKña
vabegáIt)ankrNIBiessmYyEdleKeGayeQμaHfa kMlaMgbgVil. ¬rUbTI15¦



ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                   12
T.Chhay




           C = ± F .d
     lkçN³rbs;kMlaMgbgVilRtUv)ankMNt;cMNaMdUcxageRkam³
     - kMlaMgbgVilminTak;TgcMnucNamYyrbs;bøg;eLIy
     - GaMgtg;sIueténkMlaMgpÁÜbrbs;kMlaMgbgVilesμIsUnü
     - kMlaMgbgVilpÁÜbmantMélesμIplbUkBiCKNiténkMlaMgbgVilTaMgGs;
     - kMlaMgbgVilmansBaØabUk enAeBlNaEdlvaeFVIeGayGgÁFatumYyvilRcasRTnicnaLika
]TahrN_³
     kMNt;GaMgtg;sIueténm:Um:g;énkMlaMgBIrRsbKñadUcbgðajkñúgrUbTI16. edayeFobnwg
     k> cMnuc O
     x> cMnuc B




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                    13
T.Chhay


dMeNaHRsay³
       edaysarkMlaMgTaMgBIrmanGaMgtg;sIuetesμIKña TisedApÞúyKña dUcenHvabegáIt)ankMlaMgbgVilmYy
           C = 100 × 4 = +400 N .m
           KNnaGaMgtg;sIueténplbUkm:Um:g;eFobcMnuc O
           ∑ M o = −100 × 2 + 100 × 6 = +400 N .m
           KNnaGaMgtg;sIueténplbUkm:Um:g;eFobcMnuc B
           ∑ M A = +100 × 7 − 100 × 3 = +400 N .m


        cMNaM³ eyIgGacCMnYskMlaMgmYyEdlmanGMeBIRtg;cMnuc A edaykMlaMgmYyEdlmanGaMgtg;sIuetesμIKña
TisedAdUcKña ExSskmμRsbKña Rtg;cMnuc B edayRKan;EtbEnßmkMlaMgbgVilmYyEdlmantMélesμIplKuNrvag
kMlaMgenaHCamYycMgayrvagBIcMnuc A dl;cMnuc B . ¬rUbTI17¦




     7> kMlaMgpÁÜbénRbBn§½kMlaMgminRbsBVkñúgbøg; (Resultant of no concurrent force system)
        GaMgtg;sIuet Tis TisedArbs;kMlaMgpÁÜbénRbBn§½kMlaMgminRbsBVkñúgbøg; RtUv)anKNnadUckrNIRbBn§½
kMlaMgRbsBVkñúgbøg;Edr edayeRbInUvRbBn§½kUGredaen X − Y nigeRbInUvplbUkBiCKNiténbgÁúMkMlaMgtamGkS½
X nigGkS½ Y . ehIyeRbIpleFobénplbUkm:Um:g;énkMlaMgeFobcMnucNamYyelIkMlaMgpÁÜbenaH edIm,IkMNt;TI

taMgrbs;kMlaMgpÁÜbenaH.
]TahrN_³
kMNt;GaMgtg;sIuet Tis TisedA nigTItaMgrbs;kMlaMgpÁÜbénRbBn§½kMlaMgminRbsBVkñúgbøg;dUcbgðajkñúgrUbTI18.




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                     14
T.Chhay




dMeNaHRsay³
snμt;TisedArbs;kMlaMg eTAelI nigeTAsþaMmansBaØaviC¢man dUcTisedArbs;GkS½ X nigGkS½ Y Edr.
kMNt;kMlaMgpÁÜbénbgÁúMkMlaMgtamGkS½ X
RX = ∑ FX = +10 cos 60o − 30 cos 75o − 40 cos 45o − 50 = −81N ←
kMNt;kMlaMgpÁÜbénbgÁúMkMlaMgtamGkS½ Y
RY = ∑ FY = −10 sin 60o − 20 − 30 sin 75o − 40 sin 45o = −85.9 N ↓
kMNt;kMlaMgpÁÜbénbgÁúMkMlaMgTaMgBIr
R = RX + RY =
     2    2
                            (− 81)2 + (− 85.9)2   = 118.1N
kMNt;muMR)ab;TiseFobGkS½ X
                  RY          − 85.9
θ X = arctan         = arctan        = 46.7 o
                  RX           − 81




kMNt;plbUkm:Um:g;eFobcMnuc O
∑ M o = −20 × 4 − 30 × sin 75o × 7 − 40 × sin 45o × 12 = −622 N .m
kMNt;TItaMgkMlaMgpÁÜbeFobcMnuc O
      ∑ M o − 622
x=         =        = 7.24m
       RY    − 85.9




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                             15
T.Chhay


lMhat;³
   1> kMNt;GaMgtg;sIuetkMlaMgxageRkamtamGkS½ X nigtamGkS½ Y
   k>                        x>                    K>                                X>



     2> kMNt;GaMgtg;sIuet Tis nigTisedAénkMlaMgpÁÜbénRbBn§½kMlaMgxageRkamedayviFIRbelLÚRkam³
     k>                                     x>




     K>




     3> kMNt;GaMgtg;sIuet Tis nigTisedAénkMlaMgpÁÜbénRbBn§½kMlaMgxageRkamedayviFIbgÁúMkMlaMg³
     k>                      x>                                   K>




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                                 16
T.Chhay


X>                                                 g>




     4> kMNt;GaMgtg;sIuet Tis TisedA nigTItaMgrbs;kMlaMgpÁÜbénRbBn§½kMlaMgxageRkam³
     k>




     x>




     K>


                        A




ers‘ultg;énRbBn§½kMlaMgkñúgbøg;                                                       17

1.resultants of coplanar force system17

  • 1.
    T.Chhay ers‘ultg;¬kMlaMgpÁÜb¦énRbBn§½kMlaMgkñúgbøg; Resultants of Coplanar Force System 1> kMlaMgpÁÜbénkMlaMgBIrRbsBVKña (Resultant of two concurrent forces) RbsinebIkMlaMgBIrRbsBVKña)anmMuEkgenaHeKGacCMnYskMlaMgTaMgBIrenaHedaykMlaMgpÁÜbmYy Edlman tMélesIμ³ R = FX2 + FY2 nig tan θ X = FY FX ¬emIlrUbTI1¦ k> viFIRbelLÚRkam kñúgkrNIEdlmMupÁúMrvagkMlaMgBIrminesμI 90 enaHkMlaMgpÁÜbkMNt;tamviFIRbelLÚRkammantMélesμI³ o R 2 = F12 + F22 − 2 F1 F2 cos φ ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 1
  • 2.
    T.Chhay eday φ = 180 − α ⇒ cosφ = − cosα dUcenH R = F + F + 2F F cosα 2 1 2 2 2 1 2 nig sin θ = F sin φ ¬emIlrUbTI2¦ R 1 dUcKña φ = 180 − α ⇒ sin φ = sin α dUcenH sin θ = F sin α R 1 F1 sin α ⇒ θ = arcsin R Edl α CamMupÁúMrvagkMlaMgTaMgBIr F nig F . 1 2 cMNaMfa RbBn§½kMlaMgRbsBVkñúgbøg; ExSskmμrbs;kMlaMgpÁÜb EtgmancMnuccab;Rtg;kEnøgRbsBVKñaén RbBn§½kMlaMgenaH. dUcenH TItaMgrbs;kMlaMgpÁÜbEtgEtRtUv)andwg ehIyGVIEdlcaM)ac;RtUvkaredaHRsayKW GaMgtg;sIueténkMlaMg Tis nigTisedArbs;va. ]TahrN_³ kMNt;GaMgtg;sIuet Tis nigTisedA énkMlaMgpÁÜbrvagkMlaMgBIr F nig F RbsBVKñakñúgbøg;Rtg;cMnuc 1 2 0 dUcbgðajkñúgrUbTI3. GaMgtg;sIuetrbs;kMlaMgTaMgBIrmantMél F = 100 N nig F = 140 N . kMlaMg F 1 2 1 pÁúMCamYy GkS½ X )anmMu 30 enAEpñkxagelI ÉkMlaMg F pÁúMCamYyGkS½ X )anmMu 45 enAEpñkxageRkam. o 2 o ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 2
  • 3.
    T.Chhay dMeNaHRsay³ kMNt;mMupÁúMrvagkMlaMgTaMgBIr F nigF 1 2 α = α 1 + α 2 = 30 o + 45 o = 75 o kMNt;mMu φ φ = 180 o − α = 180 o − 75 o = 105 o kMNt;GaMgtg;sIueténkMlaMgpÁÜb R 2 = F12 + F22 − 2 F1 F2 cos φ = 100 2 + 140 2 − 2 × 100 × 140 × cos105 o ⇒ R = 192 N mMuR)ab;Tis rbs;kMlaMgpÁÜbCamYykMlaMg F 1 R F = 2 sin φ sin θ F2 sin φ 140 × sin 105 o ⇒ θ = arcsin = arcsin = 44.8 o R 192 dUcenH mMuR)ab;Tisrbs;kMlaMgpÁÜbCamYyGkS½ X θ x = θ − α 1 = 44.8 o − 30 o = 14.8 o mMuenaHmanTisRsbtamRTnicnaLika. dUcenHTisrbs;kMlaMgpÁÜb cuHeRkameTAsþaM. ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 3
  • 4.
    T.Chhay x> viFIbgÁúMkMlaMg kñúgkarkMNt;kMlaMgpÁÜb énRbBn§½kMlaMgRbsBV eRkABIviFIRbelLÚRkam eKenAmanviFImYyeTotehAfaviFI bgÁúMkMlaMg (method of components). viFIenHCaviFImYyEdlmanlkçN³TUeTAkñúgkarkMNt;kMlaMgpÁÜb. viFIbgÁúMkMlaMgCaviFIEdleRbIRbBn§½kUGredaenEkg X − Y . xageRkamCaviFIsaRsþkñúgkarGnuvtþviFIenH³ - KNnaplbUknBVn<énbgÁúMkMlaMgTaMgGs;tamGkS½ X (∑ F = R ) X X - KNnaplbUknBVn<énbgÁúMkMlaMgTaMgGs;tamGkS½ Y (∑ F = R ) Y Y - KNnaGaMgtg;sIuetkMlaMgpÁÜbtamrUbmnþ R = R + R 2 X 2 Y ∑F - KNnamMuR)ab;TisrvagkMlaMgpÁÜbCamYyGkS½ X tamrUbmnþ tan θ = R = ∑ FR X Y Y X X ]TahrN_³ kMNt;GaMgtg;sIuet Tis nigmMuR)ab;TisCamYYyGkS½ X rbs;kMlaMgpÁÜb R dUcbgðajkñúgrUbTI4. dMeNaHRsay³ 1> bgÁúMkMlaMgpÁÜbtamGkS½ X / R esμInwgplbUknBVnþénbgÁúMkMlaMg F nig F tamGkS½ X X 1 2 12 R X = ∑ FX = F1 X + F2 X = 350 − 300 × cos 60 o 13 R X = +173.1N → ¬tMél R viC¢man mann½yfaTisedArbs;kMlaMgeTAsþaM¦ X 2> bgÁúMkMlaMgpÁÜbtamGkS½ Y / R esμInwgplbUknBVnþénbgÁúMkMlaMg F nig F tamGkS½ Y Y 1 2 ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 4
  • 5.
    T.Chhay 5 RY = ∑ FY = F1Y + F2Y = −350 + 300 × sin 60 o 13 RY = +125.2 N ↑ ¬tMél R viC¢man mann½yfaTisedArbs;kMlaMgeTAelI¦ Y 3> kMlaMgpÁÜbén R nig R KW R dUcbgðajkñúgrUbTI5 X Y R = R X + RY 2 2 ⇒ R = 173.12 + 125.2 2 = 214 N TisedArbs;kMlaMgpÁÜb eTAelInigxagsþaM 4> mMuR)ab;Tisrbs;kMlaMgpÁÜbeFobnwgGkS½ X RY 125.2 θ = tan −1 = tan −1 = 35.9 o RX 173.1 cMNaM³ sMrab;RbBn§½kMlaMgeRcInelIsBIBIr RbsBVKña b¤minRbsBVKña viFIbgÁúMkMlaMgCaviFIgayRsYl kñúgkarkMNt;; kMlaMgpÁÜbénRbBn§½kMlaMgenaH. 2> m:Um:g;énkMlaMg (Moment of forces) ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 5
  • 6.
    T.Chhay kMlaMgEdlmanGMeBIeTAelIGgÁFatumYyEdlenAes¶óm GgÁFatu enaHnwgmanclnatamBIrebobxusKña mü:ag vaGacRtUvrMkil (translation) eTAelIb¤eTAeRkam eTAsþaMb¤eTAeqVg nigmü:ageTot vaGacRtUvvil (rotation) CMuvijExS b¤GkS½NamYy. dUckñúgrUbTI6 (a) eRkamGMeBIénkMlaMg P EdlGnuvtþmkelIGgÁFaturwg ÉksN§an Rtg;cMnuckNþal enaHGgÁFatunwgrMkileTAmux EdlRKb;cMnucTaMgGs; rbs;vamanKMlatesμI²Kña. RbsinebIkMlaMgdEdl manGMeBImkelIGgÁFatuRtg;cMnucepSgeRkABIcMnuckNþal enaHGgÁFatunwgmanclnapSM Kñarvag rMkilnigbgVilCamYyKña ¬rUbTI6(b)¦. EtRbsinebIeK eKcab;GgÁFatuenaH Rtg;cMnuc A eGayCab;edIm,I kMueGayvarMkileTAmuxenaH vanwgekItmanEtkarbgVilEtmYyKt; ¬rUbTI6(c)¦. GMeBIénkMlaMgEdleFVIeGayGgÁFatumYyGacvil)anRtUv)anKeGayeQμaHfam:Um:g;. m:Um:g;RtUv)ankMNt;tamrUbmnþxageRkam³ M = ± Fd Edl - m:Um:g;énkMlaMg ( N .m) M F - kMlaMg ( N ) d - cMgayEkgrvagkMlaMgnwgGkS½ b¤cMnuc ¬édXñas;¦ (m) eRkamGMeBIénkMlaMg GgÁFatumYymanclnavilRsbTisénRTnicnaLika RtUv)aneKsnμt;eGaym:Um:g;man tMélGviC¢manpÞúymkvij vanwgmansBaØabUkenAeBlNaEdlGgÁFatuvilRcasTisénRTnicnaLika ¬rUbTI7¦. ]TahrN_³ kMlaMgRbsBVkñúgbøg;bImanGMeBImkelIGgÁFatumYyRtg;cMnuc A dUcbgðajkñúgrUbTI8. k> kMNt;m:Umg;énkMlaMgnImYy²eFobcMnuc O. x> kMNt;plbUkBiCKNiténm:Um:g;énkMlaMgTaMgbIeFobcMnuc O nigkMNt;TisénrgVil. ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 6
  • 7.
    T.Chhay dMeNaHRsay³ k> kMNt;m:Umg;énkMlaMgnImYy²eFobcMnuc O edaycMnuc O sßitenAelIExSskmμénkMlaMg F dUcenHm:U:m:g;énkMlaMg F esμIsUnü. 1 1 eRBaHkMlaMg F min)aneFVIeGayGgÁFatuvileFobcMnuc O eT. 1 m:Um:g;énkMlaMg F RtUv)anKNnatamrUbmnþ 2 M 2 = F2 d = +75 × 5 = +375 N .m m:Um:g;enHmansBaØaviC¢manedaysarTisénm:Um:g;RcasTisénRTnicnaLika m:Um:g;énkMlaMg F RtUv)anKNnatamrUbmnþ 2 M 3 = F3 d = −60 × 5 × sin 30 o = −150 N .m edaysarvavilRsbRTnicnaLika dUcenHm:Um:g;RtUvmansBaØadk. x> kMNt;plbUkBicKNiténm:Um:g;énkMlaMgTaMgbIeFobcMnuc O nigkMNt;TisénrgVil ∑ M = 0 + 375 − 150 = +225 N .m m:Um:g;mansBaØabUk dUcenHrgVilmanTisRcasRTnicnaLika. 3> eKalkarN_rbs;m:Um:g;-RTwsþIbTv:arIjú:g (Principle of moments-Varignon’s theorem) m:Um:g;pÁÜb b¤m:Um:g;énkMlaMgpÁÜbénRbBn§½kMlaMgmYy eFobcMnucmYymantMélesμInwgplbUkBiCKNitén m:Um:g;rbs;bgÁúMkMlaMgnImYy²énRbBn§½enaH eFobnwgcMnucenaH. ]TahrN_³ kMNt;m:Um:g;énkMlaMg F = 200 N eFobcMnuc O EdlsßitenAelIbøg; X − Y ¬rUbTI9¦. edayeRbInUvbec©keTsxageRkamcUrKNna³ k> edayeRbIédXñas;EdlEkgeTAnwgExSskmμénkMlaMg x> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc M K> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc N ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 7
  • 8.
    T.Chhay dMeNaHRsay³ kMNt;m:Um:g;énkMlaMg F =200 N eFobcMnuc O k> edayeRbIédXñas;EdlEkgeTAnwgExSskmμénkMlaMg d KNnaRbEvgédXñas; d a = 2 × tan 30 o = 1.15m b = 3 − a = 3 − 1.15 = 1.85m d = b × cos 30 o = 1.85 × cos 30 o = 1.6m m:Um:g;énkMlaMg F RtUv)anKNnatamrUbmnþ M O = Fd = −200 × 1.6 = −320 N .m x> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc M kMlaMg F = 200 N bMEbk)anbgÁúMkMlaMgBIrKW FX = 200 × cos 30 o = 173.2 N FY = 200 × sin 30 o = 100 N plbUkBiCKNiténm:Um:g;énbgÁúMkMlaMg F Rtg;cMnuc O KW³ ∑ M O = − FX × 3 + FY × 2 = −173.2 × 3 + 100 × 2 = −320 N .m sBaØadk bgðajfam:Um:g;manTisRsbRTnicnaLika K> edayeRbIviFIbgÁúMkMlaMgRtg;cMnuc N ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 8
  • 9.
    T.Chhay tameKalkarN_énviFIrMkil eK)anbgÁúMkMlaMg F Rtg;cMnuc M mantMélesμIbgÁúMkMlaMg F Rtg;cMnuc N . edaycMnuc O sßitelIExSskmμénbgÁúMkMlaMg F dUcenHm:Um:g;énkMlaMg F esμIsUnü. dUcenHplbUkBiC Y Y KNiténm:Um:g;énkMlaMg F esμI³ ∑ M O = − FX b = −173.2 × 1.85 = −320 N .m 4> kMlaMgpÁÜbénRbBn§½kMlaMgRsb (Resultants of parallel force systems) RbBn§½kMlaMgRsb CaRbBn§½kMlaMgEdlExSskmμénkMlaMgnImYy² kñúgRbBn§½manTisRsbKña. ExSskmμénkM -laMgpÁÜbénRbBn§½kMlaMgRsb manTisRsbnwgRbBn§½kMlaMgenaH. GaMgtg;sIuet TisedA énkMlaMgpÁÜbRtUv)ankMNt; edayplbUkBiCKNiténRbBn§kMlaMgenaH. ¬rUbTI10¦ R = ∑ FY = F1 + F2 + ... + Fn cMENkÉTItaMgénExSskmμrbs;kMlaMgpÁÜbRtUv)anedaHRsayedayeRbIeKalkarN_m:Um:g; RTwsþIbTv:arIj:úg. ∑ M O = R.x = F1 .x1 + F2 .x 2 + ... + Fn .x n F .x + F2 .x 2 + ... + Fn .x n ⇒x= 1 1 R ∑M b¤x= R ]TahrN_³ kMNt;kMlaMgpÁÜbénRbBn§½kMlaMgRsb dUcbgðajkñúgrUbTI11 EdlmanGMeBIelIFñwmedk AB mYy. kMlaMgTaMgGs;suT§EtbBaÄr. edayecalTMgn;pÞal;rbs;Fñwm. ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 9
  • 10.
    T.Chhay dMeNaHRsay³ GaMgtg;sIuetkMlaMgpÁÜbénRbBn§½kMlaMgRsbesμInwgplbUlBiCKNiténkMlaMgbBaÄr edaysnμt;TisedAeLIgelI mansBaØaviC¢man R =∑ FY = −25 − 15 + 10 − 20 = −50 N ↓ TItaMgrbs;kMlaMgpÁÜb RtUv)ankMNt;edayplbUkBiCKNiténm:Um:g;énkMlaMgTaMgGs;eFobcMnuc A ∑ M A = −15 × 3 + 10 × 9 − 20 × 14 = −235 N .m ∑ M A − 235 ⇒x= = = 4.7m R − 50 dUcenHkMlaMgpÁÜnénRbBn§½kMlaMgRsb mantMél 50 N manTisbBaÄr TisedAcuHeRkam nigmanTItaMgcMgay 4.7m BI cMnuc A . 5> kMlaMgpÁÜbénbnÞúkBRgay (Resultants of distributed load) ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 10
  • 11.
    T.Chhay bnÞúkBRgay KWCakMlaMgEdlmanGMeBIelIGgÁFatu edayBRgayelIÉktaépÞ b¤rayelIÉktaRbEvg. bnÞúkBRgayRtUv)anEckCaBIrKW³ bnÞúkBRgayesμI (uniformly distributed load) CabnÞúkEdlBRgay edayGaMgtg;sIuetesμI²KñaelIÉktaRbEvgnig bnÞúkBRgayminesμI (nonuniformly distributed load) ¬rUbTI12¦. xñatrbs;bnÞúkBRgayRtUv)anKitCa N m . bnÞúkpÁÜbénbnÞúkBRgay CaGaMgetRkalénGaMgtg;sIuettam RbEvgEdlvaBRgayelI b¤CaRkLaépÞén bnÞúkBRgayenaH¬rUbTI13¦. bnÞúkpÁÜbenaHmanGMeBIkat;tamTIRbCMuTMgn; énragFrNImaRtrbs;bnÞúk BRgay enaH. R = ∫ q ( x )dx b a b ∫ q(x ).xdx x= a R ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 11
  • 12.
    T.Chhay ]TahrN_³ kMNt;GaMgtg;sIuet nigTItaMgrbs;kMlaMgpÁÜbR énRbBn§½kMlaMgRsbdUcbgðajkñúgrUbTI14. RbBn§½kMlaMgmanGMeBIelIFñwmedk AB . kMlaMgTaMgGs;manTisbBaÄr ehIyTMgn;pÞal;rbs;FñwmRtUv)anecal. dMeNaHRsay³ kMlaMgpÁÜb W énbnÞúkBRgayesμI w = 2 N / m EdlrayelIRbEvg 14m esμInwg W = 2 × 14 = 28 N ehIykMlaMgpÁÜbenHRtUvCMnYsedaykMlaMgcMcMnucEdlmanGMeBIkat;tamTIRbCMuTMgn; b¤esμInwgBak;kNþalRbEvgEdlva )anBRgayelI ¬sMrab;bnÞúkBRgayragctuekaNEkg¦. kMlaMgpÁÜbénRbBn§½kMlaMgRsb R = ∑ Fy = −3 − 8 − 28 = −39 N plbUkm:Um:g;énRbBn§½kMlaMgRsbeFobcMnuc A ∑ M A = −3 × 5 − 8 × (5 + 7) − 28 × (5 + 7 + 4 + 7) = −755 N TItaMgrbs;kMlaMgpÁÜbeFobcMnuc A ∑ M A − 755 x= = = 19.4m R − 39 6> kMlaMgbgVil (Couple) kalNakMlaMgBIrminenAelITMrEtmYy manExSskmμRsbKña GaMgtg;sIuetesμIKña b:uEnþmanTisedApÞúyKña vabegáIt)ankrNIBiessmYyEdleKeGayeQμaHfa kMlaMgbgVil. ¬rUbTI15¦ ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 12
  • 13.
    T.Chhay C = ± F .d lkçN³rbs;kMlaMgbgVilRtUv)ankMNt;cMNaMdUcxageRkam³ - kMlaMgbgVilminTak;TgcMnucNamYyrbs;bøg;eLIy - GaMgtg;sIueténkMlaMgpÁÜbrbs;kMlaMgbgVilesμIsUnü - kMlaMgbgVilpÁÜbmantMélesμIplbUkBiCKNiténkMlaMgbgVilTaMgGs; - kMlaMgbgVilmansBaØabUk enAeBlNaEdlvaeFVIeGayGgÁFatumYyvilRcasRTnicnaLika ]TahrN_³ kMNt;GaMgtg;sIueténm:Um:g;énkMlaMgBIrRsbKñadUcbgðajkñúgrUbTI16. edayeFobnwg k> cMnuc O x> cMnuc B ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 13
  • 14.
    T.Chhay dMeNaHRsay³ edaysarkMlaMgTaMgBIrmanGaMgtg;sIuetesμIKña TisedApÞúyKña dUcenHvabegáIt)ankMlaMgbgVilmYy C = 100 × 4 = +400 N .m KNnaGaMgtg;sIueténplbUkm:Um:g;eFobcMnuc O ∑ M o = −100 × 2 + 100 × 6 = +400 N .m KNnaGaMgtg;sIueténplbUkm:Um:g;eFobcMnuc B ∑ M A = +100 × 7 − 100 × 3 = +400 N .m cMNaM³ eyIgGacCMnYskMlaMgmYyEdlmanGMeBIRtg;cMnuc A edaykMlaMgmYyEdlmanGaMgtg;sIuetesμIKña TisedAdUcKña ExSskmμRsbKña Rtg;cMnuc B edayRKan;EtbEnßmkMlaMgbgVilmYyEdlmantMélesμIplKuNrvag kMlaMgenaHCamYycMgayrvagBIcMnuc A dl;cMnuc B . ¬rUbTI17¦ 7> kMlaMgpÁÜbénRbBn§½kMlaMgminRbsBVkñúgbøg; (Resultant of no concurrent force system) GaMgtg;sIuet Tis TisedArbs;kMlaMgpÁÜbénRbBn§½kMlaMgminRbsBVkñúgbøg; RtUv)anKNnadUckrNIRbBn§½ kMlaMgRbsBVkñúgbøg;Edr edayeRbInUvRbBn§½kUGredaen X − Y nigeRbInUvplbUkBiCKNiténbgÁúMkMlaMgtamGkS½ X nigGkS½ Y . ehIyeRbIpleFobénplbUkm:Um:g;énkMlaMgeFobcMnucNamYyelIkMlaMgpÁÜbenaH edIm,IkMNt;TI taMgrbs;kMlaMgpÁÜbenaH. ]TahrN_³ kMNt;GaMgtg;sIuet Tis TisedA nigTItaMgrbs;kMlaMgpÁÜbénRbBn§½kMlaMgminRbsBVkñúgbøg;dUcbgðajkñúgrUbTI18. ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 14
  • 15.
    T.Chhay dMeNaHRsay³ snμt;TisedArbs;kMlaMg eTAelI nigeTAsþaMmansBaØaviC¢mandUcTisedArbs;GkS½ X nigGkS½ Y Edr. kMNt;kMlaMgpÁÜbénbgÁúMkMlaMgtamGkS½ X RX = ∑ FX = +10 cos 60o − 30 cos 75o − 40 cos 45o − 50 = −81N ← kMNt;kMlaMgpÁÜbénbgÁúMkMlaMgtamGkS½ Y RY = ∑ FY = −10 sin 60o − 20 − 30 sin 75o − 40 sin 45o = −85.9 N ↓ kMNt;kMlaMgpÁÜbénbgÁúMkMlaMgTaMgBIr R = RX + RY = 2 2 (− 81)2 + (− 85.9)2 = 118.1N kMNt;muMR)ab;TiseFobGkS½ X RY − 85.9 θ X = arctan = arctan = 46.7 o RX − 81 kMNt;plbUkm:Um:g;eFobcMnuc O ∑ M o = −20 × 4 − 30 × sin 75o × 7 − 40 × sin 45o × 12 = −622 N .m kMNt;TItaMgkMlaMgpÁÜbeFobcMnuc O ∑ M o − 622 x= = = 7.24m RY − 85.9 ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 15
  • 16.
    T.Chhay lMhat;³ 1> kMNt;GaMgtg;sIuetkMlaMgxageRkamtamGkS½ X nigtamGkS½ Y k> x> K> X> 2> kMNt;GaMgtg;sIuet Tis nigTisedAénkMlaMgpÁÜbénRbBn§½kMlaMgxageRkamedayviFIRbelLÚRkam³ k> x> K> 3> kMNt;GaMgtg;sIuet Tis nigTisedAénkMlaMgpÁÜbénRbBn§½kMlaMgxageRkamedayviFIbgÁúMkMlaMg³ k> x> K> ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 16
  • 17.
    T.Chhay X> g> 4> kMNt;GaMgtg;sIuet Tis TisedA nigTItaMgrbs;kMlaMgpÁÜbénRbBn§½kMlaMgxageRkam³ k> x> K> A ers‘ultg;énRbBn§½kMlaMgkñúgbøg; 17