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 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

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;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

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¢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

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