STATICS. Rigid Bodies: (II) Rigid Bodies: Equivalent. Systems of Forces
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1 Seventh Edition CHATE VECTO ECHANICS O ENGINEES: STATICS 3 edinand. Bee E. ussell Johnston, J. igid Bodies: Lectue Notes: Equivalent N. EYDANLIK Taka Univesit igid Bodies: (II) Sstems of oces Vecto echanics fo Enginees: Statics İÇİNDEKİLE : Scala oduct of Two Vectos 003 The cgaw-hill Companies, Inc. All ights eseved. ied Tiple oduct of Thee Vectos ve oment of a oce About a Given Ais oment of a Couple esolution of a oce Into a oce at O and a Couple Sstem of oces: eduction to a oce and Couple uthe eduction of a Sstem of oces 003 The cgaw--hill Companies, Inc. All ights eseved. 3 -
2 Vecto echanics fo Enginees: Statics Scala oduct of Two Vectos The scala poduct o dot poduct between two vectos and Q is defined as Q Qcosθ scala esult Scala poducts: - ae commutative, - ae distibutive, - ae not associative, ( ) Skale çapım (vea iç çapım) deni Q Q ( Q1 + Q ) Q1 + Q Q S ( ) undefined Eample: Detemine the dot poduct of the two vectos shown below. A A 50 5º B B 30 YNT The cgaw--hill Companies, Inc. All ights eseved. 3-3 Vecto echanics fo Enginees: Statics Scala oduct of Two Vectos Scala poducts with Catesian unit components, Q i + j+ k Q i + Q j+ Q k ( ) ( ) v i i 1 j j 1 k k 1 i j 0 j k 0 k i 0 Q Q + Q Q Eample: If A i - 4j + 5k and B 3i + 6k, detemine A B YNT The cgaw--hill Companies, Inc. All ights eseved. 3-4
3 Vecto echanics fo Enginees: Statics Scala oduct of Two Vectos: Applications 1. Angle between two vectos: Q Q cosθ Q + Q cosθ Q + Q Q + Q + Q Eample: Detemine the angle between the two vectos in the last eample (epeated below). A i - 4j + 5k and B 3i + 6k YNT The cgaw--hill Companies, Inc. All ights eseved. 3-5 Vecto echanics fo Enginees: Statics Eample: oblem 3.37 Eample: Conside the volleball net shown below. Detemine the angle fomed b gu wies AB and AC. (YNT: ~43º) 003 The cgaw--hill Companies, Inc. All ights eseved. 3-6
4 Vecto echanics fo Enginees: Statics Scala oduct of Two Vectos: Applications. ojection of a vecto on a given ais: OL cos θ Q Qcosθ Q cosθ OL Q Q Q + Q OL Q Q pojection of + Q along OL o an ais defined b a unit vecto: λ OL cos θ + cosθ + cosθ 003 The cgaw--hill Companies, Inc. All ights eseved. 3-7 Vecto echanics fo Enginees: Statics 003 The cgaw--hill Companies, Inc. All ights eseved. 3-8
5 Vecto echanics fo Enginees: Statics Scala oduct of Two Vectos: Applications Eample: vektöünün line A-B üeindeki idüşümünü bulunu. B (10,1) (YNT: ~0,7) 0i + 10j A (3,4) Eample: If 16i + -14j + 10k N, detemine the magnitude of the pojection of along the ais of the pole. Also find the magnitude of. OL λ cos θ + cosθ + cosθ 003 The cgaw--hill Companies, Inc. All ights eseved. 3-9 Vecto echanics fo Enginees: Statics ied Tiple oduct of Thee Vectos Evaluating the mied tiple poduct, S 003 The cgaw--hill Companies, Inc. All ights eseved. Üç vektöün kaışık üçlü çapımı, S Q scala ( ) esult Kaışık üçlü çapım mutlak değe bakımından kenalaı S,, and Q olan paelel ülünün hacmine eşitti, eğe S,, and Q bi sağ üçlü oluştuuosa poitif sol üçlü oluştuuoa negatifti S ( Q) ( Q S) Q ( S ) S ( Q ) ( S Q) Q ( S) S, ve Q anı dülemde ise kaışık üçlü çapım sıfı olu. ( Q) S ( Q Q) + S ( Q Q) + S ( Q Q ) S Q S Q S Q 3-10
6 Vecto echanics fo Enginees: Statics EKSENE GÖE G OENT -UYGULAA Somunu sökmek için geekli moment A.d( ) ma. (Kol kuvveti ata) 003 The cgaw--hill Companies, Inc. All ights eseved. Kuvvet ata değilse Ugulanan A momentinin sadece bi kısmı sökmek için kullanılı. Bu da Z A.cosθ (.d).cosθ.d 3-11 Vecto echanics fo Enginees: Statics oment of a oce About a Given Ais 1. Sometimes need the component of a moment about a paticula ais. Scala analsis: a d a whee d a is the o shotest distance fom the foce line of action to the ais of inteest 3. Vecto analsis: a λ a ( ) 4. This called the tiple scala poduct oment O of a foce applied at the point A about a point O, O Scala moment OL about an ais OL is the pojection of the moment vecto O onto the ais, OL λ O λ ( ) oments of about the coodinate aes, 003 The cgaw--hill Companies, Inc. All ights eseved. OL λ OL ekseninin doğultman kosinüslei λ λ A nın koodinatlaı 3-1
7 Vecto echanics fo Enginees: Statics oment of a oce About a Given Ais Kuvvetin hehangi bi eksene göe momenti, BL A / B λ B λ A ( ) A / B B BL λ A / B λ A / B λ A / B SO UÇ eksen üeinde alınan B noktasından bağımsıdı. 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Eample : o?,?? , 53.13, The cgaw--hill Companies, Inc. All ights eseved. 3-14
8 Vecto echanics fo Enginees: Statics Sample oblem 3.5 A cube is acted on b a foce as shown. Detemine the moment of a) about A b) about the edge AB and c) about the diagonal AG of the cube. d) Detemine the pependicula distance between AG and C. 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Sample oblem 3.5 oment of about A, A A ai a j a( i j) A ( / ) j ( / ) k ( / A A A a i ( j) ( / ) j k ) j a / i + j+ k k oment of about AB, ( A nın AB üeindeki idüşümü) AB i A i a / i + j+ k AB a / 003 The cgaw--hill Companies, Inc. All ights eseved. 3-16
9 Vecto echanics fo Enginees: Statics Sample oblem 3.5 oment of about the diagonal AG, AG λ A G / A ai aj ak λ G / A a 3 a A ( i + j+ k) 1 AG λ A 3 a ( 1 1 1) 6 AG 1 3 ( i j k) a ( i j k) ( i + j+ k) a The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Sample oblem 3.5 ependicula distance between AG and C, λ 0 1 ( j k) ( i j k) ( ) Theefoe, is pependicula to AG. 3 6 a AG 6 d d a The cgaw--hill Companies, Inc. All ights eseved. 3-18
10 Vecto echanics fo Enginees: Statics 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics 003 The cgaw--hill Companies, Inc. All ights eseved. 3-0
11 Vecto echanics fo Enginees: Statics Eample 4.9 The od shown is suppoted b backets at A and B. Detemine the moment of the oce {-600 i + 00 j k} which tends to otate the od about the AB ais. ( ) AB û AB 0.4i ˆ+ 0.ˆj û AB + AB AB (0.4) (0.) û ˆ ˆ AB i j ˆ D ( 0. j )m { 600ˆi + 00ˆj 300 k} ˆ N 003 The cgaw--hill Companies, Inc. All ights eseved. 3-1 Vecto echanics fo Enginees: Statics AB AB [( 0.)( 300) (0)(00)] [(0)( 300) (0)( 600)] + 0[(0)(00) (0.)( 600)] AB N m AB N m oppposite sense of û AB ˆ ˆ AB ( 53.67)[0.894 i j] ˆ ˆ AB 48.0i 4.0 j N m 003 The cgaw--hill Companies, Inc. All ights eseved. 3 -
12 Vecto echanics fo Enginees: Statics oment of a Couple (Bi kuvvet çiftinin momenti) Şekilde göüldüğü gibi şiddetlei eşit etki çigilei paelel ama önlei ıt olan iki kuvvetten ( ve ) oluşan bi bileşkee indigenemeen kuvvet sistemine couple (kuvvet çifti) deni. Kuvvetlein toplamı 0 ama bi noktaa göe momentlei 0 değildi. Cismi döndümee çalışı. oment of the couple, + A B ( ) A B oment of the couple, ( ) sinθ d Kuvvet çiftinin moment vektöü koodinat eksenleinin ojinin seçiminden bağımsıdı, ani sebest vektö dü (moment alınan noktadan bağımsıdı). Đçinde bulunduğu dülemin hehangi bi noktasına ugulanabili. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-3 Vecto echanics fo Enginees: Statics Eşdeğe e kuvvet çiftlei Two couples will have equal moments if 1d1 d the two couples lie in paallel planes, and the two couples have the same sense o the tendenc to cause otation in the same diection. omentlei anı olan kuvvet çiftlei ijid cisme anı etkii apala. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-4
13 Vecto echanics fo Enginees: Statics Couples Can Be epesented b Vectos Bi kuvvet çifti bi ijid cisme etki ettiğinde kuvvet çiftin oluştuan kuvvetlein neee etkidiği, şiddetlei ve doğultulaının ne olduğunun önemi oktu. Sadece kuvvet çiftinin momentinin şiddeti ve doğultusu önemlidi.bi kuvvet çifti, şiddeti ve önü olan bi moment vektöü ile temsil edilebili. Couple vectos obe the law of addition of vectos. Couple vectos ae fee vectos, i.e., the point of application is not significant. Couple vectos ma be esolved into component vectos. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-5 Vecto echanics fo Enginees: Statics The cgaw--hill Companies, Inc. All ights eseved. 3-6
14 Vecto echanics fo Enginees: Statics Eample: inding the oment of a Couple Eample: a) Calculate the moments about points A, B, C, and D geneated b the two foces shown below b calculating the moment due to each foce sepaatel. b) Then teat the two foces as a couple and calculate the moment. c) Daw the object with the foces eplaced b a couple shown as a fee vecto. 10 N A B D 4 mm C 3 mm 10 N Eample: A plate in the shape of a paallelogam is acted upon b two couples. Detemine: a) The moment of the couple fomed b the two 1-N foces (8.4 m) b) The pependicula distance between the 1-N foces if the esultant of the two couples is eo. Also find the value of d. (0.7 m, 0.85 m) 003 The cgaw--hill Companies, Inc. All ights eseved. 3-7 Vecto echanics fo Enginees: Statics Veilen Bi kuvveti O noktasına na etkien Bi kuvvet ve kuvvet çiftine dönüştümed kuvvet vektöü O noktasına taşındığında cismin başlangıçtaki dengesi boulmamalıdı. Bunun için O noktasında cisme net etkisi olmaan bibiine eşit ve ıt önde iki kuvvet ugulanı. Ve bu üç kuvvet de bi kuvvet-kuvvet çifti sistemine indigenebili. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-8
15 Vecto echanics fo Enginees: Statics kuvvetini A dan faklı O noktasına taşımak istesek bu ke de O ne göe moment vektöünün ( O ) hesaplanması geeki, O ' The moments of about O and O ae elated, O' ' + s O ( + s) + s Bi kuvvet-kuvvet çifti sistemini O dan O ne taşıken O noktasındaki kuvvetin O noktasına göe momenti de eklenmelidi. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-9 Vecto echanics fo Enginees: Statics Sample oblem 3.6 Şekilde gösteilen kuvvet çiftlei (moment etkisi aatı) eine eşdeğe bi tek moment elde edini. SOLUTION: Attach equal and opposite 0 N foces in the + diection at A, theeb poducing 3 couples fo which the moment components ae easil computed. Altenativel, compute the sum of the moments of the fou foces about an abita single point. The point D is a good choice as onl two of the foces will poduce non-eo moment contibutions The cgaw--hill Companies, Inc. All ights eseved. 3-30
16 Vecto echanics fo Enginees: Statics Sample oblem 3.6 Attach equal and opposite 0 N foces in the + diection at A The thee couples ma be epesented b thee couple vectos, + + ( 30 )( 450 mm) ( 0 )( 300 mm) mm. ( 0 )( 5 mm) mm mm + ( 4.5 m. )k ( 13.5 m) i + ( 6 m. ) j 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Sample oblem 3.6 Altenativel, compute the sum of the moments of the fou foces about D. Onl the foces at C and E contibute to the moment about D. D ( 450 mm. ) j ( 30 ) k + ( 5 mm. ) j ( 300 mm) [ k] ( 0 )i + ( 4.5 m. )k ( 13.5 m) i + ( 6 m. ) j Sonuç önceki ile anıdı 003 The cgaw--hill Companies, Inc. All ights eseved. 3-3
17 Vecto echanics fo Enginees: Statics Kuvvetle sisteminin bi kuvvet-kuvvet çifti momentine indigenmesi The foce and couple vectos ma be combined into a esultant foce vecto and a esultant couple vecto, O ( ) The foce-couple sstem at O ma be moved to O with the addition of the moment of about O, + s O' O Two sstems of foces ae equivalent if the can be educed to the same foce-couple sstem. 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Eğe bileşke kuvvet ve moment bibiine dik ise sistem tek bi kuvvete indigenebili 003 The cgaw--hill Companies, Inc. All ights eseved. 3-34
18 Vecto echanics fo Enginees: Statics Kuvvetle sisteminin tek bi kuvvete indigenmesi Eğe bileşke kuvvet ve bi O noktasındaki moment bibiine dik ise bunla eni bi tesi çigisi üeinde etki eden tek bi kuvvete indigenebili Bi kuvvetle sistemi için bi bileşke kuvvet ve bi momentin bibiine dik olabilmesi için; kuvvetle 1) a bi noktada kesişmeli, ) a anı dülemde olmalı, ada 3) kuvvetle bibiine paelel olmalı. 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Coplana oce Sstems 003 The cgaw--hill Companies, Inc. All ights eseved. 3-36
19 Vecto echanics fo Enginees: Statics Dülemdeki kuvvetlein bi tek kuvvete indigenmesi Bileşke kuvvet tesi çigisi doğultusunda O noktasına göe O e eşit moment oluştuacak şekilde kadıılı. In tems of ectangula coodinates, O 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics aallel oce Sstems 003 The cgaw--hill Companies, Inc. All ights eseved. 3-38
20 Vecto echanics fo Enginees: Statics VİDA oanına vida adımı deni. Bi vektöün başka bi vektöün tesi çigisi üeindeki idüşümü hatılanısa ; o nun üeindeki idüşümü, O O vida ekseni 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Sample oblem 3.8 SOLUTION: a) Compute the esultant foce fo the foces shown and the esultant couple fo the moments of the foces about A. o the beam, educe the sstem of foces shown to (a) an equivalent foce-couple sstem at A, (b) an equivalent foce couple sstem at B, and (c) a single foce o esultant. Note: Since the suppot eactions ae not included, the given sstem will not maintain the beam in equilibium. b) ind an equivalent foce-couple sstem at B based on the focecouple sstem at A. c) Detemine the point of application fo the esultant foce such that its moment about A is equal to the esultant couple at A. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-40
21 Vecto echanics fo Enginees: Statics Sample oblem 3.8 SOLUTION: a) Compute the esultant foce and the esultant couple at A. 150 N j 600 N j+ 100 N j 50 N 600 N A ( ) ( ) ( ) ( ) j ( ) j ( ) ( 1.6i) ( 600 j) + (.8i) ( 100 j) + ( 4.8i) ( 50 j) A 1880 ( N m)k 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Sample oblem 3.8 b) ind an equivalent foce-couple sstem at B based on the foce-couple sstem at A. The foce is unchanged b the movement of the foce-couple sstem fom A to B. 600 N ( ) j The couple at B is equal to the moment about B of the foce-couple sstem found at A. B A + B A ( 1880 N m) k + ( 4.8 m) i ( 600 N) j ( 1880 N m) k + ( 880 N m)k ( N m)k B 003 The cgaw--hill Companies, Inc. All ights eseved. 3-4
22 Vecto echanics fo Enginees: Statics Sample oblem 3.10 SOLUTION: Detemine the elative position vectos fo the points of application of the cable foces with espect to A. esolve the foces into ectangula components. Compute the equivalent foce, Yukada üç kablo ile bağlı bi baket göülmektedi. Üç kablo kuvveti eine A da etki eden bi eşdeğe kuvvet-kuvvet çifti sistemi bulunu. 003 The cgaw--hill Companies, Inc. All ights eseved. Compute the equivalent couple, v A ( ) 3-43 Vecto echanics fo Enginees: Statics Sample oblem 3.10 esolve the foces into ectangula components. B ( 700 N) λ E B 75i 150 j+ 50k λ E B i j+ 0.89k 300i 600 j+ 00k B ( N) Detemine the elative position vectos with espect to A. B A 0.075i k ( m) C A 0.075i 0.050k ( m) 0.100i j D A ( m) 003 The cgaw--hill Companies, Inc. All ights eseved. C D ( 1000 N)( cos 45i cos 45 j) 707i 707 j ( 100 N)( cos60i + cos30 j) 600i j ( N) ( N) 3-44
23 Vecto echanics fo Enginees: Statics Sample oblem 3.10 Compute the equivalent foce, ( ) i + ( ) j + ( )k 1607i j 507k ( N) Compute the equivalent couple, v A ( ) i j k B A C A D A B c D 300 i i i 45k j k j j k k A 30 i j k 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Eample Equivalent Sstems Eample: (oblem in Statics, 9th Ed. b Hibbele) eplace the two foces b an equivalent esultant foce and couple moment at point O. Use 15 lb. Eample: (oblem in Statics, 9th Ed. b Hibbele) The sstem of paallel foces acts on the top of the Waen tuss. Detemine the equivalent esultant foce of the sstem and specif its location measued fom point A. 003 The cgaw--hill Companies, Inc. All ights eseved. 3-46
24 Vecto echanics fo Enginees: Statics Eample m N.m N.m N.m N.m 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Qui 003 The cgaw--hill Companies, Inc. All ights eseved. 3-48
25 Vecto echanics fo Enginees: Statics 1 m 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics 003 The cgaw--hill Companies, Inc. All ights eseved. 3-50
26 Vecto echanics fo Enginees: Statics Eample eplace the foce and couple moment sstem b an equivalent foce and couple moment acting at point. Solution: + 60cos N N + 60sin N 170 N + θ tan -1 ( ) tan + ( 170) cos30N -60sin30+ (-140)N N N 003 The cgaw--hill Companies, Inc. All ights eseved Vecto echanics fo Enginees: Statics Eample 60cos30N oment due to hoiontal component of 60N 8m 60sin30N oment due to vetical component of 60N 4m ( + 60 sin 30 ( 1 8 ) + 60 cos 30 ( 8 ) ( ) 676 N.m.68 kn.m (counteclockwise) oment due to the foce 140N 140N 15m 40Nm 003 The cgaw--hill Companies, Inc. All ights eseved. 3-5
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