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2. Statika Partikel

STATIKA STRUKTUR. I Made Gatot Karohika , ST. MT. Mechanical Engineering Udayana University. 2. Statika Partikel. Genap 2013 / 2014. Contents. Introduction Resultant of Two Forces Vectors Addition of Vectors Resultant of Several Concurrent Forces Sample Problem 2.1

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2. Statika Partikel

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  1. STATIKA STRUKTUR I Made GatotKarohika, ST. MT. Mechanical Engineering UdayanaUniversity 2. StatikaPartikel Genap 2013 / 2014

  2. Contents Introduction Resultant of Two Forces Vectors Addition of Vectors Resultant of Several Concurrent Forces Sample Problem 2.1 Sample Problem 2.2 Rectangular Components of a Force: Unit Vectors Addition of Forces by Summing Components Sample Problem 2.3 Equilibrium of a Particle Free-Body Diagrams Sample Problem 2.4 Sample Problem 2.6 Rectangular Components in Space Sample Problem 2.7

  3. Pendahuluan • Dalambabini, kitaakanmempelajaripengaruhgaya-gaya yang bekerjapadapartikel: - Mempelajaricaramenggantiduaataulebihgaya [multiple forces] yang bekerjapadasuatupartikelmenjadisebuahgayatunggal[resultant force] yang pengaruhnyasamasepertigaya-gayasemula, - Menurunkanhubungan-hubungan yang adaantaraberbagaigaya yang bekerjapadasuatupartikeldalamkondisiseimbang[equilibrium] danmemakaihubunganiniuntukmenentukanbeberapagaya-gaya yang bekerjapadapartikeltersebut. • Pemakaiankata “partikel”tidakberartibahwakitamembatasipelajarankitapadabenda yang kecil [miniscule bodies]. Yang dimaksuddisiniadalahukurandanbentukbenda yang dianalisatidakbanyak[ tidak significant] mempengaruhipenyelesaianmasalah, sehinggasemuagaya yang bekerjapadasuatubendaakandiasumsikanbekerjadititik yang sama.

  4. Force [Gaya]: aksisebuahbendapadabenda lain; ditentukanolehits point of application [titikkerjanya], magnitude [besarnya], line of action [garisaksi], and sense [arah]. • Dari Percobaanmenunjukkanbahwapengaruhkombinasidariduagayadapatdigantikandengansebuahgayaresultan. Resultant of Two Forces • Gaya resultant adalahequivalent dengan diagonal of a parallelogram [jajarangenjang] denganduagayapadakeduasisinya. • Force [ gaya ] adalahbesaranvector .

  5. Vector: parameters yang mempunyaibesardanarah, yang penjumlahannyamengikutihukumjajarangenjang[parallelogram law]. Examples: displacements [perpindahan], velocities [kecepatan], accelerations [percepatan], momen. • Vector Equal atausama: vectors mempunyaibesardanarahnyasama. • Negative vector of a given vector has the same magnitude and the opposite direction. Vectors • Scalar: parameters yang mempunyaibesartetapitidakmempunyaiarah. Ex: mass, volume, temperature • Klasifikasi Vector: • Tertentuatauterikat;vectors mempunyaititiktangkap yang pastiyaitupartikelitusendiri yang tidakdapatdipindahkantanpamerubahkondisisoal yang ditinjau/analisa [ex: gayapadapartikel]. • Bebas;vectors yang dapatdiubahdenganbebasdalamruangtanpamempengaruhianalisa [ex: kopelgaya]. • Geser;vectors yang dapatdipindahkan, ataumenggesersepanjanggarisaksi [ex: gayapadabendategar].

  6. Hukumjajarangenjanguntukpenjumlahan vector. • Hukumsegitigauntukpenjumlahanvektor • Law of cosines, C B C • Law of sines, B • Penjumlahan Vector adalahbersifatcommutative, • Vector subtraction Penjumlahan Vectors

  7. PenjumlahandaritigaataulebihvektordenganpenggunaanberulangdarihukumsegitigaPenjumlahandaritigaataulebihvektordenganpenggunaanberulangdarihukumsegitiga • Aturanpoligonuntukpenjumlahantigaataulebih vector. • Penjumlahan Vector adalahassociative, • Perkalianvektordenganskalar PenjumlahanVector______________cont…

  8. Concurrent forces: sekumpulangaya yang melaluititik yang sama. Sekumpulangaya concurrent yang diterapkanpadasebuahpartikeldapatdigantikandengansebuah resultant force yang merupakanpenjumlahanvektordarigaya-gaya yang bekerja. • Vector force components: two or more force vectors which, together, have the same effect as a single force vector. Resultant of Several Concurrent Forces

  9. Sample Problem 2.1 • SOLUTION: • Graphical solution - construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphically evaluate the resultant which is equivalent in direction and proportional in magnitude to the the diagonal. Duabuahgaya P dan Q beraksipadasuatupakuA. Tentukanresultannya • Trigonometric solution - use the triangle rule for vector addition in conjunction with the law of cosines and law of sines to find the resultant.

  10. Graphical solution – Jajarangenjangdengansisisamadengan P dan Q digambarmengikutiskala. Besardanarahgayareultandiukurdandiperoleh, • Graphical solution – hukumsegitigadapat pula digunakan. Gaya PdanQdigambarmegikutiskaladengancaramenghubungkanujungdanekorgaya. Kemudianbesardanarahgayadiukur, Sample Problem 2.1

  11. Sample Problem 2.1 • Trigonometric solution – hukumsegitigadigunakanlagidisini, duasisidengansudutnyadiketahui. Denganmemakairumuskosinusdapatdicari, Dari rumus sinus ditulis,

  12. Sample Problem 2.2 • SOLUTION: • Find a graphical solution by applying the Parallelogram Rule for vector addition. The parallelogram has sides in the directions of the two ropes and a diagonal in the direction of the barge axis and length proportional to 5000 lbf. Sebuahtongkangditarikolehduakapalpenyeret. Jikaresultangaya yang dilakukanolehkapalpenyeretsebesar5000 lbfdiarahkansepanjangsumbutongkang, tentukanlah: • Find a trigonometric solution by applying the Triangle Rule for vector addition. With the magnitude and direction of the resultant known and the directions of the other two sides parallel to the ropes given, apply the Law of Sines to find the rope tensions. • gayapadamasing-masingtali, denganmengetahuia= 45o, • Hargadaria agar gayapadatali2 minimum. • The angle for minimum tension in rope 2 is determined by applying the Triangle Rule and observing the effect of variations in a.

  13. Gaya untuka = 45o • Graphical solution – denganmenggunakanhukumjajarangenjangdapatdiperoleh diagonal(resultan) sebesar 5000 lbfdenganarahkekanan. Sisi-sisinyadigambarkansejajardengankeduatali. Bilagambardilakukanmengikutiskala, kitaperoleh : • Trigonometric solution - hukumsegitigadanhukum sinus , Sample Problem 2.2

  14. Mencarihargaauntukgaya T2 minimum • Untukmenentukanhargaa agar tali 2 (T2) minimum, hukumsegitigakitagunakan. Dalamgambar yang ditunjukkangaris 1 – 1’ adalaharah T1 yang diketahui. Beberapakemungkinanarah T2 ditunjukkanolehgaris 2 – 2’. • Dapatkitalihatbahwaharga T2 minimum bilaT1danT2salingtegaklurus. Hargamimimum T2 : Sample Problem 2.2

  15. Quis (10 menit) Duabuahgaya P dan Q beraksipadasuatupaku hook A. Tentukanbesardanarahresultannya secaragrafisdenganhukumjajarangenjangdansegitiga secaratrigonometik

  16. May resolve a force vector into perpendicular components so that the resulting parallelogram is a rectangle. are referred to as rectangular vector components and • Define perpendicular unit vectors which are parallel to the x and y axes. • Vector components may be expressed as products of the unit vectors with the scalar magnitudes of the vector components.Fx and Fyare referred to as the scalar components of Rectangular Components of a Force: Unit Vectors

  17. Wish to find the resultant of 3 or more concurrent forces, • Resolve each force into rectangular components • The scalar components of the resultant are equal to the sum of the corresponding scalar components of the given forces. • To find the resultant magnitude and direction, Addition of Forces by Summing Components

  18. Sample Problem 2.3 • SOLUTION: • Uraikanmasing-masinggayamenjadikomponentegaklurusnya. • Tentukankomponen-komponendariresultandenganmenambahkanberdasarkankomponengayanya. • Hitungbesardanarahresultan. Empatgayabekerjapadatitik A, sepertipadagambar. Tentukanresultangaya-gaya yang bekerjapadabaut.

  19. SOLUTION: • Uraikanmasing-masinggayamenjadikomponentegaklurus. • Tentukankomponen-komponendariresultandenganmenambahkanberdasarkankomponengayanya. • Hitungbesardanarahresultan. Sample Problem 2.3

  20. Particle acted upon by three or more forces: • graphical solution yields a closed polygon • algebraic solution • Particle acted upon by two forces: • equal magnitude • same line of action • opposite sense Equilibrium of a Particle • When the resultant of all forces acting on a particle is zero, the particle is in equilibrium. • Newton’s First Law: If the resultant force on a particle is zero, the particle will remain at rest or will continue at constant speed in a straight line.

  21. Free-Body Diagram: A sketch showing only the forces on the selected particle. Free-Body Diagrams Space Diagram: A sketch showing the physical conditions of the problem.

  22. Sample Problem 2.4 • SOLUTION: • Construct a free-body diagram for the particle at the junction of the rope and cable. • Apply the conditions for equilibrium by creating a closed polygon from the forces applied to the particle. • Apply trigonometric relations to determine the unknown force magnitudes. Dalamsuatuoperasibongkarmuatkapal, sebuahmobilseberat3500-lb diangkatolehseutaskabel. Seutastalidiikatkanpadakabeltersebutdititik A danditarik agar mobilsampaiketempat yang dikehendaki. Sudutantarakabeldanarahvertikaladalah 2o, sedangsudutantaratalidanarahhorisontal 30o. Berapagayatali

  23. Solve for the unknown force magnitudes. Sample Problem 2.4 • SOLUTION: • Construct a free-body diagram for the particle at A. • Apply the conditions for equilibrium.

  24. Sample Problem 2.6 • SOLUTION: • Choosing the hull as the free body, draw a free-body diagram. • Express the condition for equilibrium for the hull by writing that the sum of all forces must be zero. It is desired to determine the drag force at a given speed on a prototype sailboat hull. A model is placed in a test channel and three cables are used to align its bow on the channel centerline. For a given speed, the tension is 40 lb in cable AB and 60 lb in cable AE. Determine the drag force exerted on the hull and the tension in cable AC. • Resolve the vector equilibrium equation into two component equations. Solve for the two unknown cable tensions.

  25. SOLUTION: • Choosing the hull as the free body, draw a free-body diagram. • Express the condition for equilibrium for the hull by writing that the sum of all forces must be zero. Sample Problem 2.6

  26. Sample Problem 2.6 • Resolve the vector equilibrium equation into two component equations. Solve for the two unknown cable tensions.

  27. This equation is satisfied only if each component of the resultant is equal to zero Sample Problem 2.6

  28. The vector is contained in the plane OBAC. • Resolve into horizontal and vertical components. • Resolve into rectangular components Rectangular Components in Space

  29. With the angles between and the axes, • is a unit vector along the line of action ofand are the direction cosines for Rectangular Components in Space

  30. Direction of the force is defined by the location of two points, Rectangular Components in Space

  31. Sample Problem 2.7 • SOLUTION: • Based on the relative locations of the points A and B, determine the unit vector pointing from A towards B. • Apply the unit vector to determine the components of the force acting on A. • Noting that the components of the unit vector are the direction cosines for the vector, calculate the corresponding angles. The tension in the guy wire is 2500 N. Determine: a) components Fx, Fy, Fz of the force acting on the bolt at A, b) the angles qx, qy, qzdefining the direction of the force

  32. SOLUTION: • Determine the unit vector pointing from A towards B. • Determine the components of the force. Sample Problem 2.7

  33. Sample Problem 2.7 • Noting that the components of the unit vector are the direction cosines for the vector, calculate the corresponding angles.

  34. thanks

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