ES 101 Mechanics of Particles and Rigid Bodies Additional Sample Problems (Cluster ABC) INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN Prepared by: Stephanie Joy B. Carag ADDITION OF VECTORS • Parallelogram Law o This states that two forces acting on a particle may be replaced by a single force, called their resultant , obtained by drawing the diagonal of the parallelogram which has sides equal to the given forces. • Triangle Rule INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 2 RECTANGULAR COMPONENTS OF A FORCE • In many problems it will be found desirable to resolve a force into two components which are perpendicular to each other. • Scalar: Rectangular components (Fx and Fy) INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 3 UNIT VECTORS • In many problems it will be found desirable to resolve a force into two components which are perpendicular to each other. • Unit vectors (i and j) πΉπ₯ = πΉcosπ πΉπ¦ = πΉsinπ πΉ = πΉπ₯ i + πΉπ¦ j INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 4 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 5 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 6 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 7 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 8 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 9 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 10 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 11 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 12 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 13 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 14 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 15 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 16 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 17 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 18 PROBLEM 2.36 Knowing that the tension in rope AC is 365 N, determine the resultant of the three forces exerted at point C of post BC. INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 19 PROBLEM 2.36 Knowing that the tension in rope AC is 365 N, determine the resultant of the three forces exerted at point C of post BC. Free-Body Diagram 25 C 24 3 7 500N 5 4 TAC= 365N 200N INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 20 PROBLEM 2.36 Knowing that the tension in rope AC is 365 N, determine the resultant of the three forces exerted at point C of post BC. Free Body Diagram Cable force AC: πΉπ₯ = − 365π 500N πΉπ¦ = − 365π 960 = −240N 1460 1100 = −275N 1460 500N Force: πΉπ₯ = 500π TAC= 365N 200N πΉπ¦ = 500π 24 = 480N 25 7 = 140N 25 200N Force: πΉπ₯ = 200π πΉπ¦ = − 200π INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN 4 = 160N 5 3 = −120N 5 ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 21 PROBLEM 2.36 Knowing that the tension in rope AC is 365 N, determine the resultant of the three forces exerted at point C of post BC. Cable force AC: πΉπ₯ = − 365π πΉπ¦ = − 365π Resultant: 960 = −240N 1460 1100 = −275N 1460 π π₯ = ΰ· πΉπ₯ = −240π + 480π + 160π = 400N π π¦ = ΰ· πΉπ¦ = −275π + 140π − 120π = −255N 500N Force: πΉπ₯ = 500π πΉπ¦ = 500π 24 = 480N 25 24 = 140N 25 200N Force: πΉπ₯ = 200π πΉπ¦ = − 200π 4 = 160N 5 3 = −120N 5 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN R = 400N i − 255N j π = π π₯ 2 + π π¦ 2 = (400π)2 + −255π 2 = πππ. πππ΅ 255 400 ∝= 32.5° tan ∝ = Rx = 400N i ∝ Ry = -255N j ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC R = 474.37N 22 Note: Three force body, i.e., a rigid body subjected to three forces or, more generally, a rigid body subjected to forces acting at only three points INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 23 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 24 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 25 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 26 RECTILINEAR MOTION OF PARTICLES • Uniform Rectilinear Motion π₯ = π₯0 + π£π‘ • Uniformly Accelerated Rectilinear Motion π£ = π£0 + ππ‘ INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 27 MOTION OF SEVERAL PARTICLES • Consider two particles A and B moving along the same straight line • If the position coordinates xA and xB are measured from the same origin, the difference xB - xA defines the relative position coordinate of B with respect to A and is denoted by xB/A. INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 28 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 29 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 30 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 31 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 32 Problem 3.35 Given the vectors P = 3i - j + 2k, Q = 4i + 5j - 3k, and S = -2i + 3j - k, compute the scalar products P · Q, P · S, and Q · S. INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 33 Problem 3.35 Given the vectors P = 3i - j + 2k, Q = 4i + 5j - 3k, and S = -2i + 3j - k, compute the scalar products P · Q, P · S, and Q · S. π β π = (3π − π + 2π) β (4π + 5π − 3π) π β π = 3 4 + −1 5 + 2 −3 π β π = 12 − 5 − 6 = 1 π β π = (3π − π + 2π) β (−2π + 3π − π) π β π = 3 −2 + −1 3 + 2 −1 π β π = −6 − 3 − 2 = −11 π β π = (4π + 5π − 3π) β (−2π + 3π − π) π β π = 4 −2 + 5 3 + −3 −1 π β π = −8 + 15 + 3 = 10 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 34 RECTANGULAR COMPONENTS OF 3D FORCES: Using 2 points on its LOA Consider a force F along line of action, line MN. • When a segment of the line of action is given, π = ππ₯ πΤ¦ + ππ¦ πΤ¦ + ππ§ π where ππ₯ = π₯2 − π₯1 π =π= ππ¦ = π¦2 − π¦1 ππ§ = π§2 − π§1 ππ₯ 2 + ππ¦ 2 + ππ§ 2 • π the unit vector of its line of action is then determined as π π₯2 − π₯1 πΤ¦ + π¦2 − π¦1 πΤ¦ + π§2 − π§1 π π = = π π₯2 − π₯1 2 + π¦2 − π¦1 2 + π§2 − π§1 2 • The force can then be expressed as πΉ π = πΉπ = ππ₯ πΤ¦ + ππ¦ πΤ¦ + ππ§ π π ππ¦ ππ₯ ππ§ π = πΉ πΤ¦ + πΉ πΤ¦ + πΉ π π π π INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN REFLECT: This means that the direction cosines are related to the line of action such that π = πΉπ₯ πΤ¦ + πΉπ¦ πΤ¦ + πΉπ§ π ππ₯ cos ππ₯ = π ππ¦ cos ππ¦ = π ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC ππ§ cos ππ§ = π INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 36 Coordinates: A (16, 0, -11) ft B (0, 8, 0) ft C (0, 8, -27) ft INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 37 π =π= A (16, 0, -11) ft B (0, 8, 0) ft C (0, 8, -27) ft ππ₯ 2 + ππ¦ 2 + ππ§ 2 AB < -16, 8, 11 > AC < -16, 8, -16 > INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 38 A (16, 0, -11) ft B (0, 8, 0) ft C (0, 8, -27) ft AB < -16, 8, 11 > AC < -16, 8, -16 > INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 39 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 40 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. A (960, 240, 0) mm B (0, 0, 380) mm C (0, 0, -320) mm D (0, 960, -220) mm ΰ· πΉπ΄ = 0 βΆ ππ΄π΅ + ππ΄πΆ + ππ΄π· + π = 0 INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 41 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. A (960, 240, 0) mm B (0, 0, 380) mm C (0, 0, -320) mm D (0, 960, -220) mm Express the vectors into components <dx, dy, dz> or dxiΜ + dyjΜ + dzkΜ : π΄π΅ < −960, −240, 380 > ππ π΄πΆ < −960, −240, −320 > ππ π΄π· < −960, 720, −220 > ππ INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 42 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. A (960, 240, 0) mm B (0, 0, 380) mm C (0, 0, -320) mm D (0, 960, -220) mm π΄π΅ < −960, −240, 380 > ππ π΄πΆ < −960, −240, −320 > ππ π΄π· < −960, 720, −220 > ππ Identify the magnitude of the vector π΄π΅ = −960 2 + −240 2 + 380 π =π= 2 π π₯ 2 + ππ¦ 2 + ππ§ 2 = ππππ ππ AB = 1060 mm AC = 1040 mm AD = 1220 mm INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 43 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. A (960, 240, 0) mm B (0, 0, 380) mm C (0, 0, -320) mm D (0, 960, -220) mm π΄π΅ < −960, −240, 380 > ππ π΄πΆ < −960, −240, −320 > ππ π΄π· < −960, 720, −220 > ππ AB = 1060 mm AC = 1040 mm AD = 1220 mm π π π ΰ·‘ Determine the unit vector πΰ΄± = π₯ πΖΈ + π¦ πΖΈ + π§ π π π π 960 240 380 ΰ·‘ πΖΈ − πΖΈ + π 1060 1060 1060 960 240 320 ΰ·‘ ππ¨πͺ = − πΖΈ − πΖΈ − π 1040 1040 1040 960 720 220 ΰ·‘ ππ¨π« = − πΖΈ + πΖΈ − π 1220 1220 1220 ππ¨π© = − INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 44 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. Force components: 960 240 380 ΰ·‘ πΖΈ − πΖΈ + π 1060 1060 1060 960 240 320 ΰ·‘ π»π¨πͺ = ππ΄πΆ ππ΄πΆ = ππ΄πΆ − πΖΈ − πΖΈ − π 1040 1040 1040 960 720 220 ΰ·‘ π»π¨π« = ππ΄π· ππ΄π· = 305 π − πΖΈ + πΖΈ − π 1220 1220 1220 π»π¨π© = ππ΄π΅ ππ΄π΅ = ππ΄π΅ − INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 45 Problem 2.107 Three cables are connected at A, where the forces P and Q are applied as shown. Knowing that Q = 0, find the value of P for which the tension in cable AD is 305 N. Force components: 960 240 380 ΰ·‘ πΖΈ − πΖΈ + π 1060 1060 1060 960 240 320 ΰ·‘ π»π¨πͺ = ππ΄πΆ ππ΄πΆ = ππ΄πΆ − πΖΈ − πΖΈ − π 1040 1040 1040 960 720 220 ΰ·‘ π»π¨π« = ππ΄π· ππ΄π· = 305 π − πΖΈ + πΖΈ − π 1220 1220 1220 π»π¨π© = ππ΄π΅ ππ΄π΅ = ππ΄π΅ − Summation of forces: ΰ· πΉπ΄ = 0 βΆ ππ΄π΅ + ππ΄πΆ + ππ΄π· + π = 0 ΰ· πΉπ₯ = 0: − 960 960 −960 ππ΄π΅ − ππ΄πΆ + 1060 1040 1220 305 + π = 0 ΰ· πΉπ¦ = 0: − 240 240 720 ππ΄π΅ − ππ΄πΆ + 1060 1040 1220 305 = 0 ΰ· πΉπ§ = 0: 380 320 −220 ππ΄π΅ − ππ΄πΆ + 1060 1040 1220 305 = 0 π»π¨π© = πππ. πππ΅ π»π¨πͺ = πππ. πππ΅ π· = πππ π΅ INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 46 Problem 2.72 Determine (a) the x, y, and z components of the 750-N force, (b) the angles θx, θy, and θz that the force forms with the coordinate axes INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 47 Problem 2.72 Determine (a) the x, y, and z components of the 750-N force, (b) the angles θx, θy, and θz that the force forms with the coordinate axes πΉβ = πΉ sin 35 πΉβ = 750π sin 35 πΉβ = 430.18 π πΉπ₯ = πΉβ cos 25 πΉπ₯ = 430.18π cos 25 πΉπ₯ = 389.88 π πΉπ¦ = πΉ cos 35 πΉπ¦ = 750π cos 35 πΉπ¦ = 614.36π πΉπ₯ 389.88π = πΉ 750π πΉπ¦ 614.36π cos ππ¦ = = πΉ 750π πΉπ§ 181.802π cos ππ§ = = πΉ 750π cos ππ₯ = ππ₯ = 58.7° ππ¦ = 35.0° ππ§ = 76.0° Fh πΉπ§ = πΉβ sin 25 πΉπ§ = 430.18π sin 25 πΉπ§ = 181.802π INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 48 STATIC AND KINETIC FRICTION Fs πΉπ = ππ π πΉπ = ππ π μs = coefficient of static friction μk =coefficient of kinetic friction INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 49 ANGLES OF FRICTION tan ∅π = ππ tan ∅π = ππ Οs = angle of static friction Οk = angle of kinetic friction INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 50 Problem 8.6 Knowing that the coefficient of friction between the 25-kg block and the incline is μs = 0.25, determine (a) the smallest value of P required to start the block moving up the incline, (b) the corresponding value of b. INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 51 Problem 8.6 Knowing that the coefficient of friction between the 25-kg block and the incline is μs = 0.25, determine (a) the smallest value of P required to start the block moving up the incline, (b) the corresponding value of b. Free-Body Diagram (Impending motion up) P P β β 30° FS 30° W Οs R W N INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 52 Problem 8.6 Knowing that the coefficient of friction between the 25-kg block and the incline is μs = 0.25, determine (a) the smallest value of P required to start the block moving up the incline, (b) the corresponding value of b. P β Ο 30° s W P a 30° R b π = ππ π = 25 ππ 9.81π/π 2 π = 245.25π tan ∅π = ππ ∅π = tan−1 0.25 ∅π = 14° W = mg d R For minimum P, P ⊥ R π π π = π sin 30° + ∅π π = 245.25 π sin 30° + 14.04° sin 30° + ∅π = π·πππ = ππππ΅ Consider β³acd, 90 = (30 + Οs )+ [90 - (30+ β)] β = Οs β = ππ° c INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 53 Problem 8.11 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between two blocks and zero between block B and incline, determine the value of π for which motion is impending. Block A Impending motion: y T x π ΰ· πΉπ¦ = 0: π1 − π1 cos π = 0 π1 = 20 9.81 cos π π1 = 196.2 cos π F1= ππ N1 N1 W1 = m1g ΰ· πΉπ₯ = 0: −π + πΉ1 + π1 sππ π = 0 π = πΉ1 + π1 sππ π π = 0.15 196.2 cos π + 196.2 sin θ π = 29.43 cos π + 196.2 sin θ INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 54 Problem 8.11 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between two blocks and zero between block B and incline, determine the value of π for which motion is impending. Block B Impending motion: y T x ΰ· πΉπ¦ = 0: π2 − π1 − π2 cos π = 0 F1= ππ N1 π π2 = 196.2 cos π + 30(9.81) cos π π2 = 490.5 cos θ F2= 0 N2 ΰ· πΉπ₯ = 0: −π − πΉ1 − πΉ2 + π2 sππ π = 0 W2 = m2g INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN π = −πΉ1 − 0 + π2 sππ π π = −0.15 196.2 cos π + 30(9.81) sin θ π = −29.43 cos π + 294.3 sin θ ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 55 Problem 8.11 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between two blocks and zero between block B and incline, determine the value of π for which motion is impending. Eq (1): π = 29.43 cos π + 196.2 sin θ Eq (2): π = −29.43 cos π + 294.3 sin θ Eq (2) – Eq (1): 0 = −58.86 cos π + 98.1 sin θ 58.86 cos π = 98.1 sin θ 58.86 98.1 π = ππ° tan θ = INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 56 Problem 8.12 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between all surfaces of contact, determine the value of π for which motion is impending. INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 57 Problem 8.12 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between all surfaces of contact, determine the value of π for which motion is impending. Block A Impending motion: y T x π ΰ· πΉπ¦ = 0: π1 − π1 cos π = 0 π1 = 20 9.81 cos π π1 = 196.2 cos π F1= ππ N1 N1 W1 = m1g ΰ· πΉπ₯ = 0: −π + πΉ1 + π1 sππ π = 0 π = πΉ1 + π1 sππ π π = 0.15 196.2 cos π + 196.2 sin θ π = 29.43 cos π + 196.2 sin θ INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 58 Problem 8.12 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between all surfaces of contact, determine the value of π for which motion is impending. Block B Impending motion: y T x π N2 ΰ· πΉπ¦ = 0: π2 − π1 − π2 cos π = 0 F1= ππ N1 π2 = 196.2 cos π + 30(9.81) cos π π2 = 490.5 cos θ F2= ππ N2 ΰ· πΉπ₯ = 0: −π − πΉ1 − πΉ2 + π2 sππ π = 0 W2 = m2g π = −πΉ1 − πΉ2 + π2 sππ π π = −0.15 196.2 cos π − 0.15(490.5 cos θ) + 30(9.81) sin θ π = −29.43 cos π − 73.575 cos θ + 294.3 sin θ INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 59 Problem 8.12 The 20-kg block A and the 30-kg block B are supported by an incline that is held in the position shown. Knowing that the coefficient of static friction is 0.15 between all surfaces of contact, determine the value of π for which motion is impending. Eq (1): π = 29.43 cos π + 196.2 sin θ Eq (2): π = −29.43 cos π − 73.575 cos π + 294.3 sin θ Eq (2) – Eq (1): 0 = −132.435 cos π + 98.1 sin θ 132.435 cos π = 98.1 sin θ 132.435 98.1 π = ππ° tan θ = INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 60 REFERENCE • Vector mechanics for engineers: statics and dynamics / Ferdinand Beer . . . [et al.]. — 10th ed (2013) INSTITUTE OF CIVIL ENGINEERING COLLEGE OF ENGINEERING UNIVERSITY OF THE PHILIPPINES DILIMAN ES 101 Mechanics of Particles and Rigid Bodies Cluster ABC 61
