3 Rigid CHATER VECTOR ECHANICS FOR ENGINEERS: STATICS Ferdinand. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Teas Tech Universit Bodies: Equivalent Sstems of Forces Contents & Objectives Introduction Define eternal and internal forces Understand the principle of transmissibilit Be able to calculate vector products of two vectors Understand and calculate the moment of a force about a point (2D and 3D) Varigon s theorem Rectangular components of the moment of a force Scalar product of two vectors Scalar product of two vectors: Applications ied triple product of three vectors Define and calculate the moment of a force about a given ais Define and calculate the moment of a couple Resolution of a force into a force at O and a couple Be able to find and equivalent forcecouple sstem of forces 3-2 1
Introduction We need to consider: the sie of the bod and the specific points of application of the forces. We assume that bodies are rigid, (actual deformations are small and do not affect the conditions of equilibrium or motion of the bod). 3-3 Introduction (main concepts) An sstem of forces can be replaced b an equivalent sstem consisting of: one force acting at a given point and one couple. Current chapter describes the effect of forces eerted on a rigid bod and how to replace a given sstem of forces with a simpler equivalent sstem. moment of a force about a point moment of a force about an ais moment due to a couple 2
Eternal and Internal Forces Forces acting on rigid bodies are divided into two groups: - Eternal forces - Internal forces Eternal forces are shown in a free-bod diagram. If unopposed, each eternal force can impart a motion of translation or rotation, or both. 3-5 rinciple of Transmissibilit: Equivalent Forces rinciple of Transmissibilit - Conditions of equilibrium or motion are not affected b transmitting a force along its line of action. Forces are sliding vectors. NOTE: F and F are equivalent forces. oving the point of application of the force F to the rear bumper does not affect the motion or the other forces acting on the truck. rinciple of transmissibilit ma not alwas appl in determining internal forces and deformations. 3-6 3
Vector roduct of Two Vectors The moment of a force about a point is a vector product or cross product. Vector product of two vectors and is defined as the vector V which satisfies the following conditions: 1. Line of action of V is perpendicular to plane containing and. 2. agnitude of V is V sin 3. Direction of V is obtained from the right-hand rule. Vector products: - are not commutative, - are distributive, - are not associative, 1 2 1 2 S S 3-7 Vector roducts: Rectangular Components Vector products of Cartesian unit vectors, i i 0 j i k k i j i j k j j 0 k j i i k j j k i k k 0 Vector products in terms of rectangular coordinates V i j k i j k i j k To be mastered!!! 3-8 4
Calculate the following vector products: oment of a Force About a oint Where do ou think a moment is applied? Beams are often used to bridge gaps in walls. We have to know what the effect of the force on the beam will have on the beam supports. What do ou think those impacts are at points A and B? 5
The moment of a force about a point provides a measure of the tendenc for rotation (sometimes called a torque). 1. What is the moment of the 12 N force about point A ( A )? F = 12 N A) 3 N m B) 36 N m C) 12 N m D) (12/3) N m E) 7 N m A d = 3 m 2. The moment of force F about point O is defined as O =. A) r F B) F r C) r F D) r * F 6
oment of a Force About a oint A force vector is defined b its magnitude and direction. Its effect on the rigid bod depends on its point of application. The moment of F about O is defined as O r F The moment vector O is perpendicular to the plane containing O and the force F. agnitude of O measures the tendenc of the force to cause rotation of the bod about an ais along O. O rf sin Fd The sense of the moment ma be determined b the right-hand rule. An force F that has the same magnitude and direction as F, is equivalent if it also has the same line of action and therefore, produces the same moment. 3-13 oment of a Force About a oint Two-dimensional structures have length and breadth but negligible depth and are subjected to forces contained in the plane of the structure. The plane of the structure contains the point O and the force F. O, the moment of the force about O is perpendicular to the plane. If the force tends to rotate the structure anticlockwise, the sense of the moment vector is out of the plane of the structure and the magnitude of the moment is positive. If the force tends to rotate the structure clockwise, the sense of the moment vector is into the plane of the structure and the magnitude of the moment is negative. 3-14 7
Varignon s Theorem The moment about a give point O of the resultant of several concurrent forces is equal to the sum of the moments of the various moments about the same point O. Varigon s Theorem makes it possible to replace the direct determination of the moment of a force F b the moments of two or more component forces of F. r F F r F r F 1 2 1 2 3-15 Rectangular Components of the oment of a Force The moment of F about O, O r F, r i j k F Fi F j Fk O i j k i F j F k F F F i F F j F F k 3-16 8
Rectangular Components of the oment of a Force The moment of F about B, r F B A / B ra / B ra rb A B A F F i F j F k B i j A i F B A j F B B A F A k B B k 3-17 Rectangular Components of the oment of a Force For two-dimensional structures, F F k O O Z F F B F B A B A B F k Z F A B A B F 3-18 9
Sample roblem 3.1 A 500-N vertical force is applied to the end of a lever which is attached to a shaft at O. Determine: a) moment about O, b) horiontal force at A which creates the same moment, c) smallest force at A which produces the same moment, d) location for a 1200-N vertical force to produce the same moment, e) whether an of the forces from b, c, and d is equivalent to the original force. 3-19 Sample roblem 3.1 SOLUTION: a) oment about O is equal to the product of the force and the perpendicular distance between the line of action of the force and O. Since the force tends to rotate the lever clockwise, the moment vector is into the plane of the paper. O Fd d O 600mm 500N 0.3 m cos60 300mm 0.3m O 150 N. m 3-20 10
Sample roblem 3.1 b) Horiontal force at A that produces the same moment, 150 N. m F 0.5196m Fd d 600 mm sin 60 519.6 mm 0.5196 m O 150 N. m F 0.5196 m F 288.68 N 3-21 Sample roblem 3.1 c) The smallest force A to produce the same moment occurs when the perpendicular distance is a maimum or when F is perpendicular to OA. O 150 N. m Fd F 0.6 m 150 N. m F 0.6 m F 250 N 3-22 11
Sample roblem 3.1 d) To determine the point of application of a 1200 N force to produce the same moment, O 150 N. m Fd 1200 N d 150 N. m d 125 mm 1200 N OB cos60 125 mm OB 250 mm 3-23 Sample roblem 3.1 e) Although each of the forces in parts b), c), and d) produces the same moment as the 500-N force, none are of the same magnitude and sense, or on the same line of action. None of the forces is equivalent to the 500-N force. 3-24 12
ui 10 N 3 m 2 m 5 N 1. Using the CCW direction as positive, the net moment of the two forces about point is A) 10 N m B) 20 N m C) - 20 N m D) 40 N m E) - 40 N m 2. If r = { 5 j } m and F = { 10 k } N, the moment r F equals { } N m. A) 50 i B) 50 j C) 50 i D) 50 j E) 0 Sample roblem 3.4 SOLUTION: The moment A of the force F eerted b the wire is obtained b evaluating the vector product, r F A C A The rectangular plate is supported b the brackets at A and B and b a wire CD. Knowing that the tension in the wire is 200 N, determine the moment about A of the force eerted b the wire at C. 3-26 13
Sample roblem 3.4 SOLUTION: r F rc A rc ra 0.3 mi 0.08 mk rd C F F 200 N rd C 0.3 mi 0.24 m j 0.32 mk 200 N 0.5 m 120 N i 96 N j 128 N A A A i j k 0.3 0 0.08 120 96 128 C A k 7.68 N mi 28.8 N m j 28.8 N mk 3-27 Scalar roduct of Two Vectors The scalar product or dot product between two vectors and is defined as Scalar products: - are commutative, - are distributive, - are not associative, 1 2 1 2 S undefined Scalar products with Cartesian unit components, i j k i j k i i 1 j j 1 k k 1 i j 0 j k 0 k i 0 2 2 2 2 3-28 14
Scalar roduct of Two Vectors: Applications Angle between two vectors: cos cos rojection of a vector on a given ais: OL cos projection of along OL cos cos OL For an ais defined b a unit vector: OL cos cos cos 3-29 Eample: Find the angle between V and vectors: V 3i 2 j 2k 1i 2 j 1k Find the unit vector in the direction of V. 15
16 ied Triple roduct of Three Vectors 3-31 ied triple product of three vectors, result scalar S The si mied triple products formed from S,, and have equal magnitudes but not the same sign, S S S S S S S S S S S S S Evaluating the mied triple product, oment of a Force About a Given Ais - Applications With the force, a person is creating a moment A. Does all of A act to turn the socket? How would ou calculate an answer to this question?
Sleeve A of this bracket can provide a maimum resisting moment of 125 N m about the -ais. How would ou determine the maimum magnitude of F before turning about the ais occurs? oment of a Force About a Given Ais oment O of a force F applied at the point A about a point O, O r F Scalar moment OL about an ais OL is the projection of the moment vector O onto the ais, r F OL O is the unit vector along OL oments of F about the coordinate aes, F F F F F F Component of O r F on OL 3-34 17
oment of a Force About a Given Ais oment of a force about an arbitrar ais, BL A B A B B r F r r r AB The result is independent of the point B along the given ais. 3-35 ui 1. When determining the moment of a force about a specified ais, the ais must be along. A) the ais B) the ais C) the ais D) an line in 3-D space E) an line in the - plane 2. The triple scalar product u ( r F ) results in A) a scalar quantit ( + or - ). B) a vector quantit. C) ero. D) a unit vector. E) an imaginar number. 18
Sample roblem 3.5 A cube is acted on b a force as shown. Determine the moment of a) about A b) about the edge AB and c) about the diagonal AG of the cube. d) Determine the perpendicular distance between AG and FC. 3-37 Sample roblem 3.5 Note that: 1. is i 2. Vector r is AF oment of about AB, 3-38 19
Sample roblem 3.5 oment of about the diagonal AG, 3-39 Sample roblem 3.5 erpendicular distance between AG and FC, 2 0 1 j k i j k 0 11 3 6 Therefore, is perpendicular to AG. AG a 6 d d a 6 3-40 20
Eample Given: Sleeve A can provide a maimum resisting moment of 125 N m about the -ais. Find: The maimum magnitude of F before slipping occurs at A (the sleeve rotating around the -ais). lan: 1) We need to use 2) Find r B/A. AB F 3) Find F in Cartesian vector form. 4) Complete the triple scalar product & solve for F! r B/A = {( 0.15 0) i + (0.30 0) j + (0.10 0) k} m F = F {( cos 60 i + cos 60 j + cos 45 k)} N = { 0.5 F i + 0.5 F j + 0.707 F k} N 21
Now find the triple product, X = 1 0 0-0.15 0.3 0.1-0.5F 0.5F 0.707F. AB F N.m = 1 {0.3 (0.707F) 0.1 (0.5F)} + 0 + 0 = 0.162 F N.m = 125 N.m = maimum moment along -ais 125 = 0.162 F F ma = 771 N Couples: ui 1. In statics, a couple is defined as separated b a perpendicular distance. A) two forces in the same direction B) two forces of equal magnitude C) two forces of equal magnitude acting in the same direction D) two forces of equal magnitude acting in opposite directions 2. The moment of a couple is called a vector. A) Free B) Spin C) Romantic D) Sliding 22
Couples - Applications A torque or moment of 12 N.m is required to rotate the wheel. Wh does one of the two grips of the wheel above require less force to rotate the wheel? Couples - Applications When ou grip a vehicle s steering wheel with both hands, a couple moment is applied to the wheel. Would older vehicles without power steering have larger or smaller steering wheels than new X5? 23
oment of a Couple Two forces F and -F having the same magnitude, parallel lines of action, and opposite sense are said to form a couple. oment of the couple, r F r F A A B B r r F r F rf sin Fd 3-47 oment of a Couple The moment vector of the couple is independent of the choice of the origin of the coordinate aes (depends onl on d), i.e., it is a free vector that can be applied at an point with the same effect. Two couples will have equal moments if F1 d1 F2d 2 the two couples lie in parallel planes, and the two couples have the same sense or the tendenc to cause rotation in the same direction. 3-48 24
Addition of Couples Consider two intersecting planes 1 and 2 with each containing a couple 1 r F1 in plane 1 r F in plane 2 2 2 Resultants of the vectors also form a couple r R r F 1 F 2 B Varigon s theorem r F1 r F2 1 2 Sum of two couples is also a couple that is equal to the vector sum of the two couples 3-49 Couples Can Be Represented b Vectors A couple can be represented b a vector with magnitude and direction equal to the moment of the couple. Couple vectors obe the law of addition of vectors. Couple vectors are free vectors, i.e., the point of application is not significant. Couple vectors ma be resolved into component vectors. 3-50 25
ui 1. F 1 and F 2 form a couple. The moment of the couple is given b. A) r 1 F 1 B) r 2 F 1 C) F 2 r 1 D) r 2 F 2 r1 F1 F2 r2 2. If three couples act on a bod, the overall result is that A) The net force is not equal to 0. B) The net force and net moment are equal to 0. C) The net moment equals 0 but the net force is not necessaril equal to 0. D) The net force equals 0 but the net moment is not necessaril equal to 0. Eample Given: A 35 N force couple acting on the rod. Find: The couple moment acting on the rod in Cartesian vector notation. lan: 1) Use = r F to find the couple moment. 2) Set r = r B/A and F = {35 k} N. 3) Calculate the cross product to find. 26
r B/A = { 0 i (0.25) j + (0.25 tan 30) k} m r B/A = { 0.25 j + 0.1443 k} m F = {0 i + 0 j + 35 k} N = r B/A F i j k 0 0 A 0 0.25 0.1443 35 0 0 35 = {( 8.75 0) i (0 0) j (0 0) k} N m = { 8.75 i + 0 j + 0 k} N m Resolution of a Force Into a Force at O and a Couple What are the resultant effects on the person s hand when the force is applied in these four different was? Wh is understanding these difference important when designing various load-bearing structures? 27
Several forces and a couple moment are acting on this vertical section of an I- beam.?? For the process of designing the I- beam, it would be ver helpful if ou could replace the various forces and moment just one force and one couple moment at point O with the same eternal effect? How will ou do that? The effect on the vertical pole can be confined to point O. 28
Resolution of a Force Into a Force at O and a Couple Force vector F can not be simpl moved to O without modifing its action on the bod. Attaching equal and opposite force vectors at O produces no net effect on the bod. The three forces ma be replaced b an equivalent force vector and couple vector, i.e, a force-couple sstem. The couple is a free vector. 3-57 Resolution of a Force Into a Force at O and a Couple oving F from A to a different point O requires the addition of a different couple vector O r F O ' The moments of F about O and O are related, O' r 'F r s F r F s F s F O oving the force-couple sstem from O to O requires the addition of the moment of the force at O about O. 3-58 29
Sample roblem 3.6 SOLUTION: Attach equal and opposite 100-N forces in the + direction at A, thereb producing 3 couples for which the moment components are easil computed. Alternativel, compute the sum of the moments of the four forces about an arbitrar single point. The point D is a good choice as onl two of the forces will produce non-ero moment contributions. Determine the components of the single couple equivalent to the couples shown. 3-59 Sample roblem 3.6 The three couples ma be represented b three couple vectors, 3-60 30
Sample roblem 3.6 Alternativel, compute the sum of the moments of the four forces about D. 3-61 Sstem of Forces: Reduction to a Force and Couple A sstem of forces ma be replaced b a collection of force-couple sstems acting a given point O The force and couple vectors ma be combined into a resultant force vector and a resultant couple vector, R F R O r F The force-couple sstem at O ma be moved to O with the addition of the moment of R about O, R R s R O' O Two sstems of forces are equivalent if the can be reduced to the same force-couple sstem. 3-62 31
Further Reduction of a Sstem of Forces Conditions under which a given sstem of forces can be reduced to a single force. If the resultant force and couple at O are mutuall perpendicular, the can be replaced b a single force acting along a new line of action. The resultant force-couple sstem for a sstem of forces will be mutuall perpendicular if: 1) the forces are concurrent, 2) the forces are coplanar, or 3) the forces are parallel. 3-63 Further Reduction of a Sstem of Forces Sstem of coplanar forces is reduced to a R force-couple sstem R and O that is mutuall perpendicular. Sstem can be reduced to a single force b moving the line of action of until its moment about O becomes R R O In terms of rectangular coordinates, R R R O and are the coordinates of the point of application of the new position of R. 3-64 32
Sample roblem 3.8 SOLUTION: a) Compute the resultant force for the forces shown and the resultant couple for the moments of the forces about A. For the beam, reduce the sstem of forces shown to (a) an equivalent force-couple sstem at A, (b) an equivalent force couple sstem at B, and (c) a single force or resultant. Note: Since the support reactions are not included, the given sstem will not maintain the beam in equilibrium. b) Find an equivalent force-couple sstem at B based on the forcecouple sstem at A. c) Determine the point of application for the resultant force such that its moment about A is equal to the resultant couple at A. 3-65 Sample roblem 3.8 SOLUTION: a) Compute the resultant force and the resultant couple at A. R F 150 N j 600 N j 100 N j 250 N j R 600 N j R A r F 1.6 i 600 j 2.8 i 100 j 4.8 i 250 j R A 1880 N mk 3-66 33
Sample roblem 3.8 b) Find an equivalent force-couple sstem at B based on the force-couple sstem at A. The force is unchanged b the movement of the force-couple sstem from A to B. R 600 N j The couple at B is equal to the moment about B of the force-couple sstem found at A. R R B A rb A R 1880 N mk 4.8 mi 600 N j 1880 N mk 2880 N mk 1000 N mk R B 3-67 Sample roblem 3.10 SOLUTION: Determine the relative position vectors for the points of application of the cable forces with respect to A. Resolve the forces into rectangular components. Three cables are attached to the bracket as shown. Replace the forces with an equivalent forcecouple sstem at A. Compute the equivalent force, R F Compute the equivalent couple, r F R A 3-68 34
Sample roblem 3.10 SOLUTION: Determine the relative position vectors with respect to A. rb A 0.075i 0.050k m rc A 0.075i 0.050k m r 0.100i 0.100 j m D A Resolve the forces into rectangular components. FB 700 N re B 75i 150 j 50k re B 175 0.429i 0.857 j 0.289k F 300i 600 j 200k N B F C F D 1000 Ncos 45i cos 45k 707i 707k N 1200 Ncos 60i cos 30 j 600i 1039 j N 3-69 Sample roblem 3.10 Compute the equivalent force, R F 300 707 600i 600 1039 j 200 707k R 1607i 439 j 507k N Compute the equivalent couple, R A r F i j k r F 0.075 0 0.050 30i 45k B A r C A r D A R A 300 i F 0.075 F B c D 707 i 0.100 600 600 200 j k 0 0.050 17.68 j 0 707 j k 0.100 0 163.9k 1039 30i 17.68 j 118.9k 0 3-70 35
Replace each loading with an equivalent force-couple sstem at end A of the beam. Which ones are equivalent? Reduce (a) to a single force. 36
The resultant is simpl: R F ( 180 54 36 90 )j 360 j At O we can reduce the sstem to a single force and couple. r F 324i 378k R O The resultant force and couple are mutuall perpendicular. We can reduce it further to a single force. R O r R i k 360 j 324i 378k 360k 360i 324i 378k 360 378 360 324 1. 05m 0. 9m 37