VIETNAM OIL AND GAS GROUP PETROVIETNAM UNIVERSITY THEORETICAL MECHANICS CHAPTER 2: MOMENT OF A FORCE AND A COUPLE Lecturer : Dr Vo Quoc Thang Email : thangvq@pvu.edu.vn Website : http://www.pvu.edu.vn Contents • Scalar and Vector Products of Vectors • Principle of Transmissibility • Moment of a Force About a Point • Varignon’s Theorem • Rectangular Components of the Moment of a Force • Moment of a Force About a Given Axis • Moment of a Couple • Addition of Couples • Resolution of a Force Into a Force and a Couple • Reduction of a System of Forces Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Scalar Product of Two Vectors • Scalar product between two vectors P and Q: P  Q  PQ cos  • Properties: - Commutative: - Distributive: - NOT associative: PQ  QP P  Q1  Q   P  Q1  P  Q P  Q   S  P  Q  S  • Scalar products with Cartesian unit vectors: i  i 1 j j 1 k  k 1 i  j  j k  k  i  P  Q  Px i  Py j  Pzk  Q x i  Q y j  Q zk   Px Q x  Py Q y  Pz Q z P  P  Px2  Py2  Pz2  P Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Scalar Product of Two Vectors • Angle between two vectors: P  Q Px Q x  Py Q y  Pz Q z cos    PQ PQ • Projection POL of a vector P on a given axis OL: PQ Q  P  λ  P  cos  x i  cos  y j  cos zk  POL  P cos    Px cos  x  Py cos  y  Pz cos z Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Vector Product of Two Vectors • Vector product V of two vectors P and Q: V  P Q 1.Line of action: perpendicular to plane containing P and Q 2.Magnitude: V  PQ sin 3.Direction: right-hand rule or • Properties: - NOT commutative: Q  P   P  Q  - Distributive: P  Q1  Q   P  Q1  P  Q - NOT associative: P  Q   S  P  Q  S  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Vector Product of Two Vectors • Vector products of Cartesian unit vectors: ii  j  i  k k  i  j i j  k j j  k  j  i i  k  j j k  i k k  • Vector products in Cartesian coordinates: V  Px i  Py j  Pzk  Q x i  Q y j  Q zk   Py Q z  Pz Q y i  Px Q z  Pz Q x j  Px Q y  Py Q x k Px Qx i  Py Qy j Pz Qz k Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Mixed Triple Product of Three Vectors • Mixed triple product of three vectors: S  P  Q   Scalar • The six mixed triple products formed from S, P and Q have equal magnitudes but not the same sign: S  P  Q   P  Q  S   Q  S  P   S  Q  P    P  S  Q   Q  P  S  • Mixed triple product in Cartesian coordinates: Sx S  P  Q   S y Sz  Px Qx Py Qy Pz Qz  Sx Py Q z  Pz Q y   S y Px Q z  Pz Q x   Sz Px Q y  Py Q x  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Principle of Transmissibility Conditions of equilibrium or motion of a rigid body are not affected by transmitting a force along its line of action NOTE: F and F’ are equivalent forces Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Principle of Transmissibility • NOT always applied Ex: in determining internal forces and deformations Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE Moment of a Force About a Point The moment of F applied at the point A about a point O is defined as: M O  rA / O  F Line of action: perpendicular to the plane containing O and the force F Magnitude: M O  rF sin   Fd N.m  or lb.ft  Direction: right-hand rule Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 10 Sample Problem c) Moment of P about the diagonal AG: M AG  λ AG  M A AG  aj  ak i  j  k    AG a 3 aP i  j  k  MA  λ AG  aP   i  j  k  M AG  i  jk  aP 1   1  aP  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 28 Sample Problem d) Perpendicular distance between AG and FC: P  j  k   i  j  k   P 0   1 0 Pλ  Therefore, P is perpendicular to AG M AG  d Dr Vo Quoc Thang aP  Pd a MOMENT OF A FORCE AND A COUPLE 29 Moment 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 • Moment of the couple: M  rA  F  rB   F   rA  rB   F  rA / B  F M  rF sin   Fd Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 30 Moment of a Couple • The moment vector of the couple is a free vector that can be applied at any point with the same effect • Two couples will have equal moments if: - F1d1=F2d2 - Parallel planes - Same sense Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 31 Addition of Couples • Consider two intersecting planes P1 and P2 with each containing a couple: M1  r  F1 in plane P1 M  r  F2 in plane P2 • Resultants of the vectors also form a couple: M  r  R  r  F1  F2  • By Varignon’s theorem: M  r  F1  r  F2  M1  M • Sum of two couples is also a couple that is equal to the vector sum of the two couples Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 32 Resolution of a Force Into a Force and a Couple Force vector F applied at A can be moved to O by replacing it with an equivalent force vector F at O and a couple vector MO=rxF Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 33 Resolution of a Force Into a Force and a Couple Moving F from A to a different point O’ The moments of F about O and O’ are related: M O'  r 'F  r  s   F  r  F  s  F  MO  s  F Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 34 Sample Problem Determine the components of the single couple equivalent to the couples shown Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 35 Sample Problem 1) Attach equal and opposite 20 lb forces in the +x direction at A: The three couples may be represented by three couple vectors, M x  30.18  540 lb  in M y  20.12  240lb  in M z  20.9  180 lb  in M  540 i  240 j  180k lb.in. Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 36 Sample Problem 2) Compute the sum of the moments of the four forces about D: Only the forces at C and E contribute to the moment about D M  M D  18 j   30 k  9 j  12k   20  i M  540 i  240 j  180k lb.in. Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 37 Reduction of a System of Forces A system of forces may be replaced by a collection of force-couple systems acting a given point O R   Fi R MO   ri  Fi  NOTE: Two systems of forces are equivalent if they can be reduced to the same force-couple system Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 38 Sample Problem For the beam, reduce the system of forces shown to (a) an equivalent force-couple system at A, (b) an equivalent force couple system at B, (c) a single force or resultant Note: Since the support reactions are not included, the given system will not maintain the beam in equilibrium Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 39 Sample Problem SOLUTION: a) Equivalent force-couple system at A: R   Fi  150 j  600 j  100 j  250 j R  600j  N  MR A   ri  Fi   1.6 i    600 j  2.8 i   100 j  4.8 i    250 j MR A  1880k  N.m  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 40 Sample Problem b) Equivalent force-couple system at B: The force is unchanged by the movement of the force-couple system from A to B: R  600j  N  The couple at B is equal to the moment about B of the force-couple system found at A: R MR  M B A  rB A  R  1880 k   4.8 i   600 j  1880 k  2880k MR B  1000k  N.m  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 41 Sample Problem c) Single force or resultant: The force is unchanged by the movement of the force-couple system from A: R  600j  N  The moment of R about A is equal to M R A rR  MR A xi  600 j   1880k  600 xk   1880k x  3.13 m Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 42 [...]... Thang MOMENT OF A FORCE AND A COUPLE 16 Sample Problem 1 a) Moment of force applied at A about O: M O  Fd d  24 cos 60  12 in M O  100. 12 M O  120 0 lb  in Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 17 Sample Problem 1 b) Horizontal force at A that produces the same moment: d  24 .sin 60  20 .8 in M O  Fd 120 0  20 .8F F Dr Vo Quoc Thang 120 0 20 .8 F  57.7 lb MOMENT OF A FORCE AND A COUPLE. .. is 20 0 N, determine the moment about A of the force exerted by the wire at C Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 22 Sample Problem 2 SOLUTION: M A  rC A  F rC A  rC  rA  0.3i  0.08k m  F  Fλ CD  20 0 rD C rD C  20 0  0.3i  0 .24 j  0.32k 0 5   120 i  96 j  128 k  N  0 3  120 i 96 j  7.68 i  28 .8 j  28 .8k  N.m  0.08  128 k MA  0 Dr Vo Quoc Thang MOMENT OF A FORCE AND. .. distance between AG and FC MOMENT OF A FORCE AND A COUPLE 26 Sample Problem 3 a) Moment of P about A: M A  rF A  P rF A  rF  rA  a i  j FC P  j k P  Pλ FC  P  FC 2 a MA   a 0 0 i P 2 j P 2 k  aP   i  j  k   2 b) Moment of P about AB: M AB  λ AB  M A aP  aP     i  i  j  k   2 2   Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 27 Sample Problem 3 c) Moment of P about... the same moment occurs when the perpendicular distance is a maximum or when F is perpendicular to OA: M O  Fd 120 0  24 F F Dr Vo Quoc Thang 120 0 24 F  50 lb MOMENT OF A FORCE AND A COUPLE 19 Sample Problem 1 d) To determine the point of application of a 24 0 lb force to produce the same moment: M O  Fd 120 0  24 0d 120 0  5 in 24 0 OB cos60  5 in d Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE. .. Thang MOMENT OF A FORCE AND A COUPLE 31 Addition of Couples • Consider two intersecting planes P1 and P2 with each containing a couple: M1  r  F1 in plane P1 M 2  r  F2 in plane P2 • Resultants of the vectors also form a couple: M  r  R  r  F1  F2  • By Varignon’s theorem: M  r  F1  r  F2  M1  M 2 • Sum of two couples is also a couple that is equal to the vector sum of the two couples... Quoc Thang MOMENT OF A FORCE AND A COUPLE 34 Sample Problem 4 Determine the components of the single couple equivalent to the couples shown Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 35 Sample Problem 4 1) Attach equal and opposite 20 lb forces in the +x direction at A: The three couples may be represented by three couple vectors, M x  30.18  540 lb  in M y  20 . 12  24 0lb  in M z  20 .9  180... MA  2 λ AG  1 aP   i  j  k  M AG  i  jk  3 2 aP 1  1  1  6 aP  6 Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 28 Sample Problem 3 d) Perpendicular distance between AG and FC: P  j  k   1 i  j  k   P 0  1  1 2 3 6 0 Pλ  Therefore, P is perpendicular to AG M AG  d Dr Vo Quoc Thang aP  Pd 6 a 6 MOMENT OF A FORCE AND A COUPLE 29 Moment of a Couple Two forces F and. .. Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 32 Resolution of a Force Into a Force and a Couple Force vector F applied at A can be moved to O by replacing it with an equivalent force vector F at O and a couple vector MO=rxF Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 33 Resolution of a Force Into a Force and a Couple Moving F from A to a different point O’ The moments of F about O and O’ are related:... lines of action, and opposite sense are said to form a couple • Moment of the couple: M  rA  F  rB   F   rA  rB   F  rA / B  F M  rF sin   Fd Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 30 Moment of a Couple • The moment vector of the couple is a free vector that can be applied at any point with the same effect • Two couples will have equal moments if: - F1d1=F2d2 - Parallel planes... AND A COUPLE 23 Moment of a Force About a Given Axis • The moment MO of a force F applied at the point A about a point O: MO  r  F • Scalar moment MOL about an axis OL is the projection of the moment vector MO onto the axis OL: M OL  λ  M O  λ  r  F  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 24 Moment of a Force About a Given Axis • Moments of F about the coordinate axes x, y and z: ... 20 0 rD C rD C  20 0  0.3i  0 .24 j  0.32k   120 i  96 j  128 k  N   120 i 96 j  7.68 i  28 .8 j  28 .8k  N.m  0.08  128 k MA  Dr Vo Quoc Thang MOMENT OF A FORCE AND A COUPLE 23 Moment. .. Thang 120 0 24 F  50 lb MOMENT OF A FORCE AND A COUPLE 19 Sample Problem d) To determine the point of application of a 24 0 lb force to produce the same moment: M O  Fd 120 0  24 0d 120 0  in 24 0... a MOMENT OF A FORCE AND A COUPLE 29 Moment 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 • Moment of the couple: