1. Lecture #6 Moments, Couples, and Force Couple Systems

2. Equivalent Forces  We defined equivalent forces as being forces with the same magnitude acting in the same direction and acting along the same line of action (this is through the Principle of Transmissibility), but why do the forces need to act along the same line?

3. 4.1 Introduction to Moments The tendency of a force to rotate a rigid body about any defined axis is called the Moment of the force about the axis

4. MOMENT OF A FORCE - SCALAR FORMULATION (Section 4.1) The moment, M, of a force about a point provides a measure of the tendency for rotation (sometimes called a torque). M M = F * d

5. Moment caused by a Force The Moment of Force (F) about an axis through Point (A) or for short, the Moment of F about A, is the product of the magnitude of the force and the perpendicular distance between Point (A) and the line of action of Force (F) M A = Fd

6. Units of a Moment The units of a Moment are: N·m in the SI system ft·lbs or in·lbs in the US Customary system

7. APPLICATIONS 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 you think those impacts are at points A and B?

8. APPLICATIONS Carpenters often use a hammer in this way to pull a stubborn nail. Through what sort of action does the force F H at the handle pull the nail? How can you mathematically model the effect of force F H at point O?

9. Properties of a Moment Moments not only have a magnitude, they also have a sense to them. The sense of a moment is clockwise or counter- clockwise depending on which way it will tend to make the object rotate

10. Properties of a Moment The sense of a Moment is defined by the direction it is acting on the Axis and can be found using Right Hand Rule.

11. Varignon’s Theorem The moment of a Force about any axis is equal to the sum of the moments of its components about that axis This means that resolving or replacing forces with their resultant force will not affect the moment on the object being analyzed

12. MOMENT OF A FORCE - SCALAR FORMULATION (continued) As shown, d is the perpendicular distance from point O to the line of action of the force. In 2-D, the direction of M O is either clockwise or counter-clockwise, depending on the tendency for rotation. In the 2-D case, the magnitude of the moment is M o = F d

13. READING QUIZ 1. What is the moment of the 10 N force about point A (M A )? 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 F = 12 N

14. Example #1 A 100-lb vertical force is applied to the end of a lever which is attached to a shaft at O. Determine: a)Moment about O, b)Horizontal force at A which creates the same moment, c)Smallest force at A which produces the same moment, d)Location for a 240-lb vertical force to produce the same moment, e)Whether any of the forces from b, c, and d is equivalent to the original force.

15. Example #1 a) Moment 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.

16. Example #1 b) Horizontal force at A that produces the same moment,

17. Example #1 c) The smallest force at A to produce the same moment occurs when the perpendicular distance is a maximum or when F is perpendicular to OA.

18. Example #1 d) To determine the point of application of a 240 lb force to produce the same moment,

19. Example #1 e)Although each of the forces in parts b), c), and d) produces the same moment as the 100 lb force, none are of the same magnitude and sense, or on the same line of action. None of the forces is equivalent to the 100 lb force.

23. 4.4 Principle of Moments Varignon’s Theorem: The moment of a force about a point is equal to the sum of moments of the components of the force about the point:

24. MOMENT OF A FORCE - SCALAR FORMULATION (continued) Often it is easier to determine M O by using the components of F as shown (Varignon’s Theorem). Then M O = (F Y a) – (F X b). Note the different signs on the terms! The typical sign convention for a moment in 2-D is that counter-clockwise is considered positive. We can determine the direction of rotation by imagining the body pinned at O and deciding which way the body would rotate because of the force. For example, M O = F d and the direction is counter-clockwise. F a b d O a b O F F x F y

25. GROUP PROBLEM SOLVING Since this is a 2-D problem: 1) Resolve the 20 lb force along the handle’s x and y axes. 2) Determine M A using a scalar analysis. Given: A 20 lb force is applied to the hammer. Find: The moment of the force at A. Plan: x y

26. GROUP PROBLEM SOLVING (cont.) Solution: +  F y = 20 sin 30° lb +  F x = 20 cos 30° lb x y + M A = {–(20 cos 30°)lb (18 in) – (20 sin 20°)lb (5 in)} = – 351.77 lb·in = 352 lb·in (clockwise)

29. Moments in 3D 4.5 Moment of a Force about a Specific Axis In 2D bodies the moment is due to a force contained in the plane of action perpendicular to the axis it is acting around. This makes the analysis very easy. In 3D situations, this is very seldom found to be the case.

30. Moments in 3D The moment about an axis is still calculated the same way (by a force in the plane perpendicular to the axis) but most forces are acting in abstract angles. By resolving the abstract force into its rectangular components (or into its components perpendicular to the axis of concern) the moment about the axis can then be found the same way it was found in 2D – M = Fd (where d is the distance between the force and the axis of concern)

31. Notation for Moments In simpler terms the Moment of a Force about the y-axis (M y ) can be found by using the projection of the Force on the x-z Plane The Notation used to denote Moments about the Cartesian Axes are (M x, M y, and M z )

33. Given the tension in cable BC is 700 N, find Mx, My, and Mz about point A. 3D Moments Example:

34. Force Couples A Couple is defined as two Forces having the same magnitude, parallel lines of action, and opposite sense In this situation, the sum of the forces in each direction is zero, so a couple does not affect the sum of forces equations A force couple will however tend to rotate the body it is acting on

35. Moment Due to a Force Couple By multiplying the magnitude of one Force by the distance between the Forces in the Couple, the moment due to the couple can be calculated. M = Fd c The couple will create a moment around an axis perpendicular to the plane that the couple falls in. Pay attention to the sense of the Moment (Right Hand Rule)

36. Moment of a Couple Two couples will have equal moments if the two couples lie in parallel planes, and the two couples have the same sense or the tendency to cause rotation in the same direction.

37. Why do we use Force Couples? The reason we use Force Couples to analyze Moments is that the location of the axis the Moment is calculated about does not matter The Moment of a Couple is constant over the entire body it is acting on

38. Couples are Free Vectors The point of action of a Couple does not matter The plane that the Couple is acting in does not matter All that matters is the orientation of the plane the Couple is acting in Therefore, a Force Couple is said to be a Free Vector and can be applied at any point on the body it is acting

39. Resolution of Vectors The Moment due to the Force Couple is normally placed at the Cartesian Coordinate Origin and resolved into its x, y, and z components (M x, M y, and M z ).

40. Vector Addition of Couples By applying Varignon’s Theorem to the Forces in the Couple, it can be proven that couples can be added and resolved as Vectors.

41. Force Couple System Two opposing force can be added to a rigid body without affecting the equilibrium of it. If there is a force acting at a distance from an axis, two forces of equal magnitude and opposite direction can be added at the axis with out affecting the equilibrium of the rigid body. The original force and its opposing force at the axis make a couple that equates to a moment on the rigid body. The other force at the axis results in the same force acting on the body

42. Force Couple Systems  As a result of this it can be stated that any force (F) acting on a rigid body may be moved to any given point on the rigid body as long as a moment equal to moment of (F) about the axis is added to the rigid body.

43. Force Couple Systems The reverse of this is also true, any force and moment acting at a point on a rigid body can be represented as one force at a distance from the original point where the moment of the force about the original point is equal to the original moment

44. Resultant of a System of Coplanar Forces Given a system of forces acting on a rigid body, each force can be resolved into a force-couple system acting at one point. The Forces can then be added as Vectors and the Moments can be added as scalars as long as the sense of each moment is taken into account. This will reduce the system of forces to a single force-moment system acting on the rigid body. This can be reduced further to one force at a distance Mr/Fr from the point of interest.

45. Resolution of a Force into a Force Couple System in 3D The same principles used in 2D apply in 3D also The method used to resolve a force in 3D into a Force-Couple system acting at one point is to draw the rectangular coordinate system at the point of interest. Resolve the force in the plane it is in into the force acting on one axis and moment about the axis for each axis one at a time.

46. Resolution of a System of Forces in 3D Each Force can be Resolved into a Force and Moment at the point of interest using the method just discussed. The Resultant Force can then be found by Vector Addition. The Resultant Moment must also be found using Vector Addition.