Equilibrium

1. Equilibrium • Equilibrium refers to a condition in which an object is at rest originally at rest (static equilibrium) or has a constant velocity if originaly in motion. • Equilibrium requires; • Balance of forces (prevent the body from translating with accelerated motion.) • Balance of moments (prevent the body from rotating.)

2. Equilibrium To maintain a state of equilibrium, it is necessary to satisfy Newton’s first law of motion. moves with constant velocity or remains at rest

3. Equations of Equilibrium A body will be in equilibrium provided that the sum of all the external forces acting on the body is equal to zero and the sum of the moments of the external forces about a point is equal to zero. Necessary and Sufficient Conditions

4. Free Body Diagrams The best way to specify for all known and unknown external forces acting on a body is to draw the body’s free-body diagram. Free body diagram is a sketch of the outlined shape of the body, which represents it as being isolated or free from its surroundings. On this sketch it is necessary to show all the forces and couple moments that the surroundings exert on the body.

5. Support Reactions If a support prevents translation in a given direction, then a force is developed on the contacting member in that direction. Likewise, if rotation is prevented a couple moment is exerted on the member.

6. Supports for Bodies Subjected to Two Dimensional Force Systems One unknown. The reaction is a tension force, which acts away from the member in direction of the cable. One unknown. The reaction is a force which acts along the axis of the link.

7. Supports for Bodies Subjected to Two Dimensional Force Systems One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact. One unknown. The reaction is a force which acts perpendicular to the slot.

8. Supports for Bodies Subjected to Two Dimensional Force Systems One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact. One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact.

9. Supports for Bodies Subjected to Two Dimensional Force Systems One unknown. The reaction is a force which acts perpendicular to the rod Two unknowns. The reactions are two components of force, or the magnitude and direction ø of the resultant force. Note that øand θ are not necessarily equal.

10. Supports for Bodies Subjected to Two Dimensional Force Systems Two unknowns. The reactions are the couple moment and the force which acts perpendicular to the rod Three unknowns. The reactions are the couple moment and the two force components or the couple moment and the magnitude and direction Φ of the resultant force

11. Free Body Diagrams The forces that are internal to the body are not represented on the free-body diagram. These forces always occur in equal but opposite collinear pairs, and therefore their net effect on the body is zero.

12. Free Body Diagrams Internal forces act between particles which are located within a specified system which is contained within the boundary of the free body diagram. Particles or bodies outside this boundary exert external forces on the system, these alone must be shown on the free body diagram.

13. Drawing a Free Body Diagram • Step 1: Imagine the body to be isolated from its constraints and connections and draw its outlined shape. • Step 2: Identify all the external forces and couple moments. 1. Applied loadings. 2. Reactions occuring at supports or at points of contact at other bodies. 3.The weight of the body.

14. Drawing a Free Body Diagram • Step 3: Indicate the dimensions of the body necessary for computing the moments of forces.

15. Example Two smooth tubes A and B, each having a mass of 2 kg, rest between the inclined planes shown in figure. Drwa the Free Body Diagrams for tube A, tube B and tubes A and B together.

16. Example Draw free body diagrams of the objects.

17. Example Determine the reaction forces at A and B.

18. Example Determine the reaction forces at fixed support.

19. Example The 100kg uniform beam AB shown in figure is supported at A by a pin at B and C by a continuous cable, which wraps around a frictionless pulley located at D. If a maximum tension force of 800 N can be developed in the cable before it breaks, determine the greatest distance d. What is the reaction at A?

20. Supports for Bodies Subjected to Three Dimensional Force Systems One unknown. The reaction is a force which acts away from the member in the direction of the cable One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact.

21. Supports for Bodies Subjected to Three Dimensional Force Systems One unknown. The reaction is a force which acts perpendicular to the surface at the point of contact. Three unknowns. The reactions are three rectengular force components.

22. Supports for Bodies Subjected to Three Dimensional Force Systems Four unknowns. The reactions are two force and two couple moment components which act perpendicular to the shaft. Five unknowns. The reactions are two force and three couple moment components.

23. Supports for Bodies Subjected to Three Dimensional Force Systems Five unknowns. The reactions are three force and two couple moment components. Five unknowns. The reactions are three force and two couple moment components.

24. Supports for Bodies Subjected to Three Dimensional Force Systems Five unknowns. The reactions are three force and two couple moment components. Six unknowns. The reactions are three force and three couple moment components.

25. Equations of Equilibrium Vector Equations of Equilibrium Scalar Equations of Equilibrium

26. Example Determine the magnitude of the vertical force P that must be applied to the handle to maintain equilibrium. Also calculate the reactions at bearings.

27. Example FREE BODY DIAGRAM