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Objectives

Objectives. To introduce the concept of the free-body diagram for a particle. To show how to solve particle equilibrium problems using the equations of equilibrium. Definitions.

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Objectives

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  1. Objectives • To introduce the concept of the free-body diagram for a particle. • To show how to solve particle equilibrium problems using the equations of equilibrium.

  2. Definitions • A particle is in equilibrium if it is at rest if originally at rest or has a constant velocity if originally in motion. • Static equilibrium denotes a body at rest. • Newton’s first law is that a body at rest is not subjected to any unbalanced forces.

  3. Static Equilibrium

  4. Static Equilibrium

  5. The Free-Body Diagram • To apply equilibrium equations we must account for all known and unknown forces acting on the particle. • The best way to do this is to draw a free-body diagram of the particle. 3. The free-body diagram (FBD) of a body is a sketch of the body showing all forces that act on it. The term free implies that all supports have been removed and replaced by the forces (reactions) that they exert on the body.

  6. Springs

  7. Cables and Pulleys

  8. Cables and Pulleys Cables are assumed to have negligible weight and they cannot stretch. They can only support tension or pulling (you can’t push on a rope). Pulleys are assumed to be frictionless. A continuous cable passing over a frictionless pulley must have tension force of a constant magnitude. The tension force is always directed in the direction of the cable.

  9. Drawing Free-Body Diagrams • Draw Outlined Shape - Imagine the particle isolated or cut “free” from its surroundings • Show All Forces - Include “active forces” and “reactive forces” • Identify Each Force - Known forces labeled with proper magnitude and direction. Letters used for unknown quantities.

  10. Force Types • Active Forces - tend to set the particle in motion. • Reactive Forces - result from constraints or supports and tend to prevent motion.

  11. Example The sphere has a mass of 6 kg and is supported as shown. Draw a free-body diagram of the sphere, cord CE, and the knot at C

  12. Sphere There are two forces acting on the sphere. These are its weight and the force of cord CE. The weight is: W = 6 kg (9.81 m/s2)=58.9 N.

  13. Sphere FCE Free-Body Diagram 58.9 N

  14. Cord CE There are two forces acting on the cord. These are the force of the sphere, and the force of the knot. A cord is a tension only member. Newton’s third law applies.

  15. FEC C E FCE Cord CE Free-Body Diagram

  16. Knot at C There are three forces acting on the knot at C. These are the force of the cord CBA, and the force of the cord CE, and the force of the spring CD.

  17. Knot at C FCBA 60o FCD Free-Body Diagram C FCE

  18. Example

  19. Example Not a Free Body Diagram!

  20. FBD

  21. Example

  22. FBE of E

  23. FBD of C

  24. Example

  25. FBD of A

  26. Coplanar Force System • A two dimensional system. • Assumed to lie in the x-y plane. • Use i and j unit vectors.

  27. 2D Equilibrium Equations Scalar equations of equilibrium require that the algebraic sum of the x and y components of all the forces acting on a particle be equal to zero.

  28. 2D Equilibrium Equations Two equations means only two unknowns can be solved for from a single FBD.

  29. Assume a sense for an unknown force. If the equations yield a negative value for the magnitude then the sense is opposite of what was assumed. F + 10 N = 0 F = -10 N F acts to the left (opposite of direction shown).

  30. Procedure for Analysis • Free-Body Diagram • Establish the x, y axes in any suitable orientation. • Label all known and unknown force magnitudes and directions on the FBD. • The sense of an unknown force may be assumed.

  31. Procedure for Analysis • Equations of Equilibrium • Apply equations of equilibrium. • Components of force are positive if directed along a positive axis and negative if directed along a negative axis. • If solution yields a negative result the force is in the opposite sense of that shown on the FBD.

  32. Example Determine the tension in cables AB and AD for equilibrium of the 250 kg engine block.

  33. To solve this problem apply equilibrium equation at point A. The weight of the object is W = 250 kg (9.81 m/s2) = 2.452 N. This weight is supported by cable AC so TAC = 2.452 N. y TAB A 30o TAD x TAC = 2.452 kN Free-Body Diagram

  34. Equilibrium Equations

  35. Reporting our answers to three significant figures: TAB = 4.90 kN TAD = 4.25 kN

  36. Example If the sack has a weight of 20 lb, determine the weight of the sack at B and the force in each cord needed to hold the system in the equilibrium position shown.

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