1. ECOR 1101 Mechanics I Sections C and F Jack Vandenberg
Lecture 04 – Equilibrium and FBDs
(Chapter 3 – Sections )

2. 2D and 3D Equilibrium of a particle
Objectives
Learn the concept of free-body diagrams
Learn to solve problems involving particles in equilibrium
Learn to use equilibrium equations using cartesian vector coordinates
ECOR1101 –Mechanics I

3. Equilibrium of a Particle
A particle is in equilibrium if:
It remains at rest under action of a system of forces, or,
It continues in its state of motion with constant velocity under action of a system of forces.
For a particle to be in equilibrium the resultant of all forces acting on it must be zero.
Satisfies Newton’s 1’st law of motion
The above equation represents the necessary and sufficient condition for equilibrium of a particle in space.

4. Equilibrium of a Particle
According to Newton’s 2nd law of motion, if F = ma = 0,
the particle is in equilibrium since a = 0 and F = 0
i.e. the particle is under constant velocity or is at rest
The equilibrium equation can be used to solve problems dealing with equilibrium of a particle involving no more than three unknowns
ECOR1101 –Mechanics I

5. Free-body Diagrams (FBD)
To apply the equations of equilibrium to a particle all forces (known and unknown) must be accounted for.
The best way to do this is to isolate the particle from its surroundings to form a “free-body diagram (FBD)”.
Then apply all the forces (known and unknown) acting on the particle
ECOR1101 –Mechanics I

6. Procedure for drawing FBD
Isolate particle from its surrounding
Sketch outline shape of particle with all forces (active and reactive) indicated
Label all forces (known and unknown) with both their magnitudes and directions
If you know that an unknown force is in tension, do you draw it away or towards the particle?
ECOR1101 –Mechanics I

7. Free Body Diagrams (FBDs)

8. FBDs Rigid Bodies Springs Cables (assumptions) Hooke’s law F = k s
F = spring constant x displacement
Cables (assumptions)
Must be in tension
Negligible weight
Do not stretch
Frictionless pulley

9. FBDs
Draw a FBD of the cable AB and of the joint C.

10. Sample Problem
8 m
10 m
12 m
1.2 m
2 m A B C z
A 200-kg cylinder is hung by means of two cables AB and AC, which are attached to the top of a vertical wall. A horizontal force P perpendicular to the wall holds the cylinder in the position shown. Determine the magnitude of P and tension in each cable P
Introduce unit vectors i, j, k along orthogonal axes and resolve forces
P = Pi + 0j + 0k
W = 0i + 0j - mgk =
= 0i + 0j - 2009.81k = 0i + 0j -1962Nk, N
For forces TAB and TAC we need their respective unit vectors z B y x
TAB k A C
TAC P j i w y x
ECOR1101 –Mechanics I

11. rAB = (-1.2m)i - (8m)j + (10m) k
rAC = (-1.2m)i + (10m)j+ (10m)k
Since the cylinder is under equilibrium:
ECOR1101 –Mechanics I

12. Cables, Springs and Pulleys
Cables (or cords), in general, are assumed to have the following properties
Weightless
Supports only tension in the direction of the cable (cannot be pushed)
Cannot stretch (i.e. increase in length under load)
A cable passing over a frictionless pulley has a constant magnitude
ECOR1101–Mechanics I

13. Cables, Pulleys and Springs
Springs, when deformed, exert a force proportional to the amount of deformation.
Springs are often defined by the spring constant or stiffness k
The magnitude of force exerted on a linearly elastic spring with stiffness k is given by:
F = ks
s = l − lo,
lo = unstretched length,
l = stretched length
ECOR1101–Mechanics I

14. Coplanar Force Systems (2D)
If a particle subjected to a system of coplanar forces (x-y plane), then the forces can be resolved and equilibrium equations applied. y F1
F1y
F2x x
F1x
F2y F2
The two equations of equilibrium can be solved for at most two unknowns. Application of the equation must take into account direction of components of the force
ECOR1101–Mechanics I

15. Procedure For Analysis of Coplanar (2D) Force Equilibrium
Establish x-y axes
Draw a free-body diagram
Draw and label all forces (known and unknown) with magnitudes, sense and direction
Choose an arbitrary direction for unknown forces
Resolve forces in x-y axes
Apply equations of equilibrium
Assume a +ve direction for the purpose of writing your equation of equilibrium
Solve for unknown forces
Compare your answers to your original assumption (not to the +ve direction when writing your equations)
Redraw your FBD with all Forces as positive numbers
ECOR1101–Mechanics I

16. Three-Dimensional (3D) Force Systems
Conditions for equilibrium
Resolve forces into respective Cartesian components, i, j, k x y z F1
F1z
F1x
F1y F2
F2y
F2z
F2x
The three equation of equilibrium are algebraic sums of force components and can be used to find at most three unknowns (coordinate direction angles or magnitudes of forces acting on a particle)
ECOR1101–Mechanics I

17. Procedure For Analysis of 3D Force Systems
Establish x-y-z axes
Draw a free-body diagram
Draw and label all forces (known and unknown) with magnitudes, sense and direction
Choose an arbitrary direction for unknown forces
Resolve forces in x-y-z axes
Apply equations of equilibrium
Assume a +ve direction for the purpose of writing your equation of equilibrium
Solve for unknown forces
Compare your answers to your original assumption (not to the +ve direction when writing your equations)
ECOR1101–Mechanics I

18. Sample Problem
The shear leg derrick is used to haul the 200-kg net of fish onto the dock. Determine the compressive force along each of the legs AB and CB and the tension in the winch cable DB. Assume the force in each leg acts along its axis.
ECOR1101–Mechanics I

19. W
FBD x y z
FBA
FBC B A C D 4 2
5.6 4 2
Write the coordinates for points A, B, C, and D, position vectors, unit vectors, Force Vectors A(2m, 0, 0) B(0, 4m, 4m) C(-2m, 0, 0) D(0, -5.6m, 0)
ECOR1101–Mechanics I

20. rBA = 2mi − 4mj − 4mk rBC = − 2mi − 4mj − 4mk rBD = 0mi − 9
rBA = 2mi − 4mj − 4mk rBC = − 2mi − 4mj − 4mk rBD = 0mi − 9.6mj − 4mk W = 0i + 0j – (2009.81)k = 0i + 0j − 1962Nk
ECOR1101–Mechanics I

21. W
FBD x y z
FBA
FBC B A C D
ECOR1101–Mechanics I

22. Problem F3-8
Determine the tension developed in cables AB, AC, and AD.

23. Problem 3-59
Determine the maximum weight of the crate that can be supported from cables AB, AC, and AD so that the tension developed in any one of the cables does not exceed 250 lb.

24. Problem 3-61
If cable AD is tightened by a turnbuckle and develops a tension of 1,300 lb, determine the tension developed in cables AB and AC and the force developed along the antenna tower AE at point A.

25. Problem 3-77
The joint of a space frame is subjected to four member forces. Member OA lies in the x-y plane and member OB lies in the y-z plane. Determine the forces acting in each of the members required for equilibrium at the joint.