1. Chapter 12: Equilibrium and Elasticity
Conditions Under Which a Rigid Object is in Equilibrium
Problem-Solving Strategy
Elasticity

2. Equilibrium: An object at equilibrium is either ...
at rest and staying at rest (i.e., static equilibrium) , or
in motion and continuing in motion with the constant velocity and constant angular momentum.
For the object in equilibrium,
the linear momentum ( ) of its center of mass is constant.
the angular momentum ( ) about its center of mass, or any other point, is constant.

3. Conditions of Equilibrium:
Net force: Net torque:
Conditions of equilibrium:
Another requirements for static equilibrium:

4. The center or gravity:
The gravitational force on a body effectively acts at a single point, called the center of gravity (cog) of the body.
the center of mass of an object depends on its shape and its density
the center of gravity of an object depends on its shape, density, and the external gravitational field.
Does the center of gravity of the body always coincide with the center of mass (com)?
Yes, if the body is in a uniform gravitational field.

5. How is the center of gravity of an object determined?
The center of gravity (cog) of a regularly shaped body of uniform composition lies at its geometric center.
The (cog) of the body can be located by suspending it from several different points. The cog is always on the line-of-action of the force supporting the object.
cog

6. Problem-Solving Strategy:
Define the system to be analyzed
Identify the forces acting on the system
Draw a free-body diagram of the system and show all the forces acting on the system, labeling them and making sure that their points of application and lines of action are correctly shown.
Write down two equilibrium requirements in components and solve these for the unknowns

7. Sample Problem 12-1: Define the system to be analyzed: beam & block
Identify the forces acting on the system:
the gravitational forces: mg & Mg,
the forces from the left and the right scales: Fl & Fr
Draw a force diagram
Write down the equilibrium requirements in components and solve these for the unknowns

8. Elasticity Some concepts: Rigid Body: Deformable Body:
elastic body: rubber, steel, rock…
plastic body: lead, moist clay, putty…
Stress: Deforming force per unit area (N/m2)
Strain: unit deformation

9. Young’s Modulus: Elasticity in Length
The Young’s modulus, E, can be calculated by dividing the stress by the strain, i.e.
   
where (in SI units)
E is measured in newtons per square metre (N/m²).
F is the force, measured in newtons (N)
A is the cross-sectional area through which the force is applied, measured in square metres (m2)
L is the extension, measured in metres (m)
L is the natural length, measured in metres (m)

10. Some elastic properties of selected material of engineering interest
Table 12-1:
Some elastic properties of selected material of engineering interest
Material
Density 
(kg/m3)
Young’s Modulus E
(109N/m2)
Ultimate Strength Su
(106N/m2)
Yield Strength Sy
Steel
7860
200
400
250
Aluminum
2710 70
110 90
Glass
2190 65 50 
Concrete
2320 30 40
Wood
525 13
Bone
1900 9
170
Polystyrene
1050 3 48

11. Shear Modulus: Elasticity in Shape
The shear modulus, G, can be calculated by dividing the shear stress by the strain, i.e.
   
where (in SI units)
G is measured in newtons per square metre (N/m²)
F is the force, measured in newtons (N)
A is the cross-sectional area through which the force is applied, measured in square metres (m2)
x is the horizontal distance the sheared face moves, measured in metres (m)
L is the height of the object, measured in metres (m)

12. Bulk Modulus: Elasticity in Volume
The bulk modulus, B, can be calculated by dividing the hydraulic stress by the strain, i.e.
   
where (in SI units)
B is measured in newtons per square metre (N/m²)
P is measured in in newtons per square metre (N/m²)
V is the change in volume, measured in metres (m3)
V is the original volume, measured in metres (m3)

13. Young’s modulus Shear modulus Bulk modulus
Under tension and compression
Under shearing
Under hydraulic stress
Strain is

14. Summary: Requirements for Equilibrium:
The cog of an object coincides with the com if the object is in a uniform gravitational field.
Solutions of Problems:
Elastic Moduli:
tension and compression
shearing
hydraulic stress
Define the system to be analyzed
Identify the forces acting on the system
Draw a force diagram
Write down the equilibrium requirements in components and solve these for the unknowns

15. Sample Problem 12-2:
Define the object to be analyzed: firefighter & ladder
Identify the forces acting on the system:
the gravitational forces: mg & Mg,
the force from the wall: Fw
the force from the pavement: Fpx & Fpy
Draw a force diagram
Write down the equilibrium requirements in components and solve these for the unknowns

16. Sample Problem 12-3: Define the object to be analyzed: Beam
Identify the forces acting on the system:
the gravitational force (mg),
the force from the rope (Tr)
the force from the cable (Tc), and
the force from the hinge (Fv and Fh)
Draw a force diagram

17. Write down the equilibrium requirements in components and solve these for the unknowns
Balance of torques:
Balance of forces:

18. Sample Problem 12-6:
Define the system to be analyzed: table plus steel cylinder.
Identify the forces acting on the object:
the gravitational force (Mg),
the forces on legs from the floor (F1= F2= F3 and F4).
Draw a force diagram

19. Write down the equilibrium requirements in components and solve these for the unknowns
Balance of forces:
If table remains level:

20. Chapter 12: Recitation

21. Problem-Solving Tips:
Try to guess the correct direction for each force
The choice of the origin for the torque equation is arbitrary. Choose an origin that will simplify your calculation as much as possible.
You must have as many independent equations as you have unknowns in order to obtain a complete solution.

22. Homework: 12-22
Define the system to be analyzed: the rod plus the uniform square sign
Identify the forces acting on the system:
the gravitational force (mg), the force from the cable (Tv and Th), and the force from the hinge (Fv and Fh)
Draw a force diagram
Write down the equilibrium requirements 4 Tv Fv Fh O Th 2 3 mg

24. Homework: 12-34
Define the system to be analyzed: the beam plus the package
Identify the forces acting on the system:
the gravitational force (mbg & mpg),
the force from the cable (T),
and the force from the hinge (Fv and Fh) o
Tsin
mpg
mbg
Tcos Fv Fh
Draw a force diagram
Write down the equilibrium requirements

26. Homework: 12-35
Define the system to be analyzed: two sides of the ladder are considered separately
Identify the forces acting on the system:
Left side: the gravitational force of the man (mg), the tension force of the tie rod (T), the force of the floor on the left feet (FA), and the force exerted by the right side of the ladder (Fv and Fh)
Right side: the tension force of the tie rod (T), the force of the floor on the right feet (FE), and the force exerted by the right side of the ladder (Fv and Fh) O O
Draw a force diagram  

27. Write down the equilibrium requirements in components and solve these for the unknowns
0.762 
2.44
(a) First we solve for T by eliminating the other unknowns. The first equations of two sides give FA=mg-Fv and FV=FE . Substituting them into the remaining three equations to obtain:

28. The last of these gives
We substitute this expression into the second equation and solve for T. The result is:
(b) We now solve for FA.
(c) We now solve for FE. We have already obtained an expression for FE.

29. Homework: 12-40
, (b). Since the brick rests horizontally on cylinders now and the cylinders had identical length l before the brick was placed on them, then both cylinders have been compressed an equal amount l. Thus,
(c) Computing torques about the center of mass, we found,

30. Homework: 12-43   mg T x y N
Choose x-axis is parallel to the incline (positive uphill)
Balance of forces in x-direction:

31. Basic Pulley Physics:
Fixed Pulley
Movable Pulley 2P

32. Homework: 12-47
Cable 3
T=8F
Cable 2 4F 2F
Cable 1 4F 2F F mg

33. Homework: 12-54 mg
2Fr
2Ff
2fr
2ff
(a) With F=ma=-kmg, the magnitude of the deceleration is:
(b), (c) :
(d), (e) :