1. Chapter 11 Outline Equilibrium and Elasticity
Conditions for equilibrium
Center of gravity and stability
Stress and strain
Elastic moduli
Elasticity and plasticity

2. Equilibrium
If a body is not accelerating, we say that it is in equilibrium.
If it is at rest, we have static equilibrium.
Conditions for equilibrium:
Σ 𝑭 =0
Σ 𝝉 =0 about any point

3. Equilibrium Example

4. Center of Gravity and Stability
If the acceleration due to gravity is constant (good approximation on Earth) the center of gravity is the same as the center of mass.
𝒓 cm = 𝑚 1 𝒓 1 + 𝑚 2 𝒓 𝟐 + 𝑚 3 𝒓 3 +… 𝑚 1 + 𝑚 2 + 𝑚 3 +… = 𝑖 𝑚 𝑖 𝒓 𝑖 𝑖 𝑚 𝑖
If the center of gravity is outside the area bounded by the supports the object will tip.

5. Stress and Strain Real objects are not perfectly rigid.
Forces can deform the objects in a number of ways (stretching, squeezing, twisting…)
Stress characterizes the strength of the forces causing the deformation.
Measured in force per area, or pascals (N/ m 2 =Pa)
Strain describes the resulting deformation.
Generally a ratio of lengths, areas, or volumes (dimensionless)

6. Elastic Modulus
For relatively small stress and strain, the two are often directly proportional. (Hooke’s law)
This proportionality constant is the elastic modulus.
Stress Strain =Elastic modulus
This relationship is generally only valid over a limited range.

7. Tension
When forces pull on an object in opposite directions, it is in tension.
Tensile stress= 𝐹 ⊥ 𝐴
Tensile strain= ∆𝑙 𝑙 0
Young’s modulus, 𝑌, describes tension.
Tensile stress Tensile strain = Young ′ s modulus
𝑌= 𝐹 ⊥ 𝐴 𝑙 0 ∆𝑙

8. Tensile Stress Example

9. Compression
When forces push on an object in opposite directions, it is in compression.
Compressive stress= 𝐹 ⊥ 𝐴
Compressive strain= ∆𝑙 𝑙 0
Young’s modulus, 𝑌, also describes compression.
For many materials, the value is the same.

10. 𝐵=− ∆𝑝 ∆𝑉/ 𝑉 0 Bulk Stress and Strain
When the pressure is uniform, the we use bulk, or volume, stress and strain.
Bulk stress, 𝑝= 𝐹 ⊥ 𝐴
Bulk strain= ∆𝑉 𝑉 0
The Bulk modulus, 𝐵, describes bulk stress and strain.
𝐵=− ∆𝑝 ∆𝑉/ 𝑉 0

11. Shear Stress and Strain
When the forces act tangentially to opposite sides, we have shear.
Shear stress= 𝐹 ∥ 𝐴
Shear strain= 𝑥 ℎ
The shear modulus, 𝑆, describes shear stress and strain.
𝑆= 𝐹 ∥ 𝐴 h x

12. Elastic Moduli

13. Shear Stress Example

14. Elasticity and Plasticity
If the stress is too great, Hooke’s law no longer describes the situation well.
Past the proportional limit, the behavior is no longer elastic.
Plastic deformation follows.
Finally, the material fractures.

15. Chapter 11 Outline Equilibrium and Elasticity
Conditions for equilibrium
Σ 𝑭 =0
Σ 𝝉 =0 about any point
Center of gravity and stability
Stress and strain
Stress: Force/area
Strain: Deformation
Elastic moduli
Elasticity and plasticity