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

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

Equilibrium Example

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.

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)

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.

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

Tensile Stress Example

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.

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

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

Elastic Moduli

Shear Stress Example

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.

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