Equilibrium and Elasticity Notes

Equilibrium and Elasticity

12-1 Equilibrium

• Equilibrium: Constant linear momentum ($P$) and angular momentum ($\vec{L}$).

• Static Equilibrium: Object is at rest (both translational and rotational).

• Stable Static Equilibrium: Returns to equilibrium after displacement.

• Unstable Static Equilibrium: Even slight force ends equilibrium.

• Requirements of Equilibrium:

  • $\vec{F}_{net} = 0$ (balance of forces)

  • $\vec{\tau}_{net} = 0$ (balance of torques)

• In xy plane:

  • $F_{net,x} = 0$

  • $F_{net,y} = 0$

  • $\tau_{net,z} = 0$

• Requirement for Static Equilibrium: Linear momentum ($P$) must be zero.

• Center of Gravity (cog): Point where gravitational force effectively acts. If $\vec{g}$ is uniform, cog coincides with center of mass (com).

12-2 Some Examples of Static Equilibrium

• Apply force and torque conditions for static equilibrium.

• Choose origin wisely to simplify calculations by eliminating unknown forces from torque equation.

12-3 Elasticity

• Indeterminate Structures: More unknowns than equilibrium equations.

• Stress-Strain Relation: $stress = modulus \times strain$

• Tension/Compression:

  • $\frac{F}{A} = E \frac{\Delta L}{L}$, where $E$ is Young's modulus.

• Shearing:

  • $\frac{F}{A} = G \frac{\Delta x}{L}$, where $G$ is shear modulus.

• Hydraulic Stress:

  • $p = B \frac{\Delta V}{V}$, where $B$ is bulk modulus and $p$ is pressure.

• Yield Strength: Stress beyond which permanent deformation occurs.

• Ultimate Strength: Stress at which rupture occurs.

Equilibrium, in physics, is a state in which opposing forces or influences are balanced. In the context of mechanics, it specifically refers to a condition where an object's linear momentum ($P$) and angular momentum ($\vec{L}$) remain constant. This means that the object is neither accelerating linearly nor rotating with changing angular velocity. There are two types of equilibrium: static and dynamic. Static equilibrium occurs when the object is at rest, with no translational or rotational motion. Dynamic equilibrium, on the other hand, involves an object moving with constant velocity and