Chapter 11: Equilibrium and Elasticity – Structured Study Notes

Equilibrium and Elasticity

Introduction

Equilibrium and elasticity are fundamental concepts in physics, especially in the study of statics and material science. Structures such as Roman aqueducts utilize principles of equilibrium to sustain weight without accelerating, while real materials exhibit elastic behavior and deform under force.

• Equilibrium: The state in which a body does not accelerate; all forces and torques are balanced.

• Elasticity: The property of materials to return to their original shape after deformation.

Conditions for Equilibrium

Static Equilibrium

For a body or structure to be in static equilibrium, two essential conditions must be satisfied:

• First Condition (Force Equilibrium): The vector sum of all external forces acting on the body must be zero.

$\sum \vec{F} = 0$

• Second Condition (Torque Equilibrium): The sum of all external torques about any point must be zero.

$\sum \tau = 0$

Examples

• Example 1: Both conditions satisfied; body is at rest and does not rotate.

• Example 2: Net force is zero, but net torque is not; body will start rotating.

• Example 3: Net torque is zero, but net force is not; body will start moving as a whole.

Center of Gravity

Definition and Properties

The center of gravity is the point at which the entire weight of a body can be considered to act. For most practical purposes, if gravity does not vary significantly with altitude, the center of gravity coincides with the center of mass.

• Stability: A body is stable if its center of gravity is above or within the area of support.

• Equilibrium: If the center of gravity lies outside the area of support, the body will tip over.

Example: The Petronas Towers have a center of gravity only about 2 cm below the center of mass, despite their height.

Problem-Solving Strategy for Static Equilibrium

Steps for Analysis

• Sketch the physical situation and identify the body in equilibrium.

• Draw a free-body diagram showing all forces and their points of application.

• Choose coordinate axes and specify their direction, including a positive direction for rotation.

• Choose a reference point for computing torques.

• Write equations for equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum \tau = 0$.

• Use multiple reference points if necessary to solve for unknowns.

• Check results by evaluating torque equilibrium about a different point.

Strain, Stress, and Elastic Moduli

Types of Stress

• Tensile Stress: Stretching forces (e.g., guitar strings).

• Bulk Stress: Pressure from all sides (e.g., diver underwater).

• Shear Stress: Forces causing deformation parallel to a surface (e.g., scissors cutting ribbon).

Stress and Strain

• Stress: Force per unit area applied to a material.

• Strain: Fractional change in size or shape due to stress.

• Elastic Deformation: Material returns to its original shape after the force is removed.

Tensile Stress and Strain

• Tensile Stress: $\frac{F}{A}$

• Tensile Strain: $\frac{\Delta l}{l_0}$

Young's Modulus

Young's modulus quantifies the relationship between tensile stress and strain for small deformations.

• Formula: $Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta l / l_0}$

• Example: Human anterior tibial tendon: $Y = 1.2 \times 10^9$ Pa

Compressive Stress and Strain

• Compressive Stress: $\frac{F}{A}$

• Compressive Strain: $\frac{\Delta l}{l_0}$ (where $\Delta l$ is contraction)

Compression and Tension in Structures

• Beams can experience both compression (top) and tension (bottom) simultaneously.

Bulk Stress and Strain

• Pressure: $p = \frac{F_{\perp}}{A}$

• Bulk Modulus: $B = -\Delta p / (\Delta V / V_0)$

Example: Anglerfish withstands high bulk stress due to lack of internal air spaces.

Shear Stress and Strain

• Shear Stress: $\frac{F}{A}$

• Shear Strain: $\frac{x}{h}$

• Shear Modulus: $S = (F/A) / (x/h)$

Elastic Moduli of Materials

Elastic moduli quantify the stiffness of materials under different types of stress. The following table summarizes typical values:

Material	Young's Modulus, Y (Pa)	Bulk Modulus, B (Pa)	Shear Modulus, S (Pa)
Aluminum	7.0 × 1010	7.5 × 1010	2.5 × 1010
Brass	9.0 × 1010	6.0 × 1010	3.5 × 1010
Copper	11 × 1010	14 × 1010	4.4 × 1010
Iron	21 × 1010	16 × 1010	7.7 × 1010
Lead	1.6 × 1010	4.1 × 1010	0.6 × 1010
Nickel	20 × 1010	1.8 × 1011	7.8 × 1010
Silicone rubber	0.001 × 1010	0.2 × 1010	0.0002 × 1010
Steel	20 × 1010	16 × 1010	7.5 × 1010
Tendon (typical)	0.12 × 1010	—	—

Compressibility

The reciprocal of the bulk modulus is called compressibility ($k$), which measures how much a material's volume changes under pressure.

$k = \frac{1}{B} = -\frac{\Delta V / V_0}{\Delta p}$

Liquid	Compressibility, k (Pa-1)	Compressibility, k (atm-1)
Carbon disulfide	93 × 10-11	94 × 10-6
Ethyl alcohol	110 × 10-11	111 × 10-6
Glycerine	21 × 10-11	21 × 10-6
Mercury	3.7 × 10-11	3.8 × 10-6
Water	45.8 × 10-11	46.4 × 10-6

Elasticity and Plasticity

Hooke's Law and Elastic Limit

Hooke's law states that stress and strain are proportional for small deformations, but this proportionality only holds within the elastic limit. Beyond this, materials may exhibit plastic behavior and permanent deformation.

• Elastic Hysteresis: The difference in the stress-strain curve for increasing and decreasing stress, as seen in materials like vulcanized rubber.

• Plastic Deformation: Permanent change in shape after the elastic limit is exceeded.

• Fracture Point: The point at which the material breaks.

Approximate Breaking Stresses

The breaking stress is the stress required to cause actual fracture of a material. Typical values for several materials are shown below:

Material	Breaking Stress (Pa or N/m2)
Aluminum	2.2 × 108
Brass	4.7 × 108
Glass	10 × 108
Iron	3.0 × 108
Steel	5–20 × 108
Tendon (typical)	1 × 108

Additional info: These notes expand on the original slides by providing definitions, formulas, and context for each concept, ensuring a comprehensive and self-contained study guide for college-level physics students.