Static Equilibrium; Elasticity and Fracture

Introduction

This chapter explores the principles of static equilibrium, elasticity, and fracture in physical systems. These concepts are essential in understanding how objects remain at rest or in uniform motion, how materials deform under force, and the conditions under which they break. Applications span engineering, architecture, biology, and medicine.

9-1 The Conditions for Equilibrium

Definition and Requirements

For an object to be in static equilibrium, it must satisfy two main conditions:

• Translational Equilibrium: The sum of all external forces acting on the object must be zero.

• Rotational Equilibrium: The sum of all external torques about any axis must be zero.

Mathematically, these conditions are expressed as:

Objects in equilibrium do not accelerate linearly or rotationally. This is crucial for structures, machines, and biological systems to maintain stability.

Example: Straightening Teeth

• Two forces act on a tooth: (upward) and (downward).

• If and , the resultant force is zero, and the tooth remains stationary.

• If the forces are not equal and opposite, the tooth will move in the direction of the net force.

Application: Orthodontic treatments use balanced forces to move teeth gradually into desired positions.

9-2 Solving Statics Problems

General Approach

Statics problems involve analyzing forces and torques to ensure equilibrium. The steps are:

1. Draw a free-body diagram showing all forces acting on the object.

2. Choose a convenient axis for calculating torques.

3. Apply the equilibrium conditions (, ).

4. Solve the resulting equations for unknown forces or torques.

Example: Chandelier Cord Tension

• A chandelier is suspended by two cords at different angles.

• Each cord experiences a tension force, which can be resolved into horizontal and vertical components.

• By applying equilibrium conditions, the tension in each cord can be calculated.

Application: This method is used in engineering to design safe support systems for ceilings, bridges, and other structures.

9-3 Elasticity: Stress and Strain

Definitions

• Stress: The force applied per unit area on a material.

• Strain: The relative deformation produced by stress.

• Young's Modulus (Y): A measure of a material's stiffness.

Materials respond to applied forces by deforming. The relationship between stress and strain is linear for small deformations (Hooke's Law).

Example: Stretching a Wire

• A steel wire of length and cross-sectional area is stretched by a force .

• The extension can be calculated using Young's modulus.

Application: Understanding elasticity is vital for designing buildings, bridges, and medical implants.

9-4 Fracture and Failure

Fracture Mechanics

When the stress on a material exceeds a critical value, the material may fracture or break. The ability to withstand stress without breaking is called strength.

• Ultimate Strength: Maximum stress a material can withstand before breaking.

• Fracture Point: The strain at which a material breaks.

Designing structures requires knowledge of material strength to prevent catastrophic failure.

9-5 Spanning a Space: Arches and Domes

Architectural Applications

Arches and domes are structures that efficiently span large spaces by distributing forces. Their shapes allow them to support heavy loads with minimal material.

• Arches: Transfer loads into horizontal thrusts at the base, which must be counteracted by supports.

• Domes: Distribute forces in all directions, providing stability and strength.

Application: Used in bridges, cathedrals, and modern stadiums for both aesthetic and structural purposes.

Summary Table: Key Concepts in Static Equilibrium and Elasticity

Additional info: These notes expand on the brief textbook content by providing definitions, equations, and applications for each concept, ensuring a self-contained study guide suitable for exam preparation.