Chapter 12: Static Equilibrium, Elasticity, and Fracture – Study Notes

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Static Equilibrium, Elasticity, and Fracture

12.1 The Conditions for Equilibrium

Static equilibrium occurs when an object experiences no net force and no net torque. This is a foundational concept in statics, the study of objects at rest.

First Condition (Translational Equilibrium): The sum of all forces acting on the object must be zero.
Second Condition (Rotational Equilibrium): The sum of all torques about any axis must be zero.
Center of Gravity: The point at which the entire weight of an object can be considered to act. For symmetrical objects, it is at the geometric center.
Example (Rod): For a horizontal rod of mass and length attached to a wall, the torque due to gravity about the wall is: (clockwise)
System of Particles: The torque due to gravity is zero when the pivot is at the center of gravity.

12.2 Solving Statics Problems

Statics problems require systematic analysis using free-body diagrams and equilibrium equations.

Choose the object and draw a free-body diagram showing all forces and their points of application.
Choose a coordinate system and resolve forces into components.
Write equilibrium equations for the forces.
Choose an axis and write the torque equilibrium equation.
Solve the resulting system of equations.

Force at a hinge: Decompose into vertical () and horizontal () components. The resultant force is .
Example (Lamp and Plank): A lamp of mass hangs from a plank of mass and length , supported by a hinge and a string at angle .
- Equilibrium equations:
- Solutions:

12.3 Applications to Muscles and Joints

Statics principles are applied to biological systems, such as muscles and joints, to determine forces required for equilibrium.

Example: Calculating the force exerted by the biceps muscle when holding a ball with the arm horizontal or at an angle. The biceps attaches to the forearm via a tendon, and the center of gravity of the forearm and hand must be considered.
Application: Used in biomechanics and medical physics to analyze joint and muscle forces.

12.4 Stability and Balance

Stability describes how an object responds to small disturbances from equilibrium.

Stable Equilibrium: Forces tend to return the object to its equilibrium position.
Unstable Equilibrium: Forces move the object further from equilibrium.
Neutral Equilibrium: The object remains in its new position after disturbance.
Example: A ladder leaning against a wall, or a refrigerator tipping over. Stability depends on the position of the center of gravity relative to the pivot point.

12.5 Elasticity: Stress and Strain

Elasticity describes how materials deform under applied forces and return to their original shape when the force is removed.

Hooke's Law: The change in length () is proportional to the applied force ().
Atomic Model: At the atomic level, materials behave like networks of microscopic springs.
Normal Strain (): Fractional change in length due to axial force.
- Positive in tension, negative in compression.
- Dimensionless, often expressed as a percentage.
Normal Stress (): Force per unit area acting normal to the cross-section.
- Units: Pascal (Pa), N/m2, or MPa.
- Positive in tension, negative in compression.
Young's Modulus (): Material property quantifying stiffness.
- Large values indicate stiff materials.
- Given in GPa for most solids.
Spring Constant ():
Stress-Strain Curve: Shows the relationship between force and elongation.
- Proportional Limit: Region where Hooke's law applies.
- Elastic Limit: Maximum strain before permanent deformation.
- Plastic Region: Material deforms permanently.
- Breaking Point: Material fractures.

Table: Elastic Moduli for Different Materials

Material	Young's Modulus, E (N/m2)	Shear Modulus, G (N/m2)	Bulk Modulus, B (N/m2)
Iron, cast	200 × 109	80 × 109	140 × 109
Steel	200 × 109	80 × 109	160 × 109
Brass	100 × 109	35 × 109	60 × 109
Aluminum	70 × 109	25 × 109	70 × 109
Concrete	20 × 109	8 × 109	15 × 109
Brick	15 × 109	6 × 109	10 × 109
Marble	50 × 109	20 × 109	70 × 109
Wood (pine, parallel to grain)	15 × 109	8 × 109	45 × 109
Nylon	3 × 109	1 × 109	2 × 109
Bone (limb)	15 × 109	8 × 109	15 × 109
Water	2 × 109	--	2.2 × 109
Alcohol (ethyl)	1.9 × 109	--	1.9 × 109
Mercury	2.5 × 109	--	2.8 × 109
Gases (Air, H2, He, CO2)	--	--	1.01 × 105

Additional Examples and Applications

Tension in Piano Wire: Given length, diameter, and stretch, use Young's modulus to calculate tension.
Bones under Tensile and Compressive Loading: Stress-strain diagrams illustrate the limits and breaking strengths of biological materials.

Summary of Chapter 12

An object at rest is in equilibrium; statics studies such objects.
Static equilibrium requires zero net force and zero net torque.
Equilibrium can be stable, unstable, or neutral.
Materials experience compression, tension, or shear stress.
Exceeding the elastic limit leads to permanent deformation and fracture.

Practice Questions

Question 1: When a tension stress is applied to a 3 m long post, its strain is 0.001. How does its length change? Answer: Increases by m ( m)
Question 2: Young's modulus for steel is N/m2. If the stress on a post of length 0.5 m is N/m2, what is the change in length? Answer: m
Question 3: A cable is 100 m long and has a cross-sectional area of 1.0 mm2. A 1000-N force is applied to stretch the cable. The elastic modulus for the cable is N/m2. How far does it stretch? Answer: m