Equilibrium and Elasticity: Static Equilibrium, Stability, and Elastic Properties of Materials

Equilibrium and Elasticity

Introduction

This chapter explores the principles of static equilibrium and elasticity, focusing on the conditions required for objects to remain at rest, the role of torque, and how materials respond to applied forces. Applications include engineering, biomechanics, and everyday stability.

Torque and Static Equilibrium

Conditions for Static Equilibrium

• Static equilibrium occurs when an object is at rest and remains at rest under the action of applied forces and torques.

• For a particle-like object, static equilibrium is achieved when the net force is zero.

• For extended objects, both the net force and the net torque must be zero.

• Net force zero: $\sum \vec{F} = 0$

• Net torque zero: $\sum \tau = 0$

If either condition is not met, the object will accelerate linearly or rotationally.

Choosing the Pivot Point

• Torque calculations can be performed about any point, but choosing a point where unknown forces act can simplify the analysis.

• The natural axis of rotation is often a convenient pivot, especially where forces are not well specified.

Stability and Balance

Base of Support and Center of Gravity

• An object's base of support is the area over which it rests.

• A wider base and a lower center of gravity increase stability.

• As long as the center of gravity remains over the base, the object is stable; if it moves outside, the object becomes unstable and may topple.

• The critical angle for toppling depends on the track width $t$ and the height $h$ of the center of gravity:

$\tan \theta_c = \frac{t}{2h}$

Vehicles with a higher $\theta_c$ are less likely to roll over.

Stability of the Human Body

• Humans maintain stability by adjusting their limbs to keep the center of gravity over the base of support.

• Standing on tiptoes requires leaning forward to maintain balance.

Elasticity and Hooke’s Law

Elastic Deformation

• All solid objects deform under applied forces, though some deformations are very small.

• Materials that return to their original shape after deformation are called elastic.

• The restoring force acts to return the object to equilibrium.

Hooke’s Law

• The force exerted by a spring is proportional to its displacement from equilibrium:

$F_{sp} = -k \Delta x$

• $k$ is the spring constant (N/m), a measure of stiffness.

• The negative sign indicates the force is in the opposite direction of displacement.

Stretching and Compressing Materials

Atomic Model of Solids

• Solids can be modeled as atoms connected by spring-like bonds.

• Applying a force stretches these bonds, causing the material to deform.

Young’s Modulus and Material Properties

• The spring constant for a rod depends on its cross-sectional area $A$, length $L$, and the material’s Young’s modulus $Y$:

$k = \frac{YA}{L}$

• Young’s modulus is a measure of a material’s stiffness.

Stress and Strain

• Stress is the force per unit area: $\text{Stress} = \frac{F}{A}$

• Strain is the fractional change in length: $\text{Strain} = \frac{\Delta L}{L}$

• Hooke’s law for materials: $\frac{F}{A} = Y \left( \frac{\Delta L}{L} \right)$

Elastic Limit and Permanent Deformation

• Within the elastic region, deformation is reversible and follows Hooke’s law.

• Beyond the elastic limit, the material is permanently deformed and may eventually break.

Biological Materials

Structure of Bone

• Bones are composed of dense, rigid compact bone and flexible, porous spongy bone.

• This structure provides both strength and flexibility.

Tensile Strength of Biological Materials

Material	Young’s Modulus (N/m2)
Spongy bone	$5 \times 10^6$
Compact bone	$1 \times 10^8$
Tendon	$1 \times 10^8$
Spider silk	$1 \times 10^9$

Example: Forces in the Ankle Joint

Biomechanical Analysis

• When standing on tiptoe, the foot pivots about the ankle, with forces applied by the floor, the Achilles tendon, and the ankle joint.

• Static equilibrium conditions are used to solve for unknown forces.

• Given: Woman’s mass $m = 61$ kg, $F_{floor} = 600$ N (rounded), distances to pivot are 0.15 m (floor) and 0.05 m (tendon).

• Torque equilibrium about the ankle:

$0.15 \text{ m} \times 600 \text{ N} = 0.05 \text{ m} \times T$

$T = 1800 \text{ N}$

• This is three times the woman’s weight.

• Vertical force equilibrium:

$F_{floor} + T = F_{joint}$ $600 \text{ N} + 1800 \text{ N} = 2400 \text{ N}$

• The force in the ankle joint is four times her weight.

Conclusion

• Static equilibrium and elasticity principles are essential for understanding the stability of structures and the mechanical properties of materials, including biological tissues.

• These concepts are widely applicable in engineering, biomechanics, and safety analysis.