Chapter 8: Equilibrium and Elasticity – Study Notes

Equilibrium and Elasticity

This chapter explores the principles of static equilibrium, stability, elasticity, and the application of these concepts to both materials and biological systems. It covers the conditions required for equilibrium, the behavior of springs and elastic materials, and the forces and torques present in the human body.

Torque and Static Equilibrium

Static equilibrium occurs when an object is at rest and remains at rest because the net force and net torque acting on it are both zero.

• Static Equilibrium: A state where an object has no net force and no net torque acting on it.

• Conditions for Static Equilibrium:

  • The sum of all external forces must be zero: $\sum F_x = 0$ $\sum F_y = 0$

  • The sum of all external torques must be zero: $\sum \tau = 0$

• Application: These conditions apply to both particles and extended objects. For extended objects, the pivot point for torque calculation can be chosen for convenience.

• Example: A block with equal and opposite forces applied at equal distances from the center is in static equilibrium.

Stability and Balance

Stability refers to an object's ability to return to its original position after being disturbed. Balance is achieved when the center of gravity is over the base of support.

• Stable Object: Center of gravity is above the base of support.

• Unstable Object: Center of gravity moves outside the base of support, causing the object to tip.

• Critical Angle ($\theta_c$): The maximum angle at which an object can be tilted before it becomes unstable.

• Example: A man standing on a ladder is stable as long as the combined center of gravity of the man and ladder remains above the ladder's base. If he leans too far, the center of gravity shifts outside the base, and the ladder tips.

Springs and Hooke's Law

Springs and other elastic materials exert a restoring force when deformed. Hooke's Law describes the relationship between the force applied to a spring and its displacement from equilibrium.

• Restoring Force: A force that acts to return a system to its equilibrium position.

• Elastic Systems: Systems that return to their original shape after deformation.

• Hooke's Law: $F_{spring} = -k \Delta x$ Where $k$ is the spring constant (stiffness), and $\Delta x$ is the displacement from equilibrium.

• Spring Constant ($k$): A measure of the stiffness of a spring. Large $k$ means a stiff spring; small $k$ means a flexible spring.

• Example Problem: If a 20-cm spring stretches by 2 cm under a 100 N force, $k = \frac{100\,N}{0.02\,m} = 5000\,N/m$.

Stretching and Compressing Materials

Solid materials can be modeled as atoms connected by spring-like bonds. When a force is applied, these bonds stretch or compress, changing the material's dimensions.

• Elastic Materials: Return to their original shape after the force is removed (e.g., steel).

• Rigid Materials: Experience only small deformations under normal forces (e.g., steel rods).

• Pliant Materials: Can be stretched easily and show large deformations with small forces (e.g., rubber bands).

• Stress ($\sigma$): Force per unit area: $\sigma = \frac{F}{A}$

• Strain ($\varepsilon$): Relative change in length: $\varepsilon = \frac{\Delta L}{L}$

• Young's Modulus ($Y$): A measure of a material's stiffness: $Y = \frac{\sigma}{\varepsilon}$

• Restoring Force Equation: $F = Y A \frac{\Delta L}{L}$

• Tensile Stress: Stress due to stretching.

• Example: Stretching a steel rod by 1 mm requires a force of 16,000 N.

Young's Modulus Table

Material	Young's modulus
Cast iron	20
Steel	20
Silicon	13
Copper	11
Aluminum	7
Glass	7
Concrete	3
Wood (Douglas Fir)	1
Tooth enamel	6
Compact bone	1.6
Spongy bone	0.02–0.3
Spider silk	0.2
Tendon	0.15
Cartilage	0.0001
Blood vessel (aorta)	0.00005

Beyond the Elastic Limit

Materials have an elastic limit, beyond which they are permanently deformed. The stress required to break a material is called its tensile strength.

• Elastic Region: Material returns to original shape after force is removed.

• Elastic Limit: Maximum stress before permanent deformation occurs.

• Breaking Point: Stress at which the material fails.

• Tensile Strength: $\text{Tensile Strength} = \frac{F_{max}}{A}$

• Example: If a rod is stretched beyond its elastic limit, it will not return to its original length.

Forces and Torques in the Body

Muscles and tendons generate forces and torques to maintain equilibrium in the body. These forces can be much larger than the weight they support due to mechanical advantage and lever arms.

• Biomechanics: The study of forces and torques in biological systems.

• Example: When standing on tiptoe, the tension in the Achilles tendon is several times greater than the body weight due to the short lever arm of the tendon compared to the longer lever arm of the foot.

• Calculation Example: For a 61 kg woman standing on one foot, the tension in the Achilles tendon can be calculated using torque equilibrium equations.

Summary of Key Principles

• Static Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum \tau = 0$

• Stability: Center of gravity must be over the base of support for stability.

• Hooke's Law: $F_{spring} = -k \Delta x$

• Elastic Materials: Obey Hooke's law up to the elastic limit.

• Young's Modulus: $Y = \frac{\sigma}{\varepsilon}$, quantifies material stiffness.

• Biomechanics: Forces and torques in the body can be analyzed using the same principles as in mechanical systems.