Physics for Life Sciences I – Lecture 14: Static Equilibrium, Hooke’s Law, Stability, and Elastic Materials

Static Equilibrium

Definition and Conditions

Static equilibrium refers to the state of an object when it is at rest and remains at rest because the net force and net torque acting on it are both zero. This is a fundamental concept in mechanics, ensuring that objects do not accelerate or rotate unexpectedly.

• Net Force: The sum of all forces acting on the object must be zero.

• Net Torque: The sum of all torques about any axis must be zero.

Mathematically, these conditions are expressed as:

• $\sum F_x = 0$

• $\sum F_y = 0$

• $\sum \tau = 0$

Because the net torque is zero about any point, the pivot point for calculating torque can be chosen for convenience.

Springs and Hooke’s Law

Spring Force and Elasticity

When a spring is stretched or compressed, it exerts a restoring force that is proportional to the displacement from its equilibrium length. This relationship is described by Hooke’s Law.

• Hooke’s Law: $F_{sp} = -k \Delta x$

• Spring Constant (k): A measure of the stiffness of the spring. Larger values of k indicate stiffer springs.

• Direction: The force exerted by the spring acts in the direction opposite to the displacement.

Example: If a spring is stretched by 0.1 m and has a spring constant of 200 N/m, the restoring force is $F_{sp} = -200 \times 0.1 = -20$ N.

Stability

Stable and Unstable Equilibrium

An object’s stability depends on the position of its center of gravity relative to its base of support. This concept is crucial in engineering and biomechanics.

• Stable: The center of gravity is over the base of support.

• Unstable: The center of gravity is outside the base of support.

• Critical Angle ($\theta_c$): The angle at which the center of gravity is directly above the edge of the base, marking the threshold between stability and instability.

The critical angle is given by:

• $\theta_c = \tan^{-1}\left(\frac{1}{2} \frac{t}{h}\right)$

Greater stability is achieved with a lower center of gravity or a broader base of support.

Elastic Materials and Young’s Modulus

Stress, Strain, and Elastic Response

Elastic materials deform under force but return to their original shape when the force is removed. The relationship between the applied force and the resulting deformation is characterized by Young’s modulus.

• Stress: Force per unit area, $\frac{F}{A}$

• Strain: Fractional change in length, $\frac{\Delta L}{L}$

• Young’s Modulus (Y): A material property that quantifies stiffness, $Y = \frac{\text{Stress}}{\text{Strain}}$

The force required to stretch or compress a rod is:

• $F = \frac{YA}{L} \Delta L$

This equation shows that a rod obeying Hooke’s law acts like a very stiff spring, with the “spring constant” for the rod given by $k = \frac{YA}{L}$.

Example: A steel rod with $Y = 2 \times 10^{11}$ Pa, $A = 1 \times 10^{-4}$ m$^2$, $L = 2$ m, and $\Delta L = 0.001$ m experiences a restoring force of $F = \frac{2 \times 10^{11} \times 1 \times 10^{-4}}{2} \times 0.001 = 10^4$ N.

Property	Definition	Formula
Stress	Force per unit area	$\frac{F}{A}$
Strain	Fractional change in length	$\frac{\Delta L}{L}$
Young’s Modulus	Stiffness of material	$Y = \frac{\text{Stress}}{\text{Strain}}$
Spring Constant (rod)	Equivalent stiffness for rod	$k = \frac{YA}{L}$

Additional info: These topics form the foundation for understanding mechanical equilibrium, elasticity, and stability in physical systems, which are essential for further study in physics and engineering.