Chapter 8: Equilibrium and Elasticity – Torque and Static Equilibrium

Equilibrium and Elasticity

Introduction

This chapter extends the analysis of torque and equilibrium from single-force scenarios to systems with multiple forces and torques. Understanding these principles is essential for analyzing the stability and mechanical behavior of objects in static situations.

Torque and Static Equilibrium

Definition of Torque

• Torque (τ) is a measure of the tendency of a force to rotate an object about a pivot point or axis.

• Mathematically, torque is defined as: $\tau = r F \sin\phi$ where:

  • r = distance from the pivot to the point of force application

  • F = magnitude of the applied force

  • \phi = angle between the force vector and the lever arm

• The direction of torque is determined by the right-hand rule.

Static Equilibrium

An object is in static equilibrium if it is at rest and remains at rest under the action of applied forces and torques.

• For a particle (an object that cannot rotate), static equilibrium is achieved when the net force is zero: $\sum \vec{F} = 0$

• For an extended object (which can rotate), both the net force and the net torque must be zero: $\sum F_x = 0$ $\sum F_y = 0$ $\sum \tau = 0$

Conditions for Static Equilibrium

• First Condition: The sum of all external forces acting on the object must be zero (no linear acceleration).

• Second Condition: The sum of all external torques about any axis must be zero (no angular acceleration).

These conditions can be summarized as:

• $\sum F_x = 0$   (No net force in x-direction)

• $\sum F_y = 0$   (No net force in y-direction)

• $\sum \tau = 0$   (No net torque)

Examples and Applications

• Example: A wrench is used to loosen a bolt. The torque applied depends on the force exerted and the distance from the pivot (the bolt).

• Application: Engineers use static equilibrium conditions to design stable structures, such as bridges and buildings, ensuring that all forces and torques balance.

Conceptual Check: Identifying Static Equilibrium

• To determine if an object is in static equilibrium, check if all forces and torques cancel out.

• Example: In a set of diagrams showing boxes with various force vectors, only the box with equal and opposite forces on both sides (no net force, no net torque) is in static equilibrium.

Key Terms

• Torque (τ): The rotational equivalent of force.

• Static Equilibrium: The state in which an object is at rest and the sum of all forces and torques is zero.

• Pivot (or Fulcrum): The point about which rotation occurs.

• Lever Arm (r): The perpendicular distance from the pivot to the line of action of the force.

Summary Table: Conditions for Static Equilibrium

Condition	Mathematical Expression	Physical Meaning
No net force (x-direction)	$\sum F_x = 0$	Object does not accelerate horizontally
No net force (y-direction)	$\sum F_y = 0$	Object does not accelerate vertically
No net torque	$\sum \tau = 0$	Object does not rotate

Additional info:

• In more complex problems, the choice of pivot point can simplify calculations, especially when some forces act directly at the pivot (producing zero torque).

• Static equilibrium analysis is foundational for understanding mechanical stability and is widely used in engineering and biomechanics.