Comprehending equilibrium conditions is a fundamental aspect of mathematical study. This involves the analysis of how balanced forces can maintain a body in a state of static equilibrium or uniform motion. This concept is pivotal in addressing problems within the realms of physics and engineering, where the interplay of forces is a significant factor.

Introduction to Equilibrium

• Equilibrium in physics: When forces on a body balance out, causing no net force.
• Types: Static (body at rest) and Dynamic (body moving at constant speed).
• Importance: Key for solving physics problems.

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Applying Equilibrium Conditions

• Static Equilibrium: Body at rest, total forces equal zero.
• Dynamic Equilibrium: Body moves at steady speed, forces are balanced.
• Force Summation: All forces add up to zero for equilibrium.
• Directional Balance: Forces cancel out in both horizontal and vertical directions.

Resolving Forces

• Process: Split a force into horizontal and vertical parts to analyze.
• Steps:
  • 1. Identify Forces: Include gravity, tension, normal, and friction.
  • 2. Decompose Forces: Break each force into horizontal and vertical parts.
  • 3. Apply Equilibrium: Horizontal and vertical force sums must be zero.
    

Example Problem

Problem Statement

• A 5 kg particle held by two ropes, Rope A (30° to horizontal) and Rope B (45° to horizontal).
• Find tension in each rope for equilibrium.

Solution Using Static Equilibrium

• Forces:
  • Gravitational Force (Weight): $5 \, \text{kg} \times 9.81 \, \text{m/s}^2$.
  • Tension in Rope A $( T_A )$: Angle 30°.
  • Tension in Rope B $( T_B )$: Angle 45°.
• Equilibrium Conditions:
  • Vertical forces sum = 0.
  • Horizontal forces sum = 0.
• Components of Tension:
  • Rope A: Vertical = $T_A \sin(30°)$, Horizontal = $T_A \cos(30°)$.
  • Rope B: Vertical = $T_B \sin(45°)$, Horizontal = $T_B \cos(45°)$.
• Equilibrium Equations:
  • Vertical: $T_A \sin(30°) + T_B \sin(45°) = 5 \times 9.81$.
  • Horizontal: $T_A \cos(30°) = T_B \cos(45°)$.
• Tensions Found:
  • Rope A $( T_A )$: ~35.91 N.
  • Rope B $( T_B )$: ~43.98 N.