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CIE A-Level Maths Study Notes

3.1.3 Equilibrium Conditions

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.

Image courtesy of byju's

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): 5kg×9.81m/s25 \, \text{kg} \times 9.81 \, \text{m/s}^2 .
    • Tension in Rope A (TA)( T_A ): Angle 30°.
    • Tension in Rope B (TB)( T_B ): Angle 45°.
  • Equilibrium Conditions:
    • Vertical forces sum = 0.
    • Horizontal forces sum = 0.
  • Components of Tension:
    • Rope A: Vertical = TAsin(30°)T_A \sin(30°), Horizontal = TAcos(30°)T_A \cos(30°).
    • Rope B: Vertical = TBsin(45°)T_B \sin(45°), Horizontal = TBcos(45°)T_B \cos(45°).
  • Equilibrium Equations:
    • Vertical: TAsin(30°)+TBsin(45°)=5×9.81T_A \sin(30°) + T_B \sin(45°) = 5 \times 9.81.
    • Horizontal: TAcos(30°)=TBcos(45°)T_A \cos(30°) = T_B \cos(45°).
  • Tensions Found:
    • Rope A (TA)( T_A ): ~35.91 N.
    • Rope B (TB)( T_B ): ~43.98 N.
Force Diagram for Equilibrium System
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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