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.

Written by: Dr Rahil Sachak-Patwa

LinkedIn

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.