Equilibrium Equations

Introduction

An equilibrium equation is the sum of forces in a direction or the sum of moments about a point. Static Mechanics focuses on analyzing systems that do not accelerate. In Engineering, this means that the sum of forces in all directions is zero ($\Sigma\overrightarrow{F} = m\overrightarrow{a} = 0$) and the sum of moments is equal to zero ($\Sigma\overrightarrow{M} = \overrightarrow{F} \times \overrightarrow{r} = 0$).

Notation

$\rightarrow \Sigma F_{x} = 0$ $\uparrow \Sigma F_{y} = 0$ $\$ $⟲ \ \Sigma M = 0$

Steps to Solve

1. Draw a Free Body Diagram (FBD)

2. Create equilibrium equations

3. Solve reactionary forces

Example Problem:

Problem F3-1: The crate has a weight of 550 lb. Determine the force in each supporting cable.

Solution:

1. Draw a FBD around point A

1. Solve Equilibrium equations

   1. $\rightarrow \Sigma F_{x} = 0 = \frac{3}{5}T_{AC} - T_{AB}cos30$

   2. $\uparrow \Sigma F_{y} = 0 = \frac{4}{5}T_{AC\ } + T_{AB}sin30 - T_{AD}$

2. Manipulate Equations and plug in knowns

   1. $T_{AD} = 550lb$

   2. $T_{AC} = 479.76lb$

   3. $T_{AB} = 332.39lb$