---
title: "Equilibrium and its equations in 2D and 3D"
subject: "Applied Mechanics"
chapter: 2
key: "semester_1/applied_mechanics:equilibrium-equations-2d-3d"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/equilibrium-equations-2d-3d
site: BCT Engineering
---

# Equilibrium and its equations in 2D and 3D

Applied Mechanics · Chapter 2: Basic Concept in Statics and Static Equilibrium

- Asked in: 2074 Asoj, 6 marks
- Video: [Equilibrium of Rigid Bodies (2D - Coplanar Forces) | Mechanics Statics | (Solved examples)](https://www.youtube.com/watch?v=jQDEOwrR4UU) — Question Solutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/equilibrium-equations-2d-3d
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-2

## Question

Define equilibrium and its essence. What are the equations of static equilibrium for 2D and 3D analysis of particle and rigid body?

## Short answer

Equilibrium means the resultant force and moment on a body are zero, so it stays at rest or moves uniformly. For a particle: 2D ΣFx=0, ΣFy=0; 3D ΣFx=ΣFy=ΣFz=0. For a rigid body: 2D ΣFx=0, ΣFy=0, ΣM_O=0; 3D ΣFx=ΣFy=ΣFz=0 and ΣMx=ΣMy=ΣMz=0.

## Answer

#### Equilibrium
A body is in equilibrium when the resultant of all forces and moments acting on it is zero, so it remains at rest (static equilibrium) or moves with constant velocity.

#### Its essence
The essence of equilibrium is that the body has no tendency to translate or rotate. The forces are balanced, so ΣF = 0, and the moments are balanced, so ΣM = 0. In engineering, structures must be in static equilibrium under their loads. The equations are used to find unknown loads and support reactions.

#### Equations for a particle
| Case | Equations |
|---|---|
| 2D | ΣFx = 0, ΣFy = 0 |
| 3D | ΣFx = 0, ΣFy = 0, ΣFz = 0 |

A particle has no size, so moments do not arise.

#### Equations for a rigid body
| Case | Equations |
|---|---|
| 2D | ΣFx = 0, ΣFy = 0, ΣM_O = 0 |
| 3D | ΣFx = ΣFy = ΣFz = 0; ΣMx = ΣMy = ΣMz = 0 |

In 2D there are 3 independent equations, and in 3D there are 6. A properly supported body is statically determinate when the number of unknown reactions equals the number of available equations. If there are more unknowns than equations, it is statically indeterminate.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A body is in equilibrium when:
   A. Its speed is always zero
   B. The reactions are larger than the loads
   C. The resultant of all forces and moments on it is zero
   D. Only the forces are equal
   Answer: C. Equilibrium means no tendency to translate or rotate. (Key point: Meaning)

2. Why are moment equations not used for a particle?
   A. A particle has no size, so moments do not arise
   B. Particles cannot be in equilibrium
   C. Moments are always infinite
   D. Forces on it are parallel
   Answer: A. A point has no moment arms; concurrent forces need only force equations. (Key point: Particle)

3. The equations for a 3D rigid body are:
   A. SM = 0 about one axis
   B. SFx = SFy = SFz = 0 only
   C. SFx = SFy = SFz = 0 and SMx = SMy = SMz = 0
   D. SFx = SFy = 0 and SM = 0
   Answer: C. Three force and three moment equations give six. (Key point: 3D rigid body)

4. A body is statically determinate if:
   A. Unknown reactions are more than the equations
   B. There are no reactions
   C. Loads are small
   D. Unknown reactions do not exceed the available equations
   Answer: D. Reactions can be found from the equations of equilibrium alone only if unknowns do not exceed them. (Key point: Determinacy)

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