---
title: "Free body diagram; Varignon's theorem"
subject: "Applied Mechanics"
chapter: 2
key: "semester_1/applied_mechanics:free-body-diagram-varignon"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/free-body-diagram-varignon
site: BCT Engineering
---

# Free body diagram; Varignon's theorem

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

- Asked in: 2074 Chaitra, 4+4 marks
- Video: [Lecture 2: Force,equilibrium,free body diagram, Varignon theorem](https://www.youtube.com/watch?v=JZY3w0-Lk0c) — mechanotherapy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/free-body-diagram-varignon
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-2

## Question

What do you understand by Free Body Diagram? Explain with sketches. Also state Varignon's Theorem and prove it.

## Short answer

An FBD is a sketch of an isolated body showing all external forces and moments, essential for applying equilibrium. Varignon's theorem states the moment of a force about a point equals the sum of moments of its components about that point, proven by the distributive property of the cross product: r × (P+Q) = r × P + r × Q.

## Answer

#### Free body diagram
A free body diagram (FBD) is a sketch of a body, or of part of a structure, isolated from everything around it, with all external forces and moments acting on it shown: applied loads, weight, support reactions and contact forces. It is the first step in solving any statics problem.

**Sketch 1: simply supported beam with a load P**
```text
Real beam:            FBD of beam:
     P                     P
     |                     |
     v                     v
 A o-----o B           ===========
 /\    (roller)         ^         ^
                       A_y       B_y
                  (A_x to the right at A)
```
**Sketch 2: block on a rough incline.** The FBD shows weight W down, normal reaction N perpendicular to the plane and friction F along the plane up the slope.

#### Varignon's theorem
**Statement:** the moment of a force about any point equals the algebraic sum of the moments of its components about the same point. For a system of concurrent forces, the moment of the resultant about a point equals the sum of the moments of the individual forces about that point.

**Proof (two concurrent forces):**
Let forces P and Q act at point A, with position vector r from the moment centre O. Their resultant is R = P + Q.
```text
Moment of R about O:   M_O = r × R
                           = r × (P + Q)
                           = r × P + r × Q      (distributive law of the cross product)
```
So the moment of the resultant equals the sum of the moments of P and Q about O. Applying it repeatedly extends it to any number of forces: r × ΣF = Σ(r × F).

**Proof in 2D by components:** let a force F at (x, y) have components Fx and Fy. Its moment about the origin is M_O = x·Fy − y·Fx. This is the sum of the moments of Fy (= x·Fy, anticlockwise) and Fx (= −y·Fx), which proves the theorem.

**Use:** the moment of an inclined force can be found easily by resolving it into horizontal and vertical components, and the line of action of a resultant can be located.

## Check yourself

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

1. A free body diagram shows:
   A. Applied loads only
   B. Only the internal forces
   C. All external forces and moments on the isolated body, including reactions
   D. The deformed shape
   Answer: C. It shows loads, weight, support reactions and contact forces. (Key point: FBD)

2. Varignon's theorem states that the moment of a force about a point equals:
   A. The algebraic sum of the moments of its components about the same point
   B. The product of the force and the distance of its components
   C. The moment of the largest component
   D. The sum of the magnitudes of its components
   Answer: A. The moment of a force equals the sum of the moments of its components. (Key point: Varignon statement)

3. In the vector proof, r x (P + Q) = r x P + r x Q follows from:
   A. Equilibrium of the point
   B. Newton's third law
   C. The distributive law of the cross product
   D. Transmissibility
   Answer: C. The cross product is distributive over addition. (Key point: Proof step)

4. The moment of a force F at (x, y) about the origin is:
   A. x Fx + y Fy
   B. x Fy - y Fx
   C. x Fx - y Fy
   D. y Fy - x Fx
   Answer: B. M_O = x Fy - y Fx, the sum of the moments of its components. (Key point: 2D use)

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