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# What is Varignon’s Theorem?

By BYJU'S Exam Prep

Updated on: September 25th, 2023

**Varignon’s Theorem** is extremely important in calculating the centroid and centre of mass and the centre of pressure of hydrostatic force on the dam wall and submerged object. A problem or system may have two or more forces in mechanics. It is known as a “system of force” or “force system”. Varignon’s Theorem is very useful for analyzing the net effect of the system of forces on the system. Various systems of force acting on the rigid object in a plane may be a collinear system of force, a concurrent system of force, and a nonconcurrent system of force.

Varignon’s Theorem enables us to replace a nonconcurrent system of forces on a body with the help of a single force without altering the net effect. A moment is the outcome of a force applied to an object, similar to torque. It is the rotational effect produced by force about the axis of rotation. A moment is a vector quantity, and its SI unit is N-m. Here we will discuss Varignon’s Theorem, its statement, proof, and examples.

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## What is Varignon’s Theorem?

Varignon’s theorem is a method for calculating moments developed in 1687 by the French mathematician Pierre Varignon (1654–1722). This theorem is essential for the GATE ME question paper. The right-hand thumb rule can be used to determine the direction of moments. A moment can be completely defined by using the five characteristics of a moment (i.e., magnitude of force acting, direction of force, point of application of force, line of action of force, and the point of rotation).

**Vaignon’s theorem is also known as the “principle of moments.” **

Varignon’s theorem helps to figure out the location of the resultant force on an object acted upon by a non-concurrent coplanar system of forces and reduces the complications in calculating net moments on objects about the provided axis of rotation. Internal forces can also act on a system but shall be neglected as no final effect will be seen from them. This theorem can help find the resulting force easily.

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### State and Explain Varignon’s Theorem

According to Varignon’s theorem, “The total of the moments of many coplanar forces around a point equals the moment of the resultant of those forces, or the moment of a force around a point equals the sum of its components.”

### Varignon’s Theorem Equation

The equation for a system of forces acting on a body can find out by this method. The formula gives Varignon’s Theorem equation:

**M _{o} = r×R **

where M_{o} is the moment of resultant ‘R’ at point ‘O’.

## Varignon’s Theorem Proof

Varignon’s theorem is a principle that is frequently utilized in conjunction with the Principle of Transmissibility to solve systems of forces acting on and/or within a structure. Varignon’s theorem can be easily understood with the help of a given example.

Consider the force R acting in the plane of the body, as indicated in the above figure. Any two non-rectangular components of ‘R’ are represented by the forces ‘P’ and ‘Q.’ The moment of resultant ‘R’ at point ‘O’ is

M_{o} = r×R ——–(1)

R is the resultant of P and Q; hence, R = P+ Q

Put the value in equation 1;

M_{o}= r×(P+Q)

Use the distributive law of cross-product,

M_{o}= (r×P)+(r×Q) ——-(2)

From equations 1 and 2

r×R = (r×P)+(r×Q)

It states that the moment of ‘R’ about ‘O’ is the total of the moments of ‘P’ and ‘Q’ about ‘O’. This establishes the theorem.

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## Varignon’s Theorem Problem

Let us understand Varignon’s Theorem with the help of an example from the GATE exam.

Calculate the moment of the force F around point O illustrated in the image using Varignon’s theorem if the magnitude of Force is 500 N.

Varignon’s theorem makes it easier to compute the moment of the force F about the point O in the structure illustrated in the picture if the force is divided into rectangular components and the moment of each is calculated:

### Varignon’s Theorem Example

Force F can be decomposed in the following manner;

The horizontal component of force, F_{x}= F. Cosθ = 500× cos 30 = 433N

The vertical component of force, F_{y}= F. Sinθ = 500× Sin 30 = 250N

Multiplying the force and the perpendicular distance yields the moment of each component of the force about O. Both forces will spin the frame in the same direction (clockwise) with an arbitrarily assigned positive sign.

As per Varignon’s theorem;

Net moment about ‘O’ due to force F given as;

M_{o}= (F. Cosθ)×2 + (F. Sinθ)× 1

M_{o}= (433)×2 + (250)×1

M_{o}= 1116 N-m, clockwise