Varignon’s Theorem - State and Explain Varignon's Theorem

By Mohit Uniyal|Updated : August 5th, 2022

Varignon's Theorem is extremely important in calculating the centroid and center of mass and the center 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". Various types of 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 is very useful for analyzing the net effect of the system of forces on the system.

Before we state and explain Varignon’s theorem, we should have a very good idea about the basics of the moment. A moment is the outcome of a force applied to an object, which is very 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. 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 the rotation). 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.

What is Varignon’s Theorem?

Varignon's theorem is a method for calculating moments that was developed in 1687 by the French mathematician Pierre Varignon (1654–1722). Vaignon’s theorem is also known as the “principle of moments''.

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 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.

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

Mo = r×R --------(1)

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

Put the value in equation 1;

Mo= r×(P+Q)

Use the distributive law of cross product,

Mo= (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.

Varignon’s Theorem Problem

Calculate the moment of the force F around the 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:

Force F can be decomposed in the following manner;

The horizontal component of force, Fx= F. Cosθ= 500× cos 30 = 433N

The vertical component of force, Fy= 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;

Mo= (F. Cosθ)×2 + (F. Sinθ)× 1

Mo= (433)×2 + (250)×1

Mo= 1116 N-m, clockwise

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FAQ on Varignon’s Theorem

  • According to Varignon’s theorem, “The torque of a resultant of non concurrent forces about any point is equal to the algebraic sum of the torques of all forces about the same point”. Varignon's theorem allows us to replace a nonconcurrent system of forces on a body with a single force while keeping the net effect constant.

  •  Varignon's Theorem, often known as the principle of Moments, is a very important technique in calculating scalar moments. When determining the perpendicular distance is difficult, Varignon's Theorem provides an alternative method. Varignon's theorem is useful in determining the center of pressure of hydrostatic force in dam design and undersea objects.

  • In general practice, a planar rigid body may be acted upon by a system of forces. The calculation of the resultant force is important to predict the net effect of a system of forces on a system. Varignon's theorem is useful in determining the position of the resulting force on an object acted on by a non-concurrent coplanar system of forces.

  • Sometimes, calculating the perpendicular distance of the line of action from the point of rotation becomes too complicated. In such cases, Varigon’s theorem helps reduce the complications in a calculating moment due to force. By resolving the force into appropriate directions and then calculating the moments due to components, we get the same moment as that produced by the resultant force.

  • Varignon's theorem is applicable only when the forces are coplanar concurrent and coplanar non-concurrent.  Varignon's theorem can be used to calculate the position of the resultant from any point on the body for a given non-concurrent system of force.

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