Varignon theorem - mechanics, Mechanical Engineering

Varignon theorem:

State Varignon's theorem. How it helps in determining of the moments? In which type of condition is it used?

Sol.: Varignon's theorem also called Law of Moment.

The application of varignon's theorem practically is to find the position of the resultant from any point of the body.

It states "If number of coplanar forces is acting simultaneously on a particle, algebraic sum of moments of all the forces about any point is equal to moment of their resultant force about same point."

Proof: Let us consider, two concurrent forces P and Q represented in magnitude and direction by AB and AC as shown in the figure given below.

Let 'O' be the point, about which moment are taken, through O draw line OD parallel to direction of force P, to meet line of action of force Q at point C. Now with AB and AC as the two adjacent sides, complete Parallelogram ABDC as shown in the figure given below. Join the diagonal AD of the parallelogram and OA and OB. From parallelogram law of forces, we know that diagonal AD represents in the magnitude and direction, resultant of two forces P and Q. Now we see that moment of the force P about O: = 2. Area of triangle AOB  ...(i)


133_Varignon theorem.png

Likewise, moment of the force Q about O:  = 2. Area of ΔAOC                                     ...(ii)

And moment of resultant force R about O:  = 2.Area of ΔAOD                                     ...(iii)

But from the geometry of the figure, we find that

Area of ΔAOD = Area of ΔAOC + Area of ΔACD

But the Area of ΔACD = Area of ΔABD = Area of ΔAOB

(As the two "AOB and ADB are on same base AB and between same // lines)

Now Area of triangle AOD = Area of triangle AOC + Area of triangle AOB

Multiply both the side by 2 we obtain;

2. Area of triangle AOD = 2.Area of triangle AOC + 2. Area of triangle ACD, that gives

Moment of force R about O = Moment of force P about O + Moment of force Q about or,

Where R . d =     ∑M

∑M = Sum of moment of all the forces

d = Distance between resultant force and point where moment of all the forces are taken. This principle can be extended for any number of the forces.


Posted Date: 10/16/2012 7:20:04 AM | Location : United States

Related Discussions:- Varignon theorem - mechanics, Assignment Help, Ask Question on Varignon theorem - mechanics, Get Answer, Expert's Help, Varignon theorem - mechanics Discussions

Write discussion on Varignon theorem - mechanics
Your posts are moderated
Related Questions
Compare the hardenability of Ferrous Alloys vs Non-Ferrous Alloys

What is the magnitude of frictional force after perfect rolling begins and why? solution) After pure rolling begins,frictional force is 0. This is because the velocity of he c

Piston and Piston Rings - Part of combustion engine: Piston: The piston reciprocates within cylinder and transmits force exerted by expanding gases to crank by means of conn

Component s of Steam Power Plant: There are four components of steam power plant: 1. The boiler: Hot-source reservoir in which the combustion gases raise steam. 2. Engi

Find out the power - hollow shaft: A hollow shaft of internal diameter 400 mm & external diameter 450 mm is needed to transmit power at 120 rpm. Find out the power it may tran

what are the factors that affecting the fatigue behaviour of materials?

If no leak found in the above listed area remove flywheel: Remove fly wheel and check weather oil leak is from the seal or "o" ring.

Constructive Solid Geometry (Csg or C-Rep) A CSG model is based on the topological notion that a physical project object can be divided into a set of primitives that can be com

Can you prove an equation from Plane Stress and Plane strain?

Vented Causeway: Vented causeways are structures provided with vents (usually pipes) to take care of dry weather perennial flows and which allow water to overflow the paved b