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
RECTIFIER TYPE POWER SOURCES The next popular power sources are of rectifier type. Three phase transformer (step down) connected to a rectifier to give DC output is the simple

Types Of Patterns There are various types of patterns depending upon the complexity of the job, the number of castings required and the moulding procedure adopted. Singl

how do i solve for thickness of insulation when i only have the r value

Find power transmitted by cross type belt: Find power transmitted by cross type belt drive connecting two pulley of diameter 45.0cm and 20.0cm that are 1.95m apart. The max


Example of Process Capability The specifications for one characteristic of a part call for its width to be 3.000 + 0.008 centimetres. The procedure has been run under controll

Q: Determine the resultant of force system which is acting tangential to the circle of radius 1m as shown in the figure given below. Also find the direction of it and line of actio

Use three frame elements to analyse the frame structure as depicted in the image below subjected to combined loadings: a) a concentred load applied  on Element 1  at  its  1/3 leng

in the spiral or helical gears- have meshing gear different helix angle?

Projection Welding Projection welding is similar to spot welding but for the fact that the welding takes place at the projections on the surface of the metals and flat shaped e