State and prove the Theorem of Varignon.
Answers
According to Varignon’s theorem, the algebraic sum of several concurrent forces about any point is equal to the moments of the moments of their resultant about the point.
To prove Varignon’s Theorem, consider the force R acting in the plane of the body as shown in the above-left side figure (a). The forces ‘P’ and ‘Q’ represent any two non-rectangular components of ‘R’. The moment of ‘R’ about point ‘O’ is
Mo = r × R
Because R = P + Q, we can write
r × R = r × (P + Q)
Using the distributive law for cross products, we have
Mo = r × R = r × P + r × Q
which says that the moment of ‘R’ about ‘O’ equals the sum of the moments about ‘O’ of its components ‘P’ and ‘Q’. This proves the theorem.
Varignon’s theorem need not be restricted to the case of two components, but it applies equally well to three or more. Thus we could have used any number of concurrent components of R in the foregoing proof.
In the above right side figure (b) illustrates the usefulness of Varignon’s theorem. The moment of ‘R’ about point ‘O’ is ‘R × d’. However, if ‘d’ is more difficult to determine than ‘p’ and ‘q’, we can resolve ‘R’ into the components ‘P’ and ‘Q’, and compute the moment as
Mo = R × d = – p × P + q × Q
where we take the clockwise moment sense to be positive.
Answer:
Explanation:
Varignon’s theorem :
According to Varignon's Theorem, the moment of the resulting force with regard to the same point is equal to the total of the moments generated by a system of concurrent forces with respect to that point.
This means that the moment of a force about a point is equal to the algebraic sum of the moments of its component forces about that point.
As we know,
Moment = F × D
Where,
F- Force
D- Perpendicular Distance
Sum means to add a number while algebraic sum means to add positive and negative numbers.
Now, look at the figure(1) for the component of forces where R is the resultant of P and Q while P and Q are the forces component of R.
So, if find the moment of R with respect to O.
Then,
Moment Of R About O = (Moment Of P About O + Moment OF Q About O)
MR = MP + MQ
Varignon's Theorem Proof
Let F1, and F2, be the two forces represented by the lines AB and AD.
See in the figure(2),
Where,
O be the point about which moment is to be taken.
From O draw a line OC parallel to AB meeting AD at D.
Join BC to complete the parallelogram.
Now join the diagonal AC which gives the resultant R of the two forces.
Now Join OA and OB.
From the figure, we can clearly see that,
(△ABC) = (△ADC) = (△OAB )
As we know that the moment of a triangle is twice its area.
So,
Moment of force F1 about O = 2 x Area of △OAB
Moment of force F2 about O =2 x Area of △OAD
Moment of resultant force R about O = 2 x area of △OAC
Sum of moments of two forces about O =
2 Area of △OAB + 2 x Area of△OAD
Now we clearly see in the figure ADC = OAB that we can write ADC in place of OAB.
Sum of moments of two forces about O =
2 Area of △ADC + 2 x Area of △OAD
Sum of moments of two forces about O =
2 (Area of △ADC + Area of △OAD)
Now you can see well in the figure that
△OAC = △ADC + △OAD
So,
Sum of moments of two forces about O =
2 x Area of△OAC
So, proving the theorem that the sum of the moment of two forces about O is equal to the moment of the resultant force R about O.
Hence, the varignon theorem is proved.
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