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Asked by:milan-ransingh

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Aparna-Dasguptaprajwalamv

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The Principle of Moments, also known as Varignon's Theorem, states that the moment of any force is equal to the algebraic sum of the moments of the components of that force.

For a set n of vectors ${\boldsymbol {u_{1}}},{\boldsymbol {u_{2}}},...,{\boldsymbol {u_{n}}}$ that concurs at point O in space. The resultant is:

${\boldsymbol {R}}=\sum _{i}{\boldsymbol {u_{i}}}$  The moment of each vector is:

$\sum _{i}{\boldsymbol {M_{O_{1}}^{u_{i}}}}=\sum _{i}({\boldsymbol {O}}-{\boldsymbol {O_{1}}})\times {\boldsymbol {u_{i}}}$ In terms of summary, taking out the common factor $(\mathbf {O} -\mathbf {O_{1}} )$ , and considering the expression of R, results:

$\sum _{i}{\boldsymbol {M_{O_{1}}^{u_{i}}}}=({\boldsymbol {O}}-{\boldsymbol {O_{1}}})\times \left(\sum _{i}{\boldsymbol {u_{i}}}\right)=({\boldsymbol {O}}-{\boldsymbol {O_{1}}})\times {\boldsymbol {R}}={\boldsymbol {M_{O_{1}}^{R}}}$ Answerd By:Amogh

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