Question

3.1
State and prove Varignon's theorem.​

Answer 1

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3.1

State and prove Varignon's theorem.

Answer 2

Answer:

Varignon's Theorem: Moment of a force about any point is equal to the sum of the moments of the components of that force about the same point. which says that the moment of R about O equals the sum of the moments about O of its components P and Q

Explanation:

Consider a set of {\displaystyle N}N force vectors {\displaystyle \mathbf {f} _{1},\mathbf {f} _{2},...,\mathbf {f} _{N}}{\displaystyle \mathbf {f} _{1},\mathbf {f} _{2},...,\mathbf {f} _{N}} that concur at a point {\displaystyle \mathbf {O} }{\displaystyle \mathbf {O} } in space. Their resultant is:

{\displaystyle \mathbf {F} =\sum _{i=1}^{N}\mathbf {f} _{i}}

The torque of each vector with respect to some other point {\displaystyle \mathbf {O} _{1}}{\displaystyle \mathbf {O} _{1}} is

{\displaystyle \mathbf {\mathrm {T} } _{O_{1}}^{\mathbf {f} _{i}}=(\mathbf {O} -\mathbf {O} _{1})\times \mathbf {f} _{i}}{\displaystyle \mathbf {\mathrm {T} } _{O_{1}}^{\mathbf {f} _{i}}=(\mathbf {O} -\mathbf {O} _{1})\times \mathbf {f} _{i}}.

Adding up the torques and pulling out the common factor {\displaystyle (\mathbf {O} -\mathbf {O_{1}} )}{\displaystyle (\mathbf {O} -\mathbf {O_{1}} )}, one sees that the result may be expressed solely in terms of {\displaystyle \mathbf {F} }\mathbf {F} , and is in fact the torque of {\displaystyle \mathbf {F} }\mathbf {F} with respect to the point {\displaystyle \mathbf {O} _{1}}{\displaystyle \mathbf {O} _{1}}:

{\displaystyle \sum _{i=1}^{N}\mathbf {\mathrm {T} } _{O_{1}}^{\mathbf {f} _{i}}=(\mathbf {O} -\mathbf {O} _{1})\times \left(\sum _{i=1}^{N}\mathbf {f} _{i}\right)=(\mathbf {O} -\mathbf {O} _{1})\times \mathbf {F} =\mathbf {\mathrm {T} } _{O_{1}}^{\mathbf {F} }}{\displaystyle \sum _{i=1}^{N}\mathbf {\mathrm {T} } _{O_{1}}^{\mathbf {f} _{i}}=(\mathbf {O} -\mathbf {O} _{1})\times \left(\sum _{i=1}^{N}\mathbf {f} _{i}\right)=(\mathbf {O} -\mathbf {O} _{1})\times \mathbf {F} =\mathbf {\mathrm {T} } _{O_{1}}^{\mathbf {F} }}.

Proving the theorem, i.e. that the sum of torques about {\displaystyle \mathbf {O} _{1}}{\displaystyle \mathbf {O} _{1}} is the same as the torque of the sum of the forces about the same point.