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Forces of magnitude aF,bF,aF,bF and cF act respectively along the sides BA,BC,DC,DA and...

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Forces of magnitude aF,bF,aF,bF and cF act respectively along the sides BA,BC,DC,DA and the diagonal BD of a rectangle ABCD, where a=AB, b=BC and c=BD, in the directions specified by the order of letters. 

If the magnitude of the force acting DA is increased to 2bF while other forces remain the same, show that the new system is equivalent to a force of magnitude aF along CD.

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The resultant R is equal to:

R=aF-aF+bF-bF+cF

R=cF              (1)

and

cF=aF+bF       (2)

Substitute (2) in (1)

R=aF+bF


If the force acting along DA becomes 2bf, then

R=aF-aF+bF-2bF+cF

R=-bF+cF       (3)

Substitute (2) in (3):

R=-bF+aF+bF

R=aF

And the force along DC is positive.

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