Find the vector sum of `N` coplanar forces, each of the magnitude `F`,when each force makes an angle of `2pi//N` with that preceding it.

Answer

zero

Explanation

consider a regular polygon of `N` sides. Each side of the poygon is same. `C` is the centre of polygon (figure) `theta=2pi//N rarr theta` is the angle subtended at the centre of polygon by any one side.
We can prove that `alpha=theta` as
`/_ABC+ /_CBD+ alpha =180^(@)`..(i)
But `/_ABC=/_CBD=90^(@)- theta/2`..(ii)
From Eqs.(i) and (ii), we get `alpha=theta`
`rArr alpha=(2pi)/N`
So if we have `N` vector of equal magnitude and they are arranged in such a way that each vector makes an angle `2pi//N(=alpha)` with its preceding vector, we find that the head of the last vector will coincide with the tail of the first vector. Hence, the resultant of all these vector becomes zero.