Question

The vector sum of $N$ coplanar forces, having magnitude of $F$, when each force is making an angle of $\dfrac{2\pi }{N}$ with that preceding it, is:
$A)\text{ }F$
$B)\text{ }\dfrac{NF}{2}$
$C)\text{ }NF$
$D) 0$

Answer 1

Hint: The sum of all the interior angles of a polygon equals ${{360}^{0}}$ or $2\pi \text{ }rad$. If the sides of the polygon are all equal, it will become a regular polygon. If the sides of a polygon are represented as a series of vectors, their total resultant cancels out.

Complete step-by-step solution -
First let us check whether the vectors are placed in series such that the head of one joins with the tail of the next one, whether it forms a polygon or not.
It is given that each vector makes an angle of $\dfrac{2\pi }{N}$ with the preceding one. So, if $N$ vectors are arranged in the way explained above that is as the sides of a polygon of $N$ sides, the sum of the interior angles will be
$\sum\limits_{1}^{N}{\dfrac{2\pi }{N}}$
$=N\times \dfrac{2\pi }{N}=2\pi$
Therefore, the sum of the interior angles is $2\pi \text{ }rad$.
The sum of the interior angles of a polygon is also always equal to $2\pi \text{ }rad$. Hence, we can conclude that if the vectors were to be arranged as explained above, they would form a polygon of $N$ sides. Moreover, since the magnitude of each of them is equal, that is, $F$, it can be said that they will form a regular polygon with all sides equal to $F$.
Now, it is a fact that if a series of vectors when joined consecutively head to tail form a polygon, then the final vector will join to the tail of the first vector and their resultant will cancel out ultimately to give zero.
Hence, the vector sum of $N$ coplanar forces, having magnitude of $F$, when each force is making an angle of $\dfrac{2\pi }{N}$ with that preceding it, is zero.

Therefore, the correct option is (D).

Note: Students must know this concept very well. In fact this is the well known polygon law of vector addition which is the more general law for vector addition and by which the triangle law of vector addition of two vectors is derived. Sometimes questions with confusing figures are purposefully made to confuse students and test their knowledge of this concept. However, most of the time, these problems can be simplified or even solved directly by using this polygon law of vector addition.