Question

Four coplanar forces are applied at a point $O$. Each of them is equal to $k$, and the angle between two consecutive force equals ${45}^{\circ }$, then the resultant is equal to

• A. $k\sqrt{2+2\sqrt{2}}$
• B. $k\sqrt{3+2\sqrt{2}}$
• C. $k\sqrt{4+2\sqrt{2}}$
• D. None of of these

Answer 1

The correct option is C $k\sqrt{4+2\sqrt{2}}$

Let $O$ be the origin.
The four vectors would look like

$k\hat{i}$, $\frac{k}{\sqrt{2}}\hat{i}+\frac{k}{\sqrt{2}}\hat{j}$, $k\hat{j}$, $-\frac{k}{\sqrt{2}}\hat{i}+\frac{k}{\sqrt{2}}\hat{j}$

Adding these, we have $k\hat{i}+k(1+\sqrt{2})\hat{j}$

Magnitude $=k\sqrt{1+1+2+2\sqrt{2}}=k\sqrt{4+2\sqrt{2}}$