Question

Two forces of 3N and 4N are acting on a particle. The forces are acting at an angle of ${30}^{\circ }$ between them. The magnitude of the resultant force will be

• A. $\sqrt{3^2+4^2}N$
• B. $\sqrt{3^2+4^2+3\times 4cos30}N$
• C. $\sqrt{3^2+4^2+3\times 4\times sin30}N$
• D. $\sqrt{3^2+4^2+2\times 3\times 4cos30}N$

Answer 1

The correct option is D $\sqrt{3^2+4^2+2\times 3\times 4cos30}N$


The situation mentioned in the question can be depicted by the following figure.

Now to add these two forces we can use parallelogram law of addition of 2 vectors by completing the parallelogram.

The diagonal $\overline{AB}$ will give the resultant force. $|\overline{AB}|$, the magnitude of the resultant force will be equal to the length of the diagonal which is $\sqrt{\left(3^2\right)+\left(4^2\right)+2\left(3\right)\left(4\right)cos30}$.
(The length of the diagonal of a parallelogram with sides a and b is $\sqrt{\left(a^2\right)+\left(b^2\right)+2\left(a\right)\left(b\right)cos\beta }$, where $\beta$ is the angle between the 2 sides of the parallelogram)