Question

Forces $\vec{P}$ and $\vec{Q}$ have resultant $\vec{R}$ whose magnitude is $40\ \text{N}$. $\vec{R}$ makes an angle $45^\circ$ with $\vec{P}$ as well as $\vec{Q}$. The magnitude of $\vec{P}$ is $\left(\tan \dfrac{\pi}{4} = 1\right)$

• A. $20\sqrt{5}\ \text{N}$
• B. $\sqrt{20}\ \text{N}$
• C. $20\sqrt{2}\ \text{N}$
• D. $20\ \text{N}$

Hint:

If resultant makes equal angles with two forces, the forces must be equal in magnitude.

Answer 1

The Correct Option is C

Step 1: Use parallelogram law of forces.
If resultant makes equal angles with $\vec{P}$ and $\vec{Q}$, then $\vec{P} = \vec{Q}$.
Step 2: Use relation for resultant of two equal forces.
\[ R = 2P\cos\theta \] Step 3: Substitute given values.
\[ 40 = 2P \cos 45^\circ \] Step 4: Simplify.
\[ 40 = 2P \times \dfrac{1}{\sqrt{2}} \] \[ P = 20\sqrt{2}\ \text{N} \]