Question

Five sinusoidal waves have the same frequency $500Hz$ but their amplitudes are in the ratio $2:\frac{1}{2}:\frac{1}{2}:1:1$ and their phase angles $0,\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2}$ and $\pi$ respectively. The phase angle of resultant wave obtained by the superposition of these five waves is:

• A. ${30}^o$
• B. ${45}^o$
• C. ${60}^o$
• D. ${90}^o$

Answer 1

Answer: (B)

${45}^o$

The phasors of five waves can be represented as

$x_1=2\hat{i}$

$x_2=\sqrt{3}/4\hat{i}+1/4\hat{j}$

$x_3=1/4\hat{i}+\sqrt{3}/4\hat{j}$

$x_4=\hat{j}$

$x_5=-\hat{i}$

Resultant $x={\sum }_{i=1}^5{x}_i$

$x=\frac{5+\sqrt{3}}{4}\hat{i}+\frac{5+\sqrt{3}}{4}\hat{j}$

Phase angle of resultant, $\theta =ta{n}^{-1}\frac{(5+\sqrt{3})/4}{(5+\sqrt{3})/4}$

$\Rightarrow \theta ={45}^o$