Question

What will be the displacement equation of the simple harmonic motion obtained by combining the motions?
$x_1=2\sin \omega t$, $x_2=4\sin (\omega t+\frac{\pi }{6})$ and $x_3=6\sin (\omega t+\frac{\pi }{3})$

• A. $x=10.25\sin (\omega t+\varphi )$
• B. $x=10.25\sin (\omega t-\varphi )$
• C. $x=11.25\sin (\omega t+\varphi )$
• D. $x=11.25\sin (\omega t-\varphi )$

Answer 1

Answer: (C)

$x=11.25\sin (\omega t+\varphi )$

The resultant equation is
$x=A\sin (\omega t+\varphi )$
$\sum A_x=2+4\cos {30}^o+6\cos {60}^o=846$
and $\sum A_y=4\sin {30}^o+6\cos {30}^o=7.2$
$\therefore A=\sqrt{{\left(\sum A_x\right)}^2+{\left(\sum A_y\right)}^2}$
$=\sqrt{{\left(8.46\right)}^2+{\left(7.2\right)}^2}=11.25$
and $\tan \varphi =\frac{\sum A_y}{\sum A_x}=\frac{7.2}{8.46}=0.85$
$\Rightarrow \varphi ={\tan }^{-1}\left(0.85\right)={40.4}^o$
Thus, the displacement equation of combined motion is
$x=11.25\sin (\omega t+\varphi )$
where, $\varphi ={40.4}^o$