Question

What will be the displacement equation of the simple harmonic motion obtained by combining the motions?

$x_1=2sin\omega t,x_2=4sin\left(\omega t+\frac{\pi }{6}\right)and{x}_3=6sin\left(\omega t+\frac{\pi }{3}\right)$

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

Answer 1

Answer: (C)

The resultant equation is
$x=Asin\left(\omega t+\varphi \right)$
$\sum A_x=2+4cos30^{\circ }+6cos60^{\circ }=8.46$

And $\sum A_y=4sin30^{\circ }+6cos30^{\circ }=7.2$
$\therefore A=\sqrt{{\left(\sum A_x\right)}^2+{\left(\sum A_y\right)}^2}$
$=\sqrt{{\left(8.46\right)}^2+{\left(7.2\right)}^2}=11.25$
And $tan\varphi =\frac{\sum A_y}{\sum A_x}=\frac{7.2}{8.46}=0.85$
$\Rightarrow \varphi ={\tan }^{-1}(0.85)=40.4^{\circ }$
Thus, the displacement equation of combined motion is
$x=11.25sin\left(\omega t+\varphi \right)$
Where, $\varphi =40.4^{\circ }$