Question

Let $x=x_m\cos (\omega t+\varphi ).$ At $t=0,x=x_m.$ If time period is $T$, what is the time taken to reach $x=x_m/2?$

• A. $\frac{3T}{2}$
• B. $\frac{T}{3}$
• C. $\frac{2T}{3}$
• D. $\frac{T}{6}$

Answer 1

Answer: (D)

Given : $x=x_m\cos (\omega t+\varphi )$ At $t=0,x=x_m$
$\therefore x_m=x_m\cos (\omega \times 0+\varphi )$ or $\cos \varphi =1=\cos 0^{\circ }$ or $\varphi =0^{\circ }$
$\therefore x=x_m\cos \omega t$
When $x=\frac{x_m}{2},\frac{x_m}{2}=x_m\cos \omega t$
or $\cos \omega t=\frac{1}{2}=\cos \frac{\pi }{3}$ or $\omega t=\frac{\pi }{3}$
or $t=\frac{\pi }{3\omega }=\frac{\pi T}{3\times 2\pi }=\frac{T}{6}\left(\therefore \omega =\frac{2\pi }{T}\right)$