1. Tardigrade
  2. Question
  3. Physics
  4. Let x=xm cos (ω t+φ) . At t=0, x=xm . If time period is T, what is the time taken to reach x=xm / 2 ?

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Q. Let $x=x_{m} \cos (\omega t+\phi) .$ At $t=0, x=x_{m} .$ If time period is $T$, what is the time taken to reach $x=x_{m} / 2 ?$

Oscillations

Solution:

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