Question

Consider a simple harmonic motion of time period T. Calculate the time taken for the displacement to change value from half the amplitude to the amplitude.

Answer 1

$y_1=\frac{r}{2},y_2=r$ (for the two given position)

Now, $y-1=rsin\omega t_1$

$\Rightarrow \frac{r}{2}=rsin\omega t_1$

$\Rightarrow sin\omega t_1=\frac{1}{2}$

$\Rightarrow \pi t_1=\frac{\omega }{6}$

$\Rightarrow \frac{2\pi }{T\times t_1}=\frac{\pi }{6}$

$\Rightarrow t_1=\frac{T}{12}$

Again, $y_2=rsin\omega t_2$

$\Rightarrow r=rsin\omega t_2$

$\Rightarrow sin\omega t_2=1$

$\Rightarrow \omega t_2=\frac{\pi }{2}$

$\Rightarrow \left(\frac{2\pi }{T}\right)t_2=\frac{\pi }{2}$

$\Rightarrow t_2=\frac{T}{4}$

So, $t_2-t_1=\frac{T}{4}-\frac{T}{12}=\frac{T}{6}$