Question

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

Answer 1

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

Now, $y_1=r\sin \omega t_1$

$\frac{r}{2}=r\sin \omega t_1\Rightarrow \sin t_1=\frac{1}{2}\Rightarrow \omega t_1=\frac{\pi }{6}\Rightarrow \frac{2\pi }{t}\times t_1=\frac{\pi }{6}\Rightarrow t_1=\frac{t}{12}$

Again , $y_2=r\sin \omega t_2$

$\Rightarrow r=r\sin \omega t_2\Rightarrow \sin t_2=1\Rightarrow \omega t_2=\frac{\pi }{2}\Rightarrow \frac{2\pi }{t}\times t_2=\frac{\pi }{2}\Rightarrow t_2=\frac{t}{4}$

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