The angle that the string of a simple pendulum makes with the...

The given equation for the angular displacement of the simple pendulum is $\theta = \dfrac{\pi}{90} \sin(\pi t)$.

Comparing this with the standard equation of simple harmonic motion $\theta = \theta_0 \sin(\omega t)$, the angular frequency is $\omega = \pi \text{ rad/s}$.

The angular frequency of a simple pendulum is given by $\omega = \sqrt{\dfrac{g}{l}}$.

Squaring both sides gives $\omega^2 = \dfrac{g}{l}$.

Rearranging for the length $l$, we get $l = \dfrac{g}{\omega^2}$.

Substituting $g = \pi^2 \text{ m/s}^2$ and $\omega = \pi \text{ rad/s}$ into the expression:

$l = \dfrac{\pi^2}{\pi^2} = 1 \text{ m}$.