Time Period Of Torsional Pendulum - Practice Questions & MCQ

Below is the figure of the Torsional pendulum which consists of a rigid object suspended by a wire attached at the top to a fixed end.

When the object is twisted through some angle $\theta$, the twisted wired exerts on the object a restoring torque and this restoring torque is proportional to the angular position.

That is $\tau=-k \theta$ where $\kappa$ is called the torsion constant of the support wire.
Applying Newton's second law for rotational motion, we find that

$
\tau=-k \theta=I \frac{d^2 \theta}{d t^2} \Rightarrow \frac{d^2 \theta}{d t^2}=-\frac{k}{I} \theta
$

So the Time Period of Torsional pendulum is given as

$
T=2 \pi \sqrt{\frac{I}{k}}
$

where
$I=$ moment of inertia
$\mathrm{k}=$ torsional constant