A  light  stick  of  length $\ell$  rests  with  its  one  end  against  the  smooth  wall and  other  end  against  the  smooth  horizontal  floor  as  shown  in  the figure.  The  bug  starts  at  point  B  from rest and  moves  such  that  the stick always remains at rest.  $a_P$ is the magnitude of acceleration of bug of mass m, which depends upon its distance of x from the top end of the stick. Choose the CORRECT option(s)

Consider the bug at point P at any time t, moving with speed v along the stick.  

Angular momentum of bug about point O

$L=mvh=mv\left(\ell \cos \theta \right)\sin \theta$

Torque is rate of change of Angular Momentum.
$\Rightarrow \tau =\frac{dL}{dt}=\left(m\ell \sin \theta \cos \theta \right)\frac{dv}{dt}$

This torque should be balanced by the torque due to gravity.

$\Rightarrow mg(\ell \cos \theta -x\cos \theta )=\left(m\ell \sin \theta \cos \theta \right)\frac{dv}{dt}$ 
$\Rightarrow \frac{dv}{dt}=\frac{g}{\sin \theta }(1-\frac{x}{\ell })=a_P$ 
$\Rightarrow \frac{dv}{dt}+\frac{g}{\ell \sin \theta }x=\frac{g}{\sin \theta }$ 

Equation of SHM with mean position at point A$\because$  a=-${\omega }^{2}x$
$\Rightarrow T=2\pi \sqrt{\frac{\ell \sin \theta }{g}}$ 

At $x=0$, $a_P=\frac{g}{\sin \theta }\left(1-\frac{0}{\mathcal{l}}\right)=\frac{g}{\sin \theta } \left[maximum value of a_P\right]$ , means point B is extreme position.

At $x=\mathcal{l}$, $a_P=0$ , means point A is mean position.
So, Time required to travel from extreme to mean position is $=\frac{T}{4}=\frac{\pi }{2}\sqrt{\frac{\ell \sin \theta }{g}}$