A light stick of length $l$ 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 rest from point B 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)

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$a_{P} = \frac{g}{\sin θ} (1 - \frac{x}{l})$

$a_{P} = \frac{g}{\cos θ} (1 - \frac{x}{l})$

The time taken by the bug to reach the bottom of the stick having started at the top end from rest is $\frac{π}{2} \sqrt{\frac{l \sin θ}{3 g}}$

The time taken by the bug to reach the bottom of the stick having started at the top end from rest is $\frac{π}{2} \sqrt{\frac{l \sin θ}{g}}$

answer is A, D.

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Detailed Solution

Consider the bug at point P at any time t, moving with speed v along the stick, so $L = m v h = m v l \sin θ \cos θ \Rightarrow$ Angular momentum of bug about point
$\Rightarrow π = \frac{d L}{d t} = (m l \sin θ \cos θ) \frac{d v}{d t} \Rightarrow m g (l - x) \cos θ = (m l \sin θ \cos θ) \frac{d v}{d t}$
$\Rightarrow \frac{d v}{d t} = \frac{g}{\sin θ} (1 - \frac{x}{l})$
$\Rightarrow \frac{d v}{d t} = \frac{g}{l \sin θ} x = \frac{g}{\sin θ} \Rightarrow$ Equation of SHM with mean position at point A
$\Rightarrow T = 2 π \sqrt{\frac{l \sin θ}{g}}$
Time required $= \frac{T}{4} = \frac{π}{2} \sqrt{\frac{l \sin θ}{g}}$
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