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 but of mass `m`, which depends upon its distance of `x` from the top end of the stick. Choose the correct option (s)

Options

A. `a_(P)=g/(sintheta)(1- d/l)`
B. `a_(P)=g/(costheta)(1-x/l)`
C. The time taken by the but to reach the bottom of the stick having started at the top end from rest is `(pi)/2 sqrt((lsintheta)/(3g))`
D. The time taken by the but to reach the bottom of the stick having started at the top end from rest is `(pi)/2 sqrt((lsintheta)/2)`

Answer

A, D

Explanation

Consider the bug at point `P` at any time `t`, moving with speed `v` along the stick, so `L=mvhmvlsintheta costhetaimplies ` Angular momentum of bug
about pooint `O`
`impliestau=(dL)/(dt)=(mlsintheta cos theta)(dv)/(dt)`
`impliesmg(l-x)costheta=(mlsintheta costheta)(dv)/(dt)`
`implies(dv)/(dt)=g/(sintheta)(1-x/l)`
`implies(dv)/(dt)+g/(lsintheta)x =g/(sintheta)implies` Equation of SHM with mean position at point `A`
`impliesT=2pisqrt((lsintheta)/g)`
Time required `=T/4=(pi)/2 sqrt((lsintheta)/2)`