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 point B from rest and moves such that the stick always remains at rest. aP 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 $$option(s)$$

Question

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  point  B  from rest and  moves  such  that  the stick always remains at rest.  aP 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 $$option(s)$$

Infinity Learn · Accepted Answer

Consider the bug at point P at any time t, moving with speed v along the stick.  Angular momentum of bug about point $$OL=mvh=mv$$ℓ$$cos$$θ$$sin$$θTorque is rate of change of Angular Momentum.⇒   τ$$=dLdt=(m$$ℓ$$sin$$θ$$cos$$θ$$)dvdtThis$$ torque should be balanced by the torque due to gravity.⇒$$mg($$ℓ$$cos$$θ−xcosθ$$)=(m$$ℓ$$sin$$θ$$cos$$θ$$)dvdt$$ ⇒    $$dvdt=gsin$$θ$$(1$$−xℓ$$)=aP$$ ⇒    $$dvdt+g$$ℓ$$sin$$θ$$x=gsin$$θ Equation of SHM with mean position at point A∵  $$a=-$$ω$$2x$$⇒    $$T=2$$πℓ$$sin$$θg At $$x=0$$, $$aP=gsin$$θ$$1-0l=gsin$$θ   maximum value of aP , means point B is extreme position.At $$x=l$$, $$aP=0$$ , means point A is mean position.So, Time required to travel from extreme to mean position is $$=T4=$$π$$2$$ℓ$$sin$$θg

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