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Asked by Vipulgautam050 | 26 Aug, 2021, 08:57: AM
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It is assumed that ' find x ' means to find the x-coordinate of particle-A
 
We are given that speed vA of particle-A when angular displacement θ = (π/6) as
 
begin mathsize 14px style v subscript A space equals space fraction numerator x space v subscript B over denominator 3 square root of 3 end fraction end style   ............................(1)
Angular speed ωA of particle-A is
 
begin mathsize 14px style omega subscript A space equals space v subscript A over r space equals fraction numerator x space v subscript B over denominator 3 square root of 3 space r end fraction end style  ............................(2)
Whn the system of light rod of length r and particle-A starts from rest ,
angular velocity ωA when the angular displacement is θ is determined from kinematics as
 
begin mathsize 14px style omega subscript A superscript 2 space equals space 2 space alpha space theta end style
where α is angular acceleration

Using eqn.(2) , above expression is written as

begin mathsize 14px style fraction numerator x squared space v subscript B superscript 2 over denominator 27 space r squared end fraction space equals space 2 space alpha space pi over 6 space equals space alpha pi over 3 end style...............................................(3)
Above expression is simplified as
 
begin mathsize 14px style x squared space equals space alpha space fraction numerator 9 pi r squared over denominator v subscript B superscript 2 end fraction end style ............................(4)
Angular acceleration α is determined using Newtons law to the rotational motion of system
that consists of rod and particl-A about the pivot point O .
 
τ = r × T × sin [ (π/4) + (θ/2) ] = I × α 
 
where T is tension force that is giving torque τ and I is moment of inertia
 
r × T × sin [ (π/4) + (θ/2) ] = ( mA r2 )  × α  ............................ (5)
 
where mA is mass of particle-A
 
If we apply Newton's law of motion to the block-B, then we have
 
mB g - T = mB a 
 
T = mB ( g - a ) =   mB
Answered by Thiyagarajan K | 26 Aug, 2021, 07:42: PM

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