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A smooth circular track of mass M is vertically hung by a string down the ceiling. Two small rings, each of mass m, are initially at rest at the top of the track. They then slide down simultaneously along the track in opposite directions. Find the position of the rings when the tension in the string is zero. 

 

Asked by haroonrashidgkp 13th February 2018, 9:58 PM
Answered by Expert
Answer:
The problem is illustrated in the above diagram. Let M be the mass of circular track and m is mass of ring.
Tension T acting in the string is (M+2m)g. 
Since the rings are constrained to travel in circular path, there is a centripetal force due to the movement of each ring.
Let θ is the angle made by the line joining the centre point O and the left sliding ring to the vertical as shown in figure.
Force diagram is given for this particula moment in the right side of figure.
resolved horizontal componenets of centripetal forces cancel each other, while the vertical components added up and acting against tension.
 
To make the tension zero, the required condition is
 begin mathsize 12px style open parentheses M plus 2 m close parentheses g space equals space 2 m space v squared over R cos theta end style...........................(1)
where v is the tangetial speed of ring and R is radius of circular track
 
if R is circular radius, distance h = R(1-cosθ) 
since the rings are starting from rest  and if the vertical component of the speed is v×sinθ when they reach the angular position θ, then we have
 
begin mathsize 12px style v squared sin squared theta space equals space 2 g R left parenthesis 1 minus cos theta right parenthesis end style..............................(2)
substituting v2 from (2) in (1), we get

begin mathsize 12px style fraction numerator open parentheses M plus 2 m close parentheses over denominator 4 m end fraction space equals space fraction numerator open parentheses 1 minus cos theta close parentheses over denominator begin display style sin squared end style begin display style theta end style end fraction cos theta end style
 
Hence from the known values of M and m one can estimate the value of θ


Answered by Expert 15th February 2018, 3:53 PM
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