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A uniform disc of mass M and radius R is free to rotate about a axis O passing through its rim. An insect of mass m at point A such that the line OA is the diameter of the disc as shown in figure. The insect describes a complete circle relative to disc and return to starting point A. Calculate the angle moved by the disc relative to ground.
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Asked by dasherbro123 | 16 Apr, 2020, 07:49: PM
answered-by-expert Expert Answer
Let ωD be the angular velocity of disc and ωi be the angular velocity of insect as shown in figure.
 
In absence of external torque, angular momentum is conserved.
 
Hence , ID ωD = Ii ωi  ......................(1)
 
where ID is moment of inertia of disc about an axis perpendicular to its plane and passing through O.
 
We have, ID = (1/2)MR2 + MR2 = (3/2)MR2
 
When the insect moving along the rim of disc and coming to same starting point after time T,
let θ be the angular displacement of disc.
 
Then angular displacement of insect (2π - θ )
 
Hence eqn.(1) is rewritten as
 
begin mathsize 14px style 3 over 2 M R to the power of 2 to the power of space end exponent cross times fraction numerator left parenthesis 2 pi space minus space theta right parenthesis over denominator T end fraction space equals space m R squared space theta over T end style
From above equation, we get begin mathsize 14px style theta space equals space fraction numerator 4 m over denominator left parenthesis space 3 M space plus space 2 m right parenthesis end fraction pi end style
Answered by Thiyagarajan K | 16 Apr, 2020, 11:00: PM
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