A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity ω. If two objects each of mass m be attached gently to the opposite ends of a diameter of the ring. the ring will then rotate with an angular velocity

$fraction numerator straight omega space left parenthesis straight M space minus 2 straight m right parenthesis over denominator straight M space plus space 2 straight m end fraction$
$fraction numerator ωM over denominator straight M space plus space 2 straight m end fraction$
$fraction numerator straight omega left parenthesis straight M space plus 2 straight m right parenthesis over denominator straight M end fraction$
$fraction numerator ωM over denominator straight M plus straight m end fraction$

Solution

fraction numerator ωM over denominator straight M space plus space 2 straight m end fraction

Applying law of conservation of angular momentum.
I₁ω₁ = I₂ω₂
In the given case
I₁ = MR²
I₂ = MR² + 2mR²
ω₁ =ω
then ω₂ = I₁ω /I₂ = Mω / M +2m

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