A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 rads^-1 in a horizontal plane about an axis vertical to its plane and passing through the centre of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the will then rotate with an angular velocity (in rads^-1).

Correct Option - 3


Detailed Solution:

By conservation of Angular momentum

Li = Lf 

$M{R}^2\omega =\left(m{R}^2+2m{R}^2\right)\omega \text{'}$

$\frac{2M}{M+2m}=\omega \text{'}$

$\Rightarrow$ $\omega \text{'}=\frac{2M}{M+2m}$

<p>By conservation of Angular momentum</p><p>Li = Lf&nbsp;</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1758002768phpi5d5u4.jpeg" alt="" width="279" height="104" loading="lazy"></picture></div></div><p><span contenteditable="false"> <math> <mrow> <mi>M</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>ω</mi> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mi>m</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>m</mi> <msup> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ω</mi> <mo>'</mo> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> <mo>=</mo> <mi>ω</mi> <mo>'</mo> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <mo>⇒</mo> </mrow> </math> </span> <span contenteditable="false"> <math> <mrow> <mi>ω</mi> <mo>'</mo> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>M</mi> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </math> </span></p>

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