5.40 A thin circular loop of radius R rotates about its vertical diameter with an angular frequency $\mathit{\omega }$ . Show that a small bead on the wire loop remains at its lowermost point for $\le \sqrt{\mathit{g}/\mathit{R}}$ . What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for $\mathit{}\mathit{\omega }$ = $\sqrt{2\mathit{g}/\mathit{R}}$ ? Neglect friction.

<p><picture><img src="https://images.shiksha.com/mediadata/images/articles/1728382192phpulIK4Q.jpeg" alt="" width="170" height="145"></picture></p><p>Let&nbsp;<span title="Click to copy mathml"><math><mi>θ</mi></math></span>&nbsp;be the angle made by the radius vector joining the bead and the centre of the wire with the downward direction. Let, N be the normal reaction.</p><p>mg = N&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span>&nbsp;…….(1)</p><p>mr&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;= N&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span>&nbsp;……(2)</p><p>m(R&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span>&nbsp;)&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;= N&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">sin</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span></p><p>Hence N = m(R)&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span></p><p>Substituting the value on N in eqn (1)</p><p>mg = mR&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span></p><p>or&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span>&nbsp;= g/ R&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;………(3)</p><p>As&nbsp;<span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></mrow></mrow></mfenced></math></span>&nbsp;<span title="Click to copy mathml"><math><mo>≤</mo></math></span>&nbsp;1, the bead will remain at the lowermost point</p><p>g/ R&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;<span title="Click to copy mathml"><math><mo>≤</mo><mn>1</mn><mi></mi><mi>o</mi><mi>r</mi><mi></mi><mi>ω</mi><mi></mi><mo>≤</mo><mi></mi><msqrt><mrow><mi>g</mi></mrow></msqrt><mo>/</mo><mi>R</mi></math></span></p><p>For&nbsp;<span title="Click to copy mathml"><math><mi>ω</mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><msqrt><mrow><mn>2</mn><mi>g</mi></mrow></msqrt></mrow></mrow><mrow><mrow><mi>R</mi></mrow></mrow></mfrac><mo>,</mo><mi></mi><mi>e</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi></mi><mfenced separators="|"><mrow><mrow><mn>3</mn></mrow></mrow></mfenced></math></span>&nbsp;becomes</p><p><span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span>&nbsp;= g/ R&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">cos</mi></mrow><mo>?</mo><mrow><mi>θ</mi></mrow></mrow></math></span>&nbsp;=(g/R)(R/2g) = ½</p><p><span title="Click to copy mathml"><math><mi>θ</mi><mo>=</mo><mn>60</mn><mo>°</mo></math></span></p>

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