5.39 A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis with 200 rev/min. The coefficient of friction between the wall and his clothing is 0.15. What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed ?

<p><picture><img src="https://images.shiksha.com/mediadata/images/articles/1728381887phpa2ny18.jpeg" alt="" width="165" height="212"></picture></p><p>Mass of the man, m = 70 kg</p><p>Radius of the drum, r = 3 m</p><p>Coefficient of friction between the wall and his clothing, &nbsp;<span title="Click to copy mathml"><math><mi>μ</mi></math></span>&nbsp;= 0.15</p><p>Number of revs of hollow cylindrical drum = 200 rev/min = 200/60 rev/s = 3.33 rev/s</p><p>The centripetal force required is provided by the normal N of the wall on the man</p><p>N = m&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi></mi></mrow></mrow></msup><mo>/</mo><mi>R</mi></math></span>&nbsp;= m&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;R</p><p>When the floor revolves, the man sticks to the wall of the drum. Hence, the weight of the man (mg) acting downwards is balanced by the frictional force acting vertically upwards.</p><p>The man will not fall, if</p><p>mg&nbsp;<span title="Click to copy mathml"><math><mo>≤</mo><mi>μ</mi><mi>N</mi></math></span></p><p>mg&nbsp;<span title="Click to copy mathml"><math><mo>≤</mo><mi></mi><mi>μ</mi><mo> (</mo><mi>m</mi><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mi>R</mi></math></span>&nbsp;)</p><p><span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>≥</mo><mi>g</mi><mo>/</mo><mi>R</mi><mi>μ</mi></math></span></p><p><span title="Click to copy mathml"><math><mi>g</mi><mo>/</mo><mi>R</mi><mi>μ</mi></math></span>&nbsp;= 10/ (3&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mi></mi></math></span>&nbsp;0.15)</p><p><span title="Click to copy mathml"><math><mi>ω</mi><mo>=</mo><mn>4.71</mn><mi></mi><mi>r</mi><mi>a</mi><mi>d</mi><mo>/</mo><mi>s</mi></math></span></p>

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