7.23 A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m².

(a) What is his new angular speed? (Neglect friction)

(b) Is kinetic energy conserved in the process? If not, from where does the change come about?

<p><strong> (a)</strong></p><p>Moment of Inertia of the man-platform system = 7.6 kg-m²</p><p>Moment of inertia when the man stretches his hands to a distance 90 cm</p><p>= 2 x mr² = 2 x 5 x (0.9)2 = 8.1 kg-m²</p><p>Initial moment of inertia of the system, &nbsp;<span title="Click to copy mathml"><math><mo>-</mo><mi></mi></math></span> = 7.6 + 8.1 = 15.7 kg-m²</p><p>Angular speed&nbsp;<span title="Click to copy mathml"><math><mo>-</mo><mi></mi></math></span>&nbsp;= 30 rev/min</p><p>Angular momentum, &nbsp;<span title="Click to copy mathml"><math><mo>-</mo><mi></mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub></math></span>&nbsp;= 15.7 x 30 …… (i)</p><p>Moment of inertia when the man folds his hands to a distance of 20 cm</p><p>= 2 x mmr²= 2 x 5 x (0.2)2 = 0.4 kg-m²</p><p>Final moment of inertia, &nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub></math></span> = 7.6 + 0.4 = 8 kg-m²</p><p>Final angular speed =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>L</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub></math></span>&nbsp;and final angular of momentum, &nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub><msub><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mi>f</mi></mrow></mrow></msub></math></span>&nbsp;= 8&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>f</mi></mrow></mrow></msub></math></span>&nbsp;…… (ii)</p><p>From the conservation of angular momentum, we have&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>L</mi></mrow></mrow><mrow><mrow><mi>f</mi></mrow></mrow></msub></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mi>f</mi></mrow></mrow></msub><msub><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>f</mi></mrow></mrow></msub></math></span></p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>f</mi></mrow></mrow></msub></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>I</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub><msub><mrow><mrow><mi>ω</mi></mrow></mrow><mrow><mrow><mi>i</mi></mrow></mrow></msub></math></span>&nbsp;= 58.88 rev/min</p><p>&nbsp;</p><p><strong> (b)</strong> Kinetic energy is not conserved in the given process. In fact, with the decrease in the moment of inertia, kinetic energy increases. The additional kinetic energy comes from the work done by the man to fold his hands towards himself.</p>

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