5.31 A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is 10⁶ kg.

What provides the centripetal force required for this purpose — The engine or the rails ? What is the angle of banking required to prevent wearing out of the rail ?

<p>Given</p><p>Radius, r = 30m</p><p>Speed, v = 54 km/h = 15 m/s</p><p>The mass of the train, m = 106 kg</p><p><strong> (a)</strong> The required centripetal force is provided by the rails to the wheels of the train</p><p><strong> (b)</strong> The angle of banking required to prevent wearing out of the rails</p><p><span title="Click to copy mathml"><math><mrow><mrow><mi mathvariant="normal">tan</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>v</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;/ rg =&nbsp;<span title="Click to copy mathml"><math><msup><mrow><mrow><mn>15</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;/ (30&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mn>10</mn><mo>)</mo></math></span></p><p><span title="Click to copy mathml"><math><mi>θ</mi></math></span>&nbsp;= 37&nbsp;<span title="Click to copy mathml"><math><mo>°</mo></math></span></p>

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