A wheel is rotating freely with an angular speed of 800 rev/min on a shaft whose

Question

A wheel is rotating freely with an angular speed of 800 rev/min on a shaft whose rotational inertia is negligible. A second wheel, initially at rest and twice the rotational inertia of the first, is suddenly coupled to the same shaft. We have to find (a) the angular speed of the resultant combination of the shaft and the two wheels; (b) the fraction of the original rotational kinetic energy lost in the process.

Explanation / Answer

Given that w1 = 800 rev/min I2 = 2 I1 a) From conservation of angular momentum we have I1 w1 = (I1 + I2) w2 ==> w2 = I1 w1 / (I1 + I2) = 800 * I1 / 3I1 = 266.66 rev/min b) Lost in KE = (Ki - Kf ) / Ki = (0.5 I1w1^2 - 0.5(I1+ I2) w2^2 ) / 0.5 I1 w1^2 = ( I1w1^2 -3 I1 w2^2 ) / I1 w1 = (w1^2 - 3w2^2) / w1^2 = (800^2 - 3 * 266.66^2) / 800^2 = 0.6666 = 2 / 3

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