Yomega (\"The yo-yo with a brain\") is constructed with a clever clutch mechanis

Question

Yomega ("The yo-yo with a brain") is constructed with a clever clutch mechanism in its axle that allows it to rotate freely and "sleep" when its angular speed is greater than a certain critical value. When the yo-yo's angular speed falls below this value, the clutch engages, causing the yo-yo to climb the string to the user's hand. If the moment of inertia of the yo-yo is 6.1 10-5 kg·m2, its mass is 0.15 kg, and the string is 1.0 m long, what is the smallest angular speed that will allow the yo-yo to return to the user's hand?

_________rad/s

Explanation / Answer

Using the law of conservation of energy    m g h = (1/2)I 2 +(1/2) m V2    m g h = (1/2)[(1/2)mR2] (V/R)2 + (1/2) mV2               [ V = R ]    V = (4gh /3 )        = 3.61 m/s The radius        R = (2I / M) =0.02851 m The angular speed      = V /R = 126.75 rad/sec. Using the law of conservation of energy    m g h = (1/2)I 2 +(1/2) m V2    m g h = (1/2)[(1/2)mR2] (V/R)2 + (1/2) mV2               [ V = R ]    V = (4gh /3 )        = 3.61 m/s The radius        R = (2I / M) =0.02851 m The angular speed      = V /R = 126.75 rad/sec.

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