A disk with moment of inertia I 1 rotates about a frictionless, vertical axle wi

Question

A disk with moment of inertia I1 rotates about a frictionless, vertical axle with angular speed ?i. A second disk, this one having moment of inertia I2 and initially not rotating, drops onto the first disk (figure). Because of friction between the surfaces, the two eventually reach the same angular speed ?f.

(a) Calculate ?f. (Use any variable or symbol stated above as necessary.)

?f = ?

(b) Calculate the ratio of the final to the initial rotational energy. (Use any variable or symbol stated above as necessary.)

Ki = ?

Kf = ?

Kf/Ki = ?

Explanation / Answer

initially, L = I1 * i
Later, L = (I1 + I2)f = I1 * i
f = i * I1 / (I1 + I2)

KEi = ½(I1)(i)²
KEf = ½(I1 + I2)(f)² = ½(I1 + I2)(i)²(I1)² / (I1 + I2)² = (i)²(I1)²/(2(I1 + I2)

KEf/KEi = I1 / (I1 + I2)

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