How Does Moment of Inertia Affect Fan Deceleration When Ignoring Friction?

[kenlai]

Homework Statement

If a fan is disengaged from the motor, how long does it take for it to slow down to 1/2 of its original speed. Assuming the only work done during coasting is to accelerate the air the fan pulls into its exit velocity. Friction loss due to bearings, etc will be ignored.

Homework Equations

Energy of Rotor = E_k=1/2 Iω^2 where I=Inertia of rotor, ω=angular velocity
Work done by fan = E_d= 1/2 mv^2 where m = mass of air being blown out, v = air exit velocity.
Each revolution of the fan rotor moves a fixed volume of air, so v is a fuction of ω.
m is a fuction of ω and time. ∆m = kω∆t, k is a constant

The Attempt at a Solution

Rate change of energy of rotor = ∆(Work done by fan) =1/2 ∆mv^2
How do I proceed?

 

[LowlyPion]

Welcome to PF.

Figure that the force is proportional to the volume of air moved and hence the f is proportional to velocity. I would look to construct an integral that takes into account the variable force.