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

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SUMMARY

This discussion focuses on calculating the deceleration time of a fan when disengaged from its motor, specifically determining how long it takes to reduce its speed to half of its original velocity. The key equations involved are the energy of the rotor, expressed as E_k = 1/2 Iω^2, and the work done by the fan, represented as E_d = 1/2 mv^2. The relationship between the mass of air moved and angular velocity is critical, with the mass changing over time as ∆m = kω∆t. The discussion emphasizes the need to construct an integral to account for variable force in the deceleration process.

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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?
 
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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.
 

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