Moment of inertia vs 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.

 

[nlightner]

The moment of inertia of the fan will determine how quickly it slows down, as it is a measure of the fan's resistance to changes in its rotational motion. The higher the moment of inertia, the longer it will take for the fan to slow down. Friction, on the other hand, will act in the opposite direction, slowing down the fan's motion. However, since friction loss is being ignored in this scenario, it will not have a significant impact on the fan's speed.

To calculate the time it takes for the fan to slow down to half its original speed, you can use the equation for angular velocity: ω = ω0e^(-kt), where ω0 is the initial angular velocity and k is a constant determined by the moment of inertia and the work done by the fan. You can then solve for t when ω = ω0/2.

Alternatively, you can also use the equation for kinetic energy: E_k = 1/2 Iω^2, where I is the moment of inertia and ω is the angular velocity. You can set the initial kinetic energy (when the fan is disengaged from the motor) equal to half the final kinetic energy (when the fan has slowed down to half its original speed) and solve for t.

In conclusion, the moment of inertia and friction both play a role in determining the speed at which the fan will slow down. However, in this scenario where friction is being ignored, the moment of inertia will have a greater impact on the fan's speed.