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

In summary, the problem concerns how long it takes for a disengaged fan to slow down to half of its original speed, assuming that the only work done during coasting is to accelerate the air the fan pulls into its exit velocity and that friction loss due to bearings is ignored. The relevant equations involve the energy of the rotor and the work done by the fan, with the latter being dependent on the mass of air being blown out and the air exit velocity. The rate of change of energy of the rotor is also considered, with the problem ultimately requiring the construction of an integral that takes into account the variable force of the air being moved.
  • #1
kenlai
1
0

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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  • #2
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.
 
  • #3


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.
 

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

1. What is moment of inertia vs friction?

Moment of inertia and friction are both concepts related to the motion of objects. Moment of inertia refers to an object's resistance to changes in its rotational motion, while friction is the force that opposes the motion of an object.

2. How are moment of inertia and friction related?

Moment of inertia and friction are related in that they both affect an object's motion. The moment of inertia of an object can impact how much friction it experiences, and the amount of friction can also affect an object's moment of inertia.

3. How do you calculate moment of inertia and friction?

Moment of inertia is calculated by multiplying the mass of an object by the square of its distance from its axis of rotation. Friction is typically calculated using the equation F = μN, where μ is the coefficient of friction and N is the normal force.

4. What are some real-life examples of moment of inertia and friction?

An example of moment of inertia is when a figure skater pulls in their arms to spin faster. This decreases their moment of inertia, allowing them to rotate faster. An example of friction is when a car's tires grip the road to prevent it from sliding.

5. How can moment of inertia and friction be manipulated?

Moment of inertia and friction can be manipulated in various ways. For example, moment of inertia can be changed by altering an object's shape or mass distribution. Friction can be changed by adjusting the surface properties or using lubricants to reduce friction.

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