Calculating Motor Force & Torque

[gawlicbread]

Homework Statement

An airplane weighing 10,000 kg is at rest but needs to accelerate to 25 k/h, propelled by an electric motor in its landing gear. The wheel of the landing gear measures approximately 25 cm in diameter. Assume that there is no wind, the coefficient of friction for the asphalt is 0.75, and the surface is level.

1a/b. Without considering the landing gear, how much force (N) and torque (Nm) is needed to break the aircraft away from rest assuming a dead load?
2a/b. Considering the landing gear, how much force (N) and torque (Nm) is needed to break the aircraft away from rest?
3a/b. How much force (N) and torque (Nm) is needed to accelerate the aircraft to 25 k/h?

Relevant Equations

1. T = F*a
2. w = m*g
3. F = m*a
4. tau (torque) = I (moment of inertia) *alpha (angular acceleration)
5. tau (torque) = F (force) * r (radius / lever length)

Hello All,

I'm trying to figure out the above problem but don't know how to set it up and could use some help.
For starting with the first question, I was thinking of doing PE = KE:

• PE = mgh = (10,000)*(9.81)*(1) = 98,100 kJ
• KE = 0.5m (v^2) = 0.5 (10,000)* (v^2)
• Sqrt(98,100 / 5,000) = 4.43 m/s
• F = ma = 10000 * (4.43-0)/1 = 44,300 kN ==> Answer for #1a (Force)
• T = F*a = 44,300 * (4.43-0)/1 = 196,249 kNm ==> Answer for #1b (Torque) ???

That just doesn't seem right. It seems like I'm missing something.

After that, I'm lost for 2a/b and 3a/b.

Any advice?

 

[erobz]

Homework Statement: An airplane weighing 10,000 kg is at rest but needs to accelerate to 25 k/h, propelled by an electric motor in its landing gear. The wheel of the landing gear measures approximately 25 cm in diameter. Assume that there is no wind, the coefficient of friction for the asphalt is 0.75, and the surface is level.

1a/b. Without considering the landing gear, how much force (N) and torque (Nm) is needed to break the aircraft away from rest assuming a dead load?

I was thinking of doing PE = KE:

Just for starters...Why?

Secondly, I think I'm dumfounded with these problem statements and I'm going to defer to someone else to decipher it. They seem ill conceived to me, but it also could be lack of cognition on my part.

 

[gawlicbread]

@erozb - My initial thoughts were translating from zero motion (potential) to dynamic motion (kinetic). Seemed like a logical choice to me, but then again, I'm a noob and am learning. I'm not sure what the issue with the problem statements are. Please let me know, but this is basically a copy/paste.

 

[erobz]

@erozb - My initial thoughts were translating from zero motion (potential) to dynamic motion (kinetic). Seemed like a logical choice to me, but then again, I'm a noob and am learning.

Don't take it personally, but that what we call equation grabbing. The potential you are referring to in the equation is gravitational PE. I think we can safely rule out gravity doing work on the airplane in a flat tarmac.

I'm not sure what the issue with the problem statements are.

Here seems to be too much ambiguity to be overcome without a reworking of the problem. For instance, you could accelerate from $0$ to $25 \frac{ \text{km}}{\text{h}}$ in $10 \text{s}$ or maybe in $100 \text{m}$, it could be acceleration as some function of position or time?

I think there are other ambiguities, but let's just start there. Maybe someone will show up and see something I'm not seeing in the meantime.

Please let me know, but this is basically a copy/paste.

Is it basically copy and paste or is it literally copy and paste, sometimes there is a difference?

 

[gawlicbread]

@erozb - I didn't consider it equation grabbing. I guess I didn't consider PE to be purely gravitational. I'm wondering what the formula would be for non-gravitational then? Like I said, I'm learning, so I appreciate helping me out with the train of thought.

The 0 to 25 kph didn't provide a specified length. If I understand you correctly, we could use any of the following as a guideline: a length of the tarmac before hitting 25 kph, a desired rate of acceleration, or a desired timeframe for achieving the speed. Unfortunately, I have none of those.

It's a copy/paste with the exception that the coefficient of friction wasn't listed in the problem statement, but there's a table there with the coefficients and the value listed corresponds to asphalt.

 

[erobz]

@erozb - I didn't consider it equation grabbing. I guess I didn't consider PE to be purely gravitational. I'm wondering what the formula would be for non-gravitational then? Like I said, I'm learning, so I appreciate helping me out with the train of thought.

Work is the concept you could be digging for...I'm not sure its relevant since the wheels aren't to be considered in (1). A torque provided to the wheel, through some angle of wheel rotation (assuming no-slip condition is met) can change the kinetic energy of the plane, but not necessarily in general.

The 0 to 25 kph didn't provide a specified length. If I understand you correctly, we could use any of the following as a guideline: a length of the tarmac before hitting 25 kph, a desired rate of acceleration, or a desired timeframe for achieving the speed. Unfortunately, I have none of those.

Yeah, that's basically the issue I was pointing out with that one. It doesn't have to be exactly any of those, but in a basic level problem I would expect it to be one of them.

It's a copy/paste with the exception that the coefficient of friction wasn't listed in the problem statement, but there's a table there with the coefficients and the value listed corresponds to asphalt.

I don't know what to tell you. I'm strongly put off...but you should wait for other opinions. Many times, others see through the ghosts I see.

 

[Tom.G]

Since friction was included in the question, the author likely wanted you to use it somehow.

Does it make sense to treat the friction as static (break-away) friction to start movement, and then treat it as rolling friction when the airplane is moving?

In the real world they would not be the same, but that is what has been given.

Cheers,
Tom
p.s. In any case it is confusing, so please let us know how your instructor expected it to be done.

 

[gawlicbread]

Thank you for the feedback folks! I'll send some questions to the instructor to gather their insights and report back.

 

[haruspex]

Yes, the question is flawed in many ways.

- What does tarmac "friction" refer to here?
If it is static friction, just as well there is some or the plane will never move, just spin its wheels. Indeed, if we knew how quickly it needed to reach the target speed then we could calculate a minimum necessary static friction.
If kinetic friction, why is the plane skidding?
That leaves rolling resistance, but that arises from deformation of the surfaces, tyre and tarmac. Unless it is a very hot day, nearly all the deformation will be in the tyres, making the tarmac's contribution to rolling resistance negligible. Either way, 0.75 is huge. For a plane on an airstrip, a total (tyre+surface) of 0.01 to 0.04 would be expected.
See 4.1 in https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/ for more on the subject.

- Both questions 1 and 2 ask for both torque and force. In question 1, the landing gear is to be ignored, so there is no radius value to convert between torque and force. It would make a little more sense if 1 were to ask for force and 2 for torque.

- Since the friction/resistance is associated with the landing gear, Q1 is nonsensical in saying it is to be ignored. I suggest it means that in Q1 the wheel radius is ignored.

- As others have noted, once the force/torque are sufficient to get it moving 25k/h can be achieved given enough time and distance. I suspect it was intended to provide a distance.

 

[erobz]

Maybe the intent was to calculate peak acceleration based on the static friction limit? If so we need some information on how the weight of the plane is distributed on the landing gear (there are a few sets to consider) as far as I can see.

I suspect you were to ignore the mass (and/or rotational inertia) of the wheels (under consideration) w.r.t. the airplane mass throughout the problem.

If @haruspex wasn't up to the task of trying to salvage it, it's cringeworthy!