Ratio of mass moment of inertia to mass

AI Thread Summary
The discussion revolves around calculating the resultant translational force of a wheel subjected to an applied force at the axle while considering its mass moment of inertia. Participants clarify that adding mass and moment of inertia is dimensionally incompatible, emphasizing the need to focus on energy and torque rather than direct force/acceleration calculations. The correct approach involves determining the torque necessary for rotation and the force needed for linear movement, leading to a formula that incorporates both translational and rotational dynamics. A participant reflects on their earlier calculations, realizing they were close to the correct solution but had made a minor error. The conversation highlights the complexity of the problem and the importance of understanding the relationship between linear and angular forces.
WildEnergy
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Hi

Thinking about a simple stable wheel at rest on a flat surface
with a force applied at the axle parallel to the ground
trying to work out the resultant translational force

(note: no slipping, static friction is very high, no rolling resistance)

I think it is the ratio between the mass moment of inertia (I) of the wheel
and the total inertia (translational and rotational)

resultant = force * mass / (mass + moi)
-- and --
friction = force * moi / (mass + moi)

is that correct?
 
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It is not clear what you are asking. What is 'resultant translational force'?

Passing over that for the moment, there is no physical meaning to the addition of mass and moment of inertia, the dimensions are totally incompatible.
 
AJ Bentley said:
It is not clear what you are asking. What is 'resultant translational force'?

some of the force is used up overcoming the rotational inertia - this is what remains?

AJ Bentley said:
Passing over that for the moment, there is no physical meaning to the addition of mass and moment of inertia, the dimensions are totally incompatible.

that is what I was afraid of

I am trying to work out the acceleration of a wheel with a force applied
thinking it would be less than f/m because of the angular momentum of rotating the wheel
 
This is actually quite a tricky problem (to me anyway!)

In this sort of situation, one normally abandons the force/acceleration route and concentrates instead on energy/work done considerations. But here, you are specifically asking for the force.

The only way I can see of solving it is to consider how much force you need to apply (at the axle) to provide the necessary torque to get the wheel rotating at the desired speed and to add to that the force required to also get the wheel moving linearly at the same speed. You'll need to bear in mind that the wheel velocity will be it's angular rotational speed over 2*pi*r.

Your initial gut-feeling about f=ma is correct.
 
I found this idea in a book:

F = original force, aa = angular acceleration, m = mass, r = radius, moi = moment of inertia

friction = F - translation
friction = F - m * a
friction = F - m * aa * r
torque =
friction * r =
r * (F - m * aa * r) = moi * aa
F * r - m * aa * r^2 = moi * aa
aa = F * r / (moi + m * r^2)

I was so damn close yesterday but my maths let me down
being bad at both maths and physics really sucks
 
after checking my original attempt I realized I was on the right track
but had made one small mistake, in fact my solution seems superior to the one written by nasa

ratio of forces = linear force / angular force
ratio of forces = mass * radius * radius / moment of inertia

simple!
 

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