Ratio of mass moment of inertia to mass

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Discussion Overview

The discussion revolves around the relationship between the mass moment of inertia and the total inertia of a wheel subjected to a force at its axle. Participants explore the implications of this relationship on the resultant translational force and the acceleration of the wheel, considering both translational and rotational dynamics.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes that the resultant translational force can be expressed as a ratio involving mass moment of inertia (I) and total inertia, suggesting a formula: resultant = force * mass / (mass + moi).
  • Another participant questions the validity of adding mass and moment of inertia, stating that their dimensions are incompatible.
  • Some participants express uncertainty about the concept of 'resultant translational force' and discuss the need to account for rotational inertia when calculating acceleration.
  • A participant suggests that energy and work considerations might be more appropriate for solving the problem than a direct force/acceleration approach.
  • One participant shares a derived equation relating force, mass, moment of inertia, and angular acceleration, indicating a realization of being close to a correct solution despite earlier mistakes.
  • Another participant claims to have found a simpler solution, asserting a ratio of forces that relates linear and angular forces, though this claim is not universally accepted.

Areas of Agreement / Disagreement

Participants express differing views on the validity of combining mass and moment of inertia, and there is no consensus on the best approach to calculate the resultant translational force or acceleration of the wheel. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight limitations in their understanding of the relationship between translational and rotational dynamics, with some acknowledging mistakes in their mathematical reasoning. The discussion reflects varying levels of confidence in the proposed solutions and approaches.

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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