# Ratio of mass moment of inertia to mass

1. Jul 24, 2010

### WildEnergy

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?

2. Jul 24, 2010

### AJ Bentley

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.

3. Jul 24, 2010

### WildEnergy

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

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

4. Jul 24, 2010

### AJ Bentley

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.

5. Jul 24, 2010

### WildEnergy

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

6. Jul 26, 2010

### WildEnergy

after checking my original attempt I realised 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!