# Calculating the Rotational Inertia

## Homework Statement

A rigid structure consisting of a circular hoop of radius R and mass m, and a square made of four thin bars, each of length R and mass m. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of 4.0 s. Assume R = 0.90 m and m = 3.0 kg, calculate the structure's rotational inertia about the axis of rotation.

## Homework Equations

I_rods = (1/12)ML^2
I_hoop = (1/2)MR^2
I = I_rods + I_hoop

## The Attempt at a Solution

The third equation is constructed by noticing that the problem states that the structure consists of both the square made of thin rods and the hoop. The square only has two rods that have moments of inertia because two are perpendicular to the axis and the other two are parallel, which have a moment of inertia of 0 (at least that is what I have been told). So...

I = 2[I_rods] + I_hoop
I = 2[(1/12)(3 kg)(0.90 m)^2] + (1/2)(3 kg)(0.90 m)^2
I = 2.835 kg*m^2

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Doc Al
Mentor
I_rods = (1/12)ML^2
I_hoop = (1/2)MR^2
These formulas assume a perpendicular axis through their centers. But where's the axis of rotation in this problem?

A diagram would help.

Doc Al
Mentor
OK. Make use of the parallel axis theorem to find the moments of inertia about the axis of rotation for the rods (two of them, at least) and the hoop.

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The Parallel Axis Theorem states:

I = I_com + Md^2

So the rod furthest to the left will be have a moment of inertia of:

I = (1/12)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2

And the hoop's moment of inertia will be:

I = (1/2)(3 kg)(2*0.09 m)^2 + (3 kg)(0.09 m)^2

Then the solution will be the summation of those equations? Are these right?

Doc Al
Mentor
The Parallel Axis Theorem states:

I = I_com + Md^2
OK.
So the rod furthest to the left will be have a moment of inertia of:

I = (1/12)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2
No. The rod furthest to the left has all its mass equidistant from the axis.

But what about the two horizontal rods?

And the hoop's moment of inertia will be:

I = (1/2)(3 kg)(2*0.09 m)^2 + (3 kg)(0.09 m)^2
Why is there a 2 in there?

Ok, I don't why I put a 2*0.90 in the hoop's moment of inertia equation. That was a mistake. As for the horizontal rods, I have no idea and I need some explanation of them and what to do for them because I thought they had no moment of inertia. The rod furthest to the left must be then I = (1/12)(3 kg)(0.09 m)^2, but why not include its distance from the axis?

Doc Al
Mentor
As for the horizontal rods, I have no idea and I need some explanation of them and what to do for them because I thought they had no moment of inertia.
Realize that the rods are being rotated about a vertical axis. The equation you have, I_rods = (1/12)ML^2, can be used for a horizontal rod rotating about a vertical axis (through its center of mass). (Use the parallel axis theorem to move the axis.)
The rod furthest to the left must be then I = (1/12)(3 kg)(0.09 m)^2, but why not include its distance from the axis?
Just the opposite: you must include the distance from the axis. Trick question: What's the rotational inertia of a rod about an axis parallel to the rod (and through its center of mass)? Hint: Only three of the rods contribute to the total rotational inertia.

What's the rotational inertia of a rod about an axis parallel to the rod (and through its center of mass)? Hint: Only three of the rods contribute to the total rotational inertia.
I = 0, because the distance away from the axis is 0, thus I = Md^2 = 0

Doc Al
Mentor
I = 0, because the distance away from the axis is 0, thus I = Md^2 = 0
Exactly. With the left rod oriented parallel to the axis, the only thing that counts is its distance from the axis. (If its mass were concentrated at a single point, its I would be the same.)

Ok, so...

I_parallelRod = (3 kg)(0.09 m)^2

The parallel rod is the length of the rod away from the axis.

I_hoop = (1/2)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2

The hoop is the length of the radius away from the axis.

So the summation of these moments of inertia is:

I = I_horizontalRods + I_parallelRod + I_hoop

But I don't understand what you are getting at by saying the the rod furthest to the left is equidistant to the axis... how does that change the previous formula I posted?

Why is this still wrong?

Doc Al
Mentor
Ok, so...

I_parallelRod = (3 kg)(0.09 m)^2

The parallel rod is the length of the rod away from the axis.

I_hoop = (1/2)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2

The hoop is the length of the radius away from the axis.

So the summation of these moments of inertia is:

I = I_horizontalRods + I_parallelRod + I_hoop
All good. But what about the horizontal rods? What did you use for their rotational inertia?
But I don't understand what you are getting at by saying the the rod furthest to the left is equidistant to the axis... how does that change the previous formula I posted?
Realize that the formula for the vertical rod at distance R from the axis is the same as the formula for a point mass a distance R from the axis.

Oops I meant to say what is wrong with my horizontal equation that I first posted?

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Doc Al
Mentor
Oops I meant to say what is wrong with my horizontal equation that I first posted?
Tell me exactly which one. (Quote it.)

Sure...

So the rod furthest to the left will be have a moment of inertia of:

I = (1/12)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2
No. The rod furthest to the left has all its mass equidistant from the axis.

Doc Al
Mentor
The rod furthest to the left is a vertical rod, right?

If you mean that to appy to the horizontal rods, what's the distance between their centers and the axis of rotation?

Yes, it is a vertical rod. The furthest rod has a distance of 0.90 m away from axis

Doc Al
Mentor
Yes, it is a vertical rod. The furthest rod has a distance of 0.90 m away from axis
I thought we finished with the vertical rods in post #11? (Vertical = parallel to the axis.)

You might want to summarize everything that your adding to the mix, so we get it straight once and for all.

Sorry, I've been confusing myself and you over this problem.

I_parallelRod = (3 kg)(0.09 m)^2

I_hoop = (1/2)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2

I_horizontalRod = (1/12)(3 kg)(0.09 m)^2 - (3 kg)(0.09 m)^2

The horizontal rods have a moment of inertia of (1/12)(3 kg)(0.09 m)^2, but it is shifted the left of the axis by the length of a rod.

Thus...

I = I_parallelRod + I_hoop + I_horizontalRod

This is everything wrapped up to now, I apologize for running us in circles.

Doc Al
Mentor
Sorry, I've been confusing myself and you over this problem.
I'm easily confused.

I_parallelRod = (3 kg)(0.09 m)^2
Good. This applies to the left vertical rod. (What about the right one?)

I_hoop = (1/2)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2
Good.

I_horizontalRod = (1/12)(3 kg)(0.09 m)^2 - (3 kg)(0.09 m)^2

The horizontal rods have a moment of inertia of (1/12)(3 kg)(0.09 m)^2, but it is shifted the left of the axis by the length of a rod.
Careful. (1/12)(3 kg)(0.09 m)^2 is the rotational inertia for a horizontal rod rotating about a vertical axis through its center. What does the parallel axis theorem tell you to add to that to get its rotational inertia about the required axis? (Left and right don't matter--all that matters is the distance from cm to the axis.)

Thus...

I = I_parallelRod + I_hoop + I_horizontalRod
Be sure to account for all four rods. (One will be trivial.)

So if left or right of the axis does not matter then according to the Parallel Axis Theorem (I = I_com + Md^2) the (3 kg)((0.09 m)/2)^2 will be added to the moment of inertia, not subtracted. The 0.09 m is divided by two in the equation because the center of mass has only shifted half of the length of the rod, not the full length like I previously had...

I_horizontalRod = (1/12)(3 kg)(0.09 m)^2 + (3 kg)((0.09 m)/2)^2

I_parallelRod = (3 kg)(0.09 m)^2

I_hoop = (1/2)(3 kg)(0.09 m)^2 + (3 kg)(0.09 m)^2

And the final equation will be represented in the following way:

I = I_parallelRod + I_hoop + 2[I_horizontalRod]

I think that may be right...?

Doc Al
Mentor
I think you've got it now!

It is still wrong! My final answer is 0.07695 kg*m^2. http://www.webassign.net/hrw/hrw7_11-45.gif do I have to change the hoop's equation at all because the diameter is 2R... I don't know anymore. I thought I had it. Unless my calculations are incorrect, but I have checked them numerous times. If anyone else could just try to plug and chug and see what they get. I really want to solve this because it doesn't seem that hard now.

Doc Al
Mentor
My final answer is 0.07695 kg*m^2.
I get the same answer. Realize that webassign can be very picky about the number of significant figures you enter. Round it off.

I can't seem to get it no matter what I try.

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