# What shapes have the greatest/smallest moment of inertia

Gold Member
I'm basically just trying to find a list of all the shapes and their relative moment of inertia to each other. I wanna see what shapes have more and what less, just to get a more intuitive sense to this subject. Does anyone show such a list, or maybe can just tell me of basic geometrical objects and their relative moment of inertia? I'll be happy to read about it!

## Answers and Replies

Gold Member
That's great, but is there a list with numbers rather than unknowns you now have to plug in numbers to and see? (math doesn't come that intuitive to me-- I have to plug in and calculate unlike you :P )

jhae2.718
Gold Member
It depends on the numbers.

As ILSe said, the larger the radius from your axis, the larger the moment of inertia for two objects of equivalent mass.

If you're bored, you could always do a little Monte Carlo.

I like Serena
Homework Helper
That's great, but is there a list with numbers rather than unknowns you now have to plug in numbers to and see? (math doesn't come that intuitive to me-- I have to plug in and calculate unlike you :P )

Hmm, most lists will include unknowns I think (typically mass and size).

But let me try this.
If you look at the second web link, you'll find for instance 2 pictures of a rod.

The first rod has the axis at its end.
The second rod has the axis in its center.

The moments of inertia are:

$$I_{end}=\frac {mL^2} 3$$

$$I_{center}=\frac {mL^2} {12}$$

Now we have 2 numbers: $\frac 1 3$ and $\frac 1 {12}$

Which one is greater? ;)

EDIT: Now think of twirling a rod in your hands.
What will take more effort?
Turning it with one end in your hands, or twirling it with the middle in your hands?

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If you take any object and ask which direction if rotated would it be most unstable, then that principal direction has the 2nd largest (or 2nd smallest - they're the same thing) moment of inertia of the three possible axis.

Take a human. If the human does a pirouette, that is stable. If a human does a cartwheel, that is stable. If a human falls on their face, that is unstable. So for a human, the axis that goes through his or her sides has the 2nd highest moment of inertia. The axis that goes through his or her head happens to have the lowest moment of inertia, and the axis that goes through his or her stomach has the highest moment of inertia.

Gold Member
Hmm, most lists will include unknowns I think (typically mass and size).
But let me try this.
If you look at the second web link, you'll find for instance 2 pictures of a rod.

The first rod has the axis at its end.
The second rod has the axis in its center.

The moments of inertia are:

$$I_{end}=\frac {mL^2} 3$$

$$I_{center}=\frac {mL^2} {12}$$

Now we have 2 numbers: $\frac 1 3$ and $\frac 1 {12}$

Which one is greater? ;)

Well, yea, but analyizing each and every one of them takes a lot of time. You'd think someone would've bothered to make a list without unknowns. But, oh well, maybe I myself will make that list some day.
EDIT: Now think of twirling a rod in your hands.
What will take more effort?
Turning it with one end in your hands, or twirling it with the middle in your hands?
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I think twirling it in the middle.

I like Serena
Homework Helper
I think twirling it in the middle.

I guess you're right. :)
Having seen those cheer leaders, it looks very hard to twirl them in the middle.