Moment of inertia

[glass sphinx]

A cylinder spinning with any line joining 2 points on its 2 faces' circumference as axis.

[teal cloak]

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[glass sphinx]

density $\rho$

[ornate shale]

Coffeycharas Chichundarcharan

[glass sphinx]

helpt meh.

[cerulean smelt]

...with? You haven't actually said what you're supposed to find.

[glass sphinx]

"Moment of inertia"

[glass sphinx]

as the picture describes

"A cylinder spinning with any line joining 2 points on its 2 faces' circumference as axis." i forgot to mention it was homogenous sorry about that

[cerulean smelt]

...look, this'll probably go a lot smoother if I can just see the question as originally written.

[glass sphinx]

There is none I made it up

wait I'll tell you my statement in a less handwavy way

[cerulean smelt]

Wouldn't it partially depend on, like, the location and angle of the axis of rotation?

[glass sphinx]

We have a cylinder of height "h" and base radius "r". We draw a line passing through the center of mass of this very cylinder such that the line touches the circumference of the faces of the cylinder above and below. What's the moment of inertia of this cylinder along this axis?

[glass sphinx]

yes that's the main factor differentiating this from the rest of the MOIs

[cerulean smelt]

Okay, see, before you said any line joining any two points on the circumferences.

[glass sphinx]

Yes, that was my fault

[cerulean smelt]

Looking at the Wikipedia page for moment of inertia, this seems like maybe something you could solve with an integral? Maybe a double integral.

[foggy totem]

Lol

[glass sphinx]

yes.

[foggy totem]

So what is your problem

[glass sphinx]

I think I'll need some help with what double integral we talking about

[ocean saffron]

This is what I have to put up with when coffee presents a new riddle to me

[glass sphinx]

it tilted

[foggy totem]

Lmao

[glass sphinx]

I smell like whiteboard marker ink all over

[cerulean smelt]

It's also what I have to put up with whenever anyone tries to paraphrase the problem they're working on instead of just taking a picture of it. It's very frustrating.

And very common. Frustratingly common.

[ocean saffron]

https://tenor.com/view/monsters-inc-boo-hug-big-hug-gif-24350322

[glass sphinx]

Sir it's called a "schizo-rant" by our terminology

[cerulean smelt]

...feels ableist.

[glass sphinx]

can you give me some insight on how to form the integrals

i thought of integrating along the Axis of rotation (I = / dI)

but it's kinda weird how it's all set up

[cerulean smelt]

Wikipedia says the moment of inertia for a point mass is the mass times the square of the distance to the axis, and the moment of inertia for a rigid body is just the sum of the moments of inertia of the point masses.

[foggy totem]

Better: Just "rant"

[glass sphinx]

yes that's the definition.

[glass sphinx]

absta hepl.

[foggy totem]

Yea I can Hepl

[glass sphinx]

technically speaking

[foggy totem]

How hepl

[glass sphinx]

wait lemme draw it for you

[foggy totem]

,w hepl

[ornate shale]

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Requested by abstia

[foggy totem]

You can just follow this construction btw

Integrate over the volume

[glass sphinx]

see this would be easier by a ton, I'd call it trivial

but

the bottom bits

i don't know how to work with those

[cerulean smelt]

...it's a cylinder. So it's symmetrical. And the axis of rotation passes through the center. So... could the MoI just be 0?

[glass sphinx]

No

Its kind of mass but for rotation

but that's the physics part

we can get the I for the whole body by integrating over elemental parts perpendicular to the axis of rotation

[cerulean smelt]

...right, sorry. I got distracted by thinking about integration, and in particular how a negative value of a function subtracts from integration.

[glass sphinx]

the bounds are generally of the form a and -a here too

[cerulean smelt]

So I thought an MoI of a point mass on the left of the axis would be canceled out by an identical MoI on the right.

[foggy totem]

Just integrate

You need to use your hand

This is not a problem you can hand-waive easily

[jolly pasture]

$\int r^2 dm$

[ornate shale]

pratham

[glass sphinx]

oh no

[glass sphinx]

how does that help

[foggy totem]

Lol

[glass sphinx]

@jolly pasture

[foggy totem]

You just compute it

[jolly pasture]

thats the formula

[foggy totem]

You know how to represent the 3-form dm

[jolly pasture]

wait i missed a term ig

[glass sphinx]

3 form what

[glass sphinx]

don't insert dr

you got it right

[foggy totem]

Differentual form of order 3

[glass sphinx]

nope

[jolly pasture]

its ryt

it has been a while so ..

ok

need to divide the figure into small parts

[foggy totem]

You may also use a density form @glass sphinx the result will not differ

[glass sphinx]

o0 that's doable

I'll let you know if i manage to do it

+close