When to use 1/2 mr² vs mr² for moment of inertia?

[Zheng_]

Homework Statement

I was learning how to calculate moment of inertia on Youtube, and I'm confused about when to use 1/2 mr² instead of mr².


This video says that he used 1/2 mr² instead of mr² because the cone is solid

But in this video, while he was calculating a solid sphere,

he used mr².

Can someone explain to me why and when should I use mr² please?

Homework Equations

I=mr²
I=mr²

The Attempt at a Solution

Maybe it's because he was trying to calculate the solid sphere by separate it into four pieces and add them together after it? But it's still solid doesn't it?

 

[lekh2003]

Maybe it's because he was trying to calculate the solid sphere by separate it into four pieces and add them together after it? But it's still solid doesn't it?

The calculations for moment of inertia are different for different mass concentrations. You might see different equations depending on the type of shape and other contributing factors. Look at this link: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

Look at the common moments of inertia section.

 

[vela]

For the cone, the calculation is being done by leveraging earlier results. The cone is being divided it into circular slices. Because the cone is solid, each slice is a disk, which is why the factor of 1/2 is there. If the cone were hollow, each slice would be a ring, so you wouldn't have a factor of 1/2.

 

[haruspex]

in this video, while he was calculating a solid sphere,

The video you linked is for a flat disc, not a solid sphere.
The mr² formula is for a point mass, or for a collection of point masses all at the same radius in the rotation (such as a ring). All other formulae are obtained from this by suitable integration. In the case of ½mr² for a disc, that integration has been done and produced the factor ½. I am guessing that where you see the mr² in the solid sphere calculation the integration has not been done yet (and it will produce a different factor).