I feel like a div. Quite simply I've forgotten how to use integration to calculate a moment of inertia (MOI). Ok I want to calculate the MOI of say a solid disk, about an axis perpendicular to the plane of the disk, running through its center of mass.(adsbygoogle = window.adsbygoogle || []).push({});

I remember that for any point particle in the disk its MOI is going to be:

mr^{2}

I can show this because if we consider a point mass at some distance r from a point of rotation, it will still subscribe to F=ma in the tangential direction. Knowing that:

T= I [alpha]

WhereTis the torque, I the MOI, and alpha the angular acceleration.

T=Fr

Therefore:

T=mar

As

[alpha] = a / r

mar=I a/r

I=mr^{2}

Hence I'm certain of that. Thus I can appreciate that for a solid disk I am looking at a summation of the MOI's between the axis and the edge of the disk. Hence I need an integration factor such that;

I_{T}= Integ^{r}_{0}{ dm/dr r^{2}} dr

I think. Yes? Hence I need an expression for the rate of change of mass with respect to radial displacement. Hence I need an area integral, and this is where I'm left blank, because I have no idea how to express the circle's mass with respect to increasing radius. Unless...

I make my integral:

I_{T}= Integ^{2PI}_{0}{ dm/dr r^{2}} d[theta]

Where dm/dr = r^{2}PI

Therefore dm = 1/3 r^{3}PI |^{r}_{0}hence:

I_{T}= Integ^{2PI}_{0}{ 1/3 r^{3}PI r^{2}} d[theta]

= 1/3 PI r^{5}[theta] |^{2PI}_{0}

= 2/3 PI^{2}r^{5}

Which I'm certain is wrong. I'm not asking why it's wrong. But how should I construct my dm/dr and integrate that through with reasoning why that's a good method. Because I have a feeling I'm forgetting something fundamental in this part of the calculation.

Cheers,

Haths

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Rotational Dynamics: Calculating a Moment of Inertia

**Physics Forums | Science Articles, Homework Help, Discussion**