Inertia matrix of a homogeneous cylinder

[influx]

Homework Statement

[/B]

Homework Equations

N/A

The Attempt at a Solution

What I am confused about is where they got the (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from? I am guessing that these came from the integral of y'^2 + z'^2 and x'^2 +y'^2 but I don't understand how this happened exactly? Could someone point me in the right direction?

Thanks

 

[Orodruin]

Did you try actually computing the integrals? What did youget? Please show your work.

 

[influx]

Did you try actually computing the integrals? What did youget? Please show your work.

What are y'^2 and z'^2? I don't know what to sub in for these (not sure what they represent?) so can't do the integral until I know them.

 

[Orodruin]

They are coordinates as shown in the figure.

 

[influx]

They are coordinates as shown in the figure.

I understand that but I mean how would one integrate those terms with respect to m? They aren't constants?

 

[Orodruin]

They depend on where in the body you are.

 

[influx]

They depend on where in the body you are.

Sorry for the late reply. I am still confused. I just want to know how they (mathematically) got from the left hand side to the right. I don't understand how integrating the left hand side yields the right hand side.

 

[Orodruin]

Use $dm = \rho \, dx'dy'$ and integrate over $x'$ and $y'$ with the appropriate boundaries. The value of $\rho$ in terms of the total mass $m$ can be inferred by computing the volume $V$ of the body and using $\rho V = m$.