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Inertia matrix of a homogeneous cylinder

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1. Homework Statement

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2. Homework Equations
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3. 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

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

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They are coordinates as shown in the figure.
 
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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

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They depend on where in the body you are.
 
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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

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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##.
 

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