how to find the moment of inertia of a disc of variable density

for ex -

find the MOI wrt Centre of a disk of surface density which varies as (A + Br) where r is distance from centre Total radius - R?

Dear Aakash

I=∫(dm)r²

you have to take an element and find dm for that

take ring of width dr at a distance r from the centre

area of elemental ring=2∏rxdr

mass of this ring= dm= (A+Br)2∏rxdr

I=∫(A+Br)2∏r³ dr limits from 0 to R

just intrgrate

I=2∏(AR⁴/4 + BR⁵/5)

All the best.                                                           

AKASH GOYAL

AskiitiansExpert-IIT Delhi

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take elemental ring of r & thickness dr ...

mass of this ring is dm = (A+Br)2pirdr

moment of inertia = dmr²

now integrate this , take limit from 0 to R ...

this will give the total moment of inertia .....