The moment of inertia may be defined as,
I = sum m_ir_i^2I=∑mir2i and if the system is continuous, then
I = int r^2dmI=∫r2dm
If rhoρ is the mass density then, dm = rhodVdm=ρdV where dVdV is an elementary volume.
Therefore,
I = int rhor^2dVI=∫ρr2dV
Here we make the assumption that the mass density is constant
Therefore,
I = rhoint r^2dVI=ρ∫r2dV
In order to evaluate this integral, we exploit the cylindrical symmetry of the system and employ cylindrical polar coordinates.
Thus, moment of inertia along axis of cylinder which is assumed to be the zz axis is,
I_z = rho intintint r^2dVIz=ρ∫∫∫r2dV
Limits of ss is from 00 to RR where RR is the radius. Limits of thetaθ are from 00 to 2pi2π and limits of zz are from 00 to hh where hh is the height of the cylinder.
implies I_z = rho int _0^(2pi) d theta int_0^R s^3ds int_0^h dz⇒Iz=ρ∫2π0dθ∫R0s3ds∫h0dz
Where in this case, r^2 = s^2r2=s2 and dV = sds d theta dzdV=sdsdθdz in cylindrical coordinates.
implies I_z = (rho2piR^4)/4h⇒Iz=ρ2πR44h
But, rho = M/(piR^2h)ρ=MπR2h where MM is total mass.
Therefore, I_z = (MR^2)/2Iz=MR22
