Mass, Centers of Mass, and Mass Moments of Inertia

Define density function rho[x,y]:

rho[x_,y_]:=1+Sqrt[x^2+y^2];

Compute Mass M:

Mass =(∫_0^R)(∫_0^(π/2))rho[r*Cos[θ],r*Sin[θ]]r\[DifferentialD]θ\[DifferentialD]r

Simplify[Mass,R>0]

Compute x coordinate of center of mass xbar:

xbar = (1/Mass)*(∫_0^R)(∫_0^(π/2))r*Cos[θ]rho[r*Cos[θ],r*Sin[θ]]r\[DifferentialD]θ\[DifferentialD]r

FullSimplify[xbar, R>0]

Compute y coordinate of center of mass ybar:

ybar = (1/Mass)*(∫_0^R)(∫_0^(π/2))r*Sin[θ]rho[r*Cos[θ],r*Sin[θ]]r\[DifferentialD]θ\[DifferentialD]r

FullSimplify[ybar, R>0]

Compute moment of inertia Ixx of Omega about the x axis:

Ixx = (∫_0^R)(∫_0^(π/2))(r*Sin[θ])^2rho[r*Cos[θ],r*Sin[θ]]r\[DifferentialD]θ\[DifferentialD]r

Simplify[Ixx,R>0]

Compute moment of inertia Iyy of Omega about the y axis:

Iyy = (∫_0^R)(∫_0^(π/2))(r*Cos[θ])^2rho[r*Cos[θ],r*Sin[θ]]r\[DifferentialD]θ\[DifferentialD]r

Simplify[Iyy,R>0]

Compute moment of inertia of Omega about the z axis.  This is the “polar” moment of inertia.

Ir = (∫_0^R)(∫_0^(π/2))(r)^2rho[r*Cos[θ],r*Sin[θ]]r\[DifferentialD]θ\[DifferentialD]r

Simplify[Ir,R>0]