A cone-shaped volume is contained by the region r less than 4 and θ less than 35 degrees, and has a density ρ(r) = 7 + 1.8 r.
a) What is the mass of the object?
b) What is the z-component of the center of mass of this object?
c) What is the moment of inertia of this object when rotated about the z-axis?
rho(r) = 7 + 1.8r
dm = rho(r)*dV = rho(r)*2*pi*(1-cos(35))*4*pi*r^2*dr/4*pi = (1.8r +
7)*2*pi*(1-cos(35))*r^2*dr
M = 2*pi*(1-cos(35))[0.45r^4 + 2.333r^3] = 300.412
Z component of com = [integrate]dm*r/[integrate]dm = (1.8r + 7)*2*pi*(1-cos(35))*r^3*dr/(1.8r + 7)*2*pi*(1-cos(35))*r^2*dr = 2*pi*(1-cos(35))[0.36r^5 + 1.75r^4]/2*pi*(1-cos(35))[0.45r^4 + 2.333r^3] = [0.36r^5 + 1.75r^4]/[0.45r^4 + 2.333r^3] = 3.0873 m
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