Question

A compound disk of outside diameter 160 cm is made up of a uniform solid disk of radius 32.0 cm and area density 4.70 g/cm2 surrounded by a concentric ring of inner radius 32.0 cm , outer radius 80.0 cm , and area density 1.80 g/cm2 .

Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

Please make sure for the answer to be correct and show proof.

Answer 1

here,

r1 = 32 cm = 0.32 m

r2 = 80 cm = 0.8 m

density of inner ring , pi = 4.7 g/cm^2 = 47 kg/m^2

density of outer ring , po = 1.8 g/cm^2 = 18 kg/m^2

the area of inner ring , Ai = pi * r1^2 = 0.3215 m^2

mass of inner ring , Mi = Ai * density of inner ring

Mi = 0.3215 * 47 = 14.6875 kg

the area of outer ring , Ao = pi * ( r2^2 - r1^2) = 1.688 m^2

mass of outer ring , Mo = Ao * density of outer ring

Mo = 1.688 * 18 kg = 30.384 kg

the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center, I = 0.5 * Mi * r1^2 + 0.5 * Mo * ( r2^2 - r1^2)

I = 0.5 * ( 14.6875 * 0.32^2 + 30.384 * ( 0.8^2 - 0.32^2))

I = 8.92 kg.m^2