Question

A compound disk of outside diameter 130 cm is made up of a uniform solid disk of radius 35.0 cm and area density 5.10 g/cm2 surrounded by a concentric ring of inner radius 35.0 cm , outer radius 65.0 cm , and area density 2.30 g/cm2 .

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

Answer 1

for the uniform solid disk of radius 35 cm:

moment of inertia=0.5*mass*radius^2

mass=density*surface area=density*pi*radius^2

then moment of inertia=0.5*pi*density*radius^4

using density=5.1 g/cm^2 and radius=35 cm,

moment of inertia =I1=0.5*pi*5.1*35^4

=1.2022*10^7 gram.cm^2

for the other ring:

moment of inertia=0.5*mass*(outer radius^2-inner radius^2)

mass=density*pi*(outer radius^2-inner radius^2)

moment of inertia=I2=0.5*pi*density*(outer radius^2-inner radius^2)^2

using the values,

moment of inertia I2=0.5*pi*2.3*(65^2-35^2)^2

=3.2515*10^7 gram.cm^2

so total moment of inertia=I1+I2

=4.4537*10^7 gram.cm^2