Question

2. Consider a thin disk composed of two homogenous halves connected along a diameter of the disk. One half has the density ρ and the other has density 2ρ.

a) Find the position of the center-of-mass of the disk.
b) Consider an axis which is perpendicular to the plane of the disk and goes through the geometric center of the disk. Calculate the moment of inertia of the disk with respect to this axis.
c) Find the moment of inertia of the disk for an axis which is perpendicular to the plane of the disk and goes through the center of mass of the disk. Use your results from (a) and (b) for this purpose.

Answer 1

The disc is composed of two homogeneous halves of densities and . The figure is given below.

a. Here the figure has a symmetry about y axis, so the centre of mass must lie on y axis.

The center - of - mass is given by,

where gives the total mass of the disc.

we know where a is the area of the disc.

So, center of mass becomes,

Representing the integral in polar form.

elemental area becomes   

and

here R in the limit is the radius of the disc.

Therefore, the center of mass is at a distance in the y axis,

The coordinate of center of mass is   .

b. Moment of inertia of disc for an axis perpendicular to the plane of disc and passing through the geometric center is given by,

     

, Which is the required moment of inertia.

c. Moment of inertia of disc for an axis perpendicular to the plane of disc and passing through center of mass can be found by parallel axis theorem.

ie.  

Where m (total mass of disc) is given by,

Which is the required moment of inertia.