Question

There are four solid balls with their centres at the four comers of a square of side a. the mass of each sphere is m and radius is r. Find the moment of inertia of the system about (i) one of the sides of the square (ii) one of the diagonals of the square.

Answer 1

Given:

Four solid balls with their centers at the four comers of a square of side a.

Mass of each sphere is m and radius is r

To Find:

The moment of inertia of the system about

1. one of the sides of the square
2. one of the diagonals of the square.

Solution:

We know the moment of inertia of Solid Sphere about an axis passing through its center,

• Io = 2mr²/5

For any body , its moment of inertia about an axis at a distance 'a' from its center ,

• I = Io + ma²

Therefore, moment of inertia of the system about

1. one of the sides of the square

• 2 solid spheres are on the axis
• 2 solid spheres are at a distance 'a'

• I = (2mr²/5) x 4 solid spheres + ma² x 2 solid sphere not on the axis = 8mr²/5 + 2ma²

     2.  one of the diagonals of the square

About the diagonal ,

• 2 spheres are on the diagonal.
• 2 spheres are at a distance 'a/√2'
• I = 2mr²/5 x 4 solid spheres + m ( a/√2)² x 2 solid spheres  not on the axis
• I = 8mr²/5 + ma²/2 x 2 = 8mr²/5 + ma²