Question

Find the moment of inertia of a pair of spheres. each having a mass m and radius r, kept in contact about the tangent passing through the point of contact?

Answer 1

The moment of inertia of a sphere with respect to the center: I = 2/5 mr^2. Using the Parallel Axes Theorem one can calculate the moment of inertia with respect to the axis tangent to the sphere:
I = 2/5 mr^ + mr^2 = 7/5 mr^2.
Since in the problem above we have two spheres, the moment of inertia = 2(7/5 mr^2) = 14/5 mr^2 or we can write the fraction "your way" (14mr^2)/5

Answer 2

Given in the question :-

Along the tangent ,The two bodies of mass m radius r are moving.
Refer to the attachment 
Here moment of inertia of one body about the axis(XY) passing by tangent.
 I = (2/5)mr²  

Now, Moment of inertia of first axis parallel to the other axis
$r = I + mr^2$
r = (2/5) mr² + mr²
r = (7/5)mr² .

Hence the moment of inertia of a pair of sphere is (7/5) mr²



Hope it Helps :-)