Determine the moment of inertia of a pair of spheres, each...

The moment of inertia of a solid sphere of mass $m$ and radius $r$ about its diameter is given by $I_{cm} = \dfrac{2}{5}mr^2$.

The tangent passing through the point of contact of the two spheres is at a perpendicular distance $r$ from the center of each sphere.

Using the parallel axis theorem, the moment of inertia of one sphere about this tangent is $I_1 = I_{cm} + mr^2 = \dfrac{2}{5}mr^2 + mr^2 = \dfrac{7}{5}mr^2$.

Since the system consists of two identical spheres, the total moment of inertia of the pair about the common tangent is $I = I_1 + I_2 = \dfrac{7}{5}mr^2 + \dfrac{7}{5}mr^2 = \dfrac{14}{5}mr^2$.