Determine the moment of inertia of a pair of spheres, each possessing a mass m and radius r , placed in contact, about the tangent passing through their point of contact.

Step-by-step solution

The moment of inertia of a solid sphere of mass m and radius r about its diameter is given by I_ cm = 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 I₁ = I_ cm + mr^2 = 2 5 mr^2 + mr^2 = 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 = I₁ + I₂ =