A composite rigid body consists of a thin uniform rod of mass M and length 12 cm , and two identical uniform solid spheres, each of mass M and radius 15 cm . The centres of the spheres are attached to the two ends of the rod. The system rotates about an axis passing through the midpoint of the rod and perpendicular to its length. If the radius of gyration of the entire system about this axis is x cm , the value of x

Step-by-step solution

Let the axis of rotation be YY' . The moment of inertia of the thin rod about an axis passing through its midpoint and perpendicular to its length is: I_ rod = ML^2 12 = M(12)^2 12 = 12M For each solid sphere, the distance from its centre to the axis of rotation is d = L 2 = 6 cm . By the parallel axis theorem, the moment of inertia of one sphere about the axis YY' is: I_ sphere = 2 5 MR^2 + Md^2 = 2 5 M(15)^2 + M(6)^2 I_ sphere = 2 5 M(225) + 36M = 90M + 36M = 126M The total moment of inertia of the composite syst