How do you calculate the moment of inertia given frequency of oscillation?
You are going to need more than just the frequency. You will need to know the restoring force.
I think the problem you are describing is a body hung on a pivot through a point on it and the body is displaced by a small angle θ with gravity acting on the centre of mass to restore equilibrium. In the limit of small θ the body executes simple harmonic motion.
For angular displacements the equivalent of F=ma is τ = I d2θ /dt2 where τ is the torque about the pivot of gravity (mg) acting at the centre of mass and I is the moment of inertia about the pivot.
Put this together and use the approximation sin θ ≈ θ for small θ and as you would for a simple pendulum solve to find ω (= dθ/dt)
since ω (= 2 [itex]\pi[/itex] f) you have solved the problem
Hope this makes sense and helps
Separate names with a comma.