A thin rod of length \(L\) and unknown rotational inertia \(I\) is pivoted at one end and can rotate on a horizontal, frictionless surface. A small sphere of mass \(m\) is attached to the other end of the rod. A constant force of magnitude \(F\) is applied to the rod at its midpoint, directed perpendicular to the rod. The system is observed to rotate with an angular acceleration of magnitude \(\alpha\). Which of the following expressions represents the rotational inertia \(I\) of the rod about the pivot?