A uniform disk of mass \(M\) and radius \(R\) is mounted on a fixed, frictionless vertical axle passing through its center. An external horizontal force \(\vec{F}\) of constant magnitude is applied to a point on the rim of the disk at a varying angle \(\phi\) relative to the outward radial vector, where \(0 < \phi < \pi\) at all times. The axle exerts a horizontal contact force on the disk such that the center of mass of the disk remains stationary. Which of the following statements correctly describes the rotational kinetic energy of the disk as it rotates from rest through an angular displacement \(\Delta\theta\), and provides the correct justification?