A uniform thin rod of rotational inertia \(I\) is pivoted at one end and rotates in a horizontal plane about a vertical axis through the pivot. The rod is immersed in a viscous fluid that exerts a retarding torque of magnitude \(\tau = b\omega\) on the rod, where \(b\) is a positive constant and \(\omega\) is the instantaneous angular speed. At time \(t = 0\), the rod has an initial angular speed \(\omega_0\) and rotational kinetic energy \(K_0\). Which of the following expressions correctly represents the kinetic energy \(K\) of the rod as a function of its angular displacement \(\theta\) from its initial position before it comes to rest?