Accurate and comprehensive 3D sensing using LiDAR systems is crucial for various applications in photogrammetry and robotics, including facility inspection, Building Information Modeling (BIM), and robot navigation. Motorized LiDAR systems can expand the Field of View (FoV) without adding multiple scanners, but existing motorized LiDAR systems often rely on constant-speed motor control, leading to suboptimal performance in complex environments. To address this, we propose UA-MPC, an uncertainty-aware motor control strategy that balances scanning accuracy and efficiency. By predicting discrete observabilities of LiDAR Odometry (LO) through ray tracing and modeling their distribution with a surrogate function, UA-MPC efficiently optimizes motor speed control according to different scenes. Additionally, we develop a ROS-based realistic simulation environment for motorized LiDAR systems, enabling the evaluation of control strategies across diverse scenarios. Extensive experiments, conducted on both simulated and real-world scenarios, demonstrate that our method significantly improves odometry accuracy while preserving the scanning efficiency of motorized LiDAR systems. Specifically, it achieves over a 60\% reduction in positioning error with less than a 2\% decrease in efficiency compared to constant-speed control, offering a smarter and more effective solution for active 3D sensing tasks. The simulation environment for control motorized LiDAR is open-sourced at: \url{https://github.com/kafeiyin00/UA-MPC.git}.