Manipulating arbitrary objects in unstructured environments is a significant challenge in robotics, primarily due to difficulties in determining an object's center of mass. This paper introduces U-GRAPH: Uncertainty-Guided Rotational Active Perception with Haptics, a novel framework to enhance the center of mass estimation using active perception. Traditional methods often rely on single interaction and are limited by the inherent inaccuracies of Force-Torque (F/T) sensors. Our approach circumvents these limitations by integrating a Bayesian Neural Network (BNN) to quantify uncertainty and guide the robotic system through multiple, information-rich interactions via grid search and a neural network that scores each action. We demonstrate the remarkable generalizability and transferability of our method with training on a small dataset with limited variation yet still perform well on unseen complex real-world objects.