The study of rigid body dynamics, especially small-angle oscillations, is fundamental in physics education and practical problem-solving. When analyzing systems like rings and disks pivoted about a point, understanding how to determine the centre of mass (CM) and compute the moment of inertia is crucial. In the given example, the combined centre of mass lies directly below the pivot point and is calculated by considering the distances and masses of both the ring and the disk. For precise calculations, the parallel axis theorem is employed to find the moment of inertia relative to the pivot. This theorem allows us to shift the axis of rotation to one not passing through the object's centre of mass, a common requirement in rigid body rotational problems. Small-angle oscillations assume that the angular displacement is sufficiently small, enabling linear approximations of the equations of motion, which simplifies complex dynamics into solvable harmonic oscillator models. This assumption is widely used in physics to analyze pendulums, torsional oscillators, and other rotational systems. These concepts are integral in physics departments worldwide and learning to apply them accurately can sometimes be challenging, as suggested by the humorous "POV: your physics department sucks" expression. However, mastering these fundamental principles opens the door to advanced topics in physics and engineering, including vibration analysis, mechanical design, and robotics. Students preparing for exams like the Math SAT or engaging with physics problem-solving communities, such as "MathGPT" or "StudyTok," can benefit from clear, step-by-step explanations of centre of mass and moment of inertia calculations to bolster their understanding and improve problem-solving skills.