Rotational Inertia

Rotational inertia I measures how hard it is to change a body's spin about a chosen axis. You build it piece by piece: I = Σ m_ir_i² for point masses, or I = ∫ r² dm for a continuous body, where every bit of mass is weighted by the square of its distance r from the axis. That r² weighting is the whole story: mass far from the axis counts far more than mass near it. To move to a parallel axis a distance d from the center of mass, the parallel-axis theorem adds one term: I = I_cm + Md².

Setting up the integral is where rotational inertia goes wrong. Dropping the r² weighting, or forgetting that dm = λ dx or σ dA, so the changing r can't be pulled outside. Misusing the parallel-axis theorem — subtracting the shift, starting from a non-center axis, or picking the wrong distance d. And treating I as a fixed property of the body, as if equal mass meant equal inertia or all the mass acted at the center. Keep the r² weighting, add Md² only from the center-of-mass axis, and ask which axis every time.