the moment of inertia
Moment of inertia is rotational mass — a measure of how hard it is to change an object's spin. Ordinary mass tells you how stubborn something is about starting or stopping straight-line motion; the moment of inertia tells you how stubborn it is about starting or stopping a turn. A figure skater feels this directly: arms stretched out, they are hard to spin up and slow to change; arms pulled in, they whirl easily. Same skater, same mass — different moment of inertia.
The twist (pun intended) is that it depends not just on how much mass there is, but on how far that mass sits from the axis. Precisely, I = sum of m_i r_i^2 — add up each little piece of mass multiplied by the square of its distance from the axis; the units are kg m^2. Because the distance is squared, mass far from the axis counts far more. Standard shapes have standard answers: a point mass I = m r^2; a hoop about its centre I = M R^2; a solid disk I = 1/2 M R^2; a solid sphere I = 2/5 M R^2; a rod about its centre I = 1/12 M L^2.
The moment of inertia is the character that plays the role of mass everywhere in rotation: in the rotational kinetic energy 1/2 I omega^2, in Newton's law for rotation tau = I alpha, and in the angular momentum L = I omega. Master it and the whole rotational world falls into a familiar pattern.
A solid disk of mass 2 kg and radius 0.3 m spinning about its centre has I = 1/2 M R^2 = 1/2 x 2 x 0.3^2 = 0.09 kg m^2. Move the same mass out into a thin hoop of the same radius and I doubles to M R^2 = 0.18 kg m^2.
Mass far from the axis matters most — the r^2 makes distance count double.
Moment of inertia is not a fixed property of an object; it changes with the chosen axis. The same rod has different I about its centre than about its end (see the parallel-axis theorem).