Why shape changes the moment of inertia
How mass distribution, not just total mass, sets an object's resistance to spinning, and where the 1/2, 2/5 and 2/3 factors come from.
Distribution matters more than mass
Two objects can weigh exactly the same yet be very different to spin up. The reason is that moment of inertia adds up each parcel of mass multiplied by the square of its distance from the axis. Because the distance is squared, mass out near the rim counts for far more than mass near the centre. That single fact explains why a figure skater speeds up when they pull their arms in, and why a flywheel is built with a heavy rim rather than a solid slug of metal.
Where the coefficients come from
The fractions in front of m r^2 are the result of integrating that squared distance over the whole body. A thin hoop has every gram at radius r, so the coefficient is 1 and I = m r^2. A uniform disk averages the contribution from the centre out to the rim, halving it to 1/2 m r^2. A solid sphere packs even more mass near the middle, dropping the factor to 2/5, while a hollow spherical shell, with everything at the surface, sits higher at 2/3. Reading the coefficients as an average of squared distance makes their ordering easy to remember.
Rods, the centre axis and the parallel-axis theorem
A thin rod spun about its middle gives 1/12 m L^2, but pivot the same rod at one end and the value jumps to 1/3 m L^2, four times larger. The parallel-axis theorem ties the two together: moving the axis a distance d from the centre of mass adds m d^2. For a rod that shift is L/2, and 1/12 m L^2 plus m (L/2)^2 equals 1/3 m L^2 exactly. This calculator gives the two standard rod cases directly, and the theorem lets you reach any other parallel pivot from there.
Putting the number to work
Once you have I in kg x m^2 it feeds every rotational calculation. Angular momentum is I times angular velocity, so a larger I means more twist is stored for the same spin rate. Rotational kinetic energy is one half I omega squared, which sets how much energy a flywheel banks. And Newton's second law for rotation, torque equals I times angular acceleration, tells you how hard you must push to spin something up. Getting I right is the first step in all three.