When working with rotating objects, the interplay of mass and velocity used to get momentum, kinetic energy, and other values becomes treacherously complicated. Take a spinning disk with a uniform density as an example. If you want to know the kinetic energy of the spinning disk, you can't just take the mass of the disk and multiply it by some simple velocity squared and then take the half. The mass at the center of the disk will be hardly moving at all and will have very little velocity or kinetic energy. Contrast this with the mass at the perimeter of the disk, which is moving very rapidly and has a high kinetic energy. To make matters worse, there's more mass at the perimeter of the disk than at the center, so you can't simply take an average. You're forced to add up the contribution of each little part as follows:

K_i = m_iv_i²/2 = m_ir_i²ω²/2
K = Σm_iv_i²/2 = Σm_ir_i²ω²/2 = [ω²/2]Σm_ir_i²

K = ω² *

⌠ r²*dM ⌡ 2

= ω² *

⌠ (M/V)r²*dV ⌡ 2

Here M is the total mass of the object, ω is the angular velocity, and r is the radius of a specific mass at a specific point. The quantity that sticks out in the final equation is the integral. This quantity is known as the Moment of Inertia and is usually denoted by a capital I. We can define it as follows:

I ≡ Σmiri² =

⌠ r²*dM ⌡

=

⌠ (M/V)r²*dV ⌡

The units for this value, I, are mass*distance² and the value is very useful for simplifying a number of calculations. For example, the equation for the kinetic energy of a rotating object, using this value, I, for the moment of inertia becomes simply:

K = Iω²/2

The Moment of Inertia then is analogous to the mass in the non-rotational equation for kinetic energy. It can be thought of as a sort of adjusted mass for the rotating object.

As an example, we can calculate I for our rotating disk using the above equations. We'll take R to be the radius of the disk. This gives:

I = ⌠ (M/V)r²*dV ⌡  = ⌠R ⌠2*π (M/A)r² * r*dθ*dr ⌡0 ⌡0  = 2*π ⌠R (M/A)r³*dr ⌡0 I = 2*π*(M/A) ⌠R r³*dr ⌡0  = 2*π*(M/A)*R4/4  = π*M/(π*R²)*R4/2 I = M*R²/2

This can now be used for any disk of any mass and at any angular velocity assuming the disk is spinning like a record or CD. However, if the disk were spinning around a different axis we would need a different moment of inertia. In order to find the Moment of Inertia about an arbitrary axis, a Moment of Inertia Tensor can be calculated.