Comparing 2 formula derivations

[Vykan12]

Here is how my book goes about defining the moment of inertia.

To begin, we think of a body as being made up of a large number of particles, with masses m_1, m_2, ... at distances r_1, r_2, from the axis of rotation. [...] The particles don't necessarily all lie in the same plane, so we specify that r_i is the perpendicular distance from the axis to the ith particle.

Now here's how my book goes about deriving an expression for the angular momentum of a rigid body.

We can use eq (10.25) to find the total angular momentum of a rigid body rating about the z axis with angular speed w. First consider a thing slice of the body lying in the xy plane.

From this the book derives that L = Iw

We can do the same calculation for other slices of the body, all parallel to the xy-plane. For points that do not lie in the xy-plane, a complication arises because the r vectors have components in the z direction as well as the x and y directions.

My question is in regards to the bolded part. How come in one derivation we assume r_i is a measure of the perpendicular distance from the axis of rotation, and in the other we made no such distinction, but rather considered r_i a 3-D vector?

 

[Vykan12]

My physics teacher answered this for me.

The only reasons the derivations are different regarding r_i is because inertia isn't a vector whereas angular momentum is.

 

[charng]

Both of these derivations are valid and serve different purposes. In the first derivation, we are calculating the moment of inertia, which is a measure of how resistant an object is to changes in its rotational motion. Therefore, we only need to consider the perpendicular distance from the axis of rotation because that is the distance at which the particles are rotating. This is a simplified approach that allows us to easily calculate the moment of inertia for a rigid body.

In the second derivation, we are calculating the angular momentum, which is a measure of the rotational motion of an object. In this case, we need to consider the full 3-D vector for the distance from the axis of rotation because the particles can have components in the z direction as well as the x and y directions. This allows us to accurately calculate the angular momentum for a rigid body that may not be confined to a single plane.

In summary, both derivations are valid and serve different purposes. The first one is a simplified approach for calculating the moment of inertia, while the second one is a more comprehensive approach for calculating the angular momentum.