Perfectly inelastic collision moving and spinning wheels

I understand the perfectly inelastic collision of spinning wheels. Like below model.

It's a simple application of conservation of angular momentum.

We also understand the perfectly inelastic collision of two mass points from high school.

Problem

But here, what if we combine above two models. We have two moving and spinning wheels. Each has a mass of $$m_1$$ and $$m_2$$ (distributed linearly), each as a moving velocity of $$v_1$$ and $$v_2$$, radius of $$r_1$$ and $$r_2$$, angular velocity $$\omega_1$$ and $$\omega_2$$. What happens after the perfectly inelastic collision?

Can we argue their linear momentum and "angular momentum" conserve individually? How do we model this case?

This is not a homework problem, just a problem I am thinking of myself.

[EDIT]. I am thinking of modeling the problem like below. Let the reference frame be the center of wheel 1. Then wheel 1 has a angular momentum $$I_1 \omega_1$$, no linear momentum. Wheel 2 comes at colliding with wheel 1 with velocity $$v = v_2-v_1$$, angular momentum $$I_2 \omega_2$$

At the moment of the collision, we decompose $$v$$ into $$v_a$$, which is along the rotating direction of $$m_1$$, $$v_b$$, which is along the radius direction. Then wheel 2 has another angular momentum component of $$m_2 v_a (r_1+r_2)$$. The angular momentum of $$I_1 \omega_1 + I_2 \omega_2 + m_2 v_a (r_1+r_2)$$ should conserve. and the linear momentum of $$m_2 v_b$$ should conserve.

Is that right?

• The term for "moving" momentum in this context is "linear momentum". Each type of momentum won't necessarily conserve independently. The easiest example to start with is drop a spinning ball straight down - what happens? Dec 29, 2021 at 15:57
• Here's some cool demonstrations btw - not your exact question but should help illustrate the topic: youtube.com/watch?v=2ugSbej4wqQ Dec 29, 2021 at 16:03
• @SeñorO I made more edits with my thoughts. Is that idea correct? Dec 29, 2021 at 16:14

You can determine the moment of inertia as the sum of the individual moments of inertia of each disk plus $$m_i r_i^2$$, where $$r$$, for each disk, is the distance from its own center of mass the the center of mass of the combination.