Physics question

We have been doing angular motion and moments of inertia in this module but I have no idea where to start with this question. Any advice would be appreciated 🙏

A child sits on a roundabout. The roundabout is a uniform disk with of mass M and radius R and is supported by a frictionless bearing. The child is a point particle of mass M/2. Ignore air resistance.

i. Initially, the child is sitting on the rim of the roundabout, which has angular speed 𝜔1. The child crawls to radius R/2 and sits there. Show that the angular speed of the roundabout is now
𝜔2 = 8/5 𝜔2

Conservation of kinetic energy (disc and child) and does the trick

I’ve just tried it and I think I’ve gone severely wrong. Instead of getting 8/5 I got sqrt154/11 lmao.

I used K = I𝜔^2
I for the roundabout being (MR^2)/2 and the child being 2(MR^2)/5 - since the child is treated as a point particle so I assume that’s a sphere??

Worked out K for each before and after getting
Before: 7/10 • (MR^2) • (𝜔1)^2
After: 11/20 • (MR^2) • (𝜔2)^2
Then setting them equal to each other.

This module is really confusing me overall so it wouldn’t surprise me if I’ve gone completely off the rails somewhere haha.

The child is a point mass, not a sphere, so you could treat its KE as normal so 1/2mv^2 where v is the (linear) speed so v=rw. Youd get the same if you did Iw^2 so I=mr^2. So you have
1/2 Iw^2 + 1/2 m(rw)^2
and obviously r chabges betweeb R and R/2.