Mechanics Circular Motion Question

[girlwhoneedsmathhelp]

Homework Statement

Q : A small sphere S of mass m kg is moving inside a smooth hollow bowl whose axis is vertical and whose sloping side is inclined at 60◦ to the horizontal. S moves with constant speed in a horizontal circle of radius 0.6 m (attached diagram 1). S is in contact with both the plane base and the sloping side of the bowl (attached diagram 2).
(i) Given that the magnitudes of the forces exerted on S by the base and sloping side of the bowl are equal, calculate the speed of S.
(ii) Given instead that S is on the point of losing contact with one of the surfaces, find the angular
speed of S

Relevant Equations

a = v^2/r OR rw^2
F=ma

As I attempted this question I was very confused and felt my understanding of the topic is unclear and has many gaps. For both parts, I approached the question as I knew the steps I had to do to get the answer but had very little understanding as to why. I managed to get the correct answer for both parts but after looking at the mark scheme felt that my method was off.
If anyone could please help me understand the following questions I would appreciate it very much! Thank you!

Part (i)

*Pink represents mark scheme method
Part 2

However I am still confused on the diagram.

 

[PeroK]

In your "why is this wrong" diagram, you seem to have the force $mg$ coming out of the side of the bowl?

A couple of questions:

How many forces are acting on the sphere?

What does "smooth" tell you about the contact forces between the sphere and the bowl?

What is the condition for uniform circular motion (of a given speed and radius)?

PS I see @archaic has given you part of the answer below.

 

[archaic]

Maybe they meant something like this:

 

[haruspex]

The marker asked how you are defining X. To critique your work, we need to know.

 

[Lnewqban]

... As I attempted this question I was very confused and felt my understanding of the topic is unclear and has many gaps.
...
However I am still confused on the diagram.

Have you learned how to produce a free body diagram (for the ball in this case)?
Do you fully understand the terms and relations shown in the relevant equations of this problem?