Solving a Physics Problem with Friction: Directions of Forces and Torques

[sapz]

Homework Statement

Hi

Im trying to solve this question, and I have the solution of it in front of me, but I really can't understand why its true.

The question is this: A bowl with radius R is given, it can't move. In it there's a ball with radius r << R, and a mass m.
Find... (lots of subclauses for the question)
A picture is attached.

My question is this:
Am I mixing the directions of the forces/torques?

F is the friction force. I think the torque equation should be I*w_dot=F*r.
and the equation for the tangential direction should be ma=F-mgs*sin(theta).

However, in the solution I see here the torque equation appears with a minus sign: I*w_dot=-F*r, which obviously changes the answers of the things I need to find.

Should it be with a minus? Or without? Thank you.

Homework Equations

The Attempt at a Solution

 

[haruspex]

F is the friction force. I think the torque equation should be I*w_dot=F*r.

This doesn't look right to me at all. If we take moments about the centre of the ball, there is indeed a torque Fr anticlockwise, and it does tend to reduce theta. But ##\ddot{\theta}## is not the angular acceleration of the ball.
There should be another angle in the picture - the angle through which the ball has rotated. It is related to theta by a simple formula.