Check out the problem, which was to find the angle at which you can pull on the string from a spool without rolling the spool forwards or backwards.
If you pull at (straight ahead), the spool will roll forward. If , (straight up) your intuition should tell you the spool rolls backward. Somewhere in between, there’s a critical angle where the spool doesn’t roll at all.
Suppose we’re pulling at that critical angle, and the spool isn’t moving (so long as we pull lightly). There better not be any net torque on the spool.
You could go about mindlessly crunching away at this, which is what I originally did. But you can save yourself some effort by carefully choosing the axis about which you’ll measure the torque. There are two relevant forces on the spool – friction from the table and tension from the string. Choose the axis where the spool touches the table. That way, the torque from friction is automatically zero, and we just need to worry about torque from tension.
The torque from tension will vanish if the line along which the tension is applied intersects the axis. Geometrically, that looks like this:
To get the condition on , slap a triangle on there whose sides you can figure out.