Tension in a rotating loop of string

[AlwaysCurious]

Hello, I am a high school student trying to learn physics out of Kleppner and Kolenkow. Unfortunately, the solutions to some of the problems are not available online, nor is a solutions manual available, so I am unable to find out where I am wrong in some cases, such as this one. I would appreciate your clarification.

Homework Statement

A piece of string of length L and mass M is fastened into a circular loop and set spinning about the center of a circle with uniform angular velocity ω. Find the tension in the string.

Homework Equations

The answer in the book states that T = (M*L*ω^2)/(2π)^2, whereas the answer that I got was T = Mω^2/2π.

The Attempt at a Solution

Please the attached pdf - I have tried to write as clearly as possible, and am unable to find how my solution is incorrect. Thank you!

 

[haruspex]

A useful check is whether the various items you have expressions for are dimensionally correct. Your final answer has dimensions MT^-2, which is wrong for a force.
Apply the same test to your expression ΔθM/L.

 

[AlwaysCurious]

A useful check is whether the various items you have expressions for are dimensionally correct. Your final answer has dimensions MT^-2, which is wrong for a force.
Apply the same test to your expression ΔθM/L.

Thank you! I forgot to include the factor R in there, which gets me where I want to go.