K&K Question 3.5 – Mass and Axle

1. The problem statement, all variables and given/known data

A mass m is connected to a vertical revolving axle by two strings of length l, each making an angle of 45 degrees with the axle, as shown. Both the axle and mass are revolving with constant angular velocity ω. Gravity is directed downward.

(a) Draw a clear force diagram for m
(b) Find the tension in the upper string, [itex]T_{up}[/itex] and the tension in the lower string, [itex]T_{low}[/itex].

2. Relevant equations

[itex]ma=mr\omega^2=ml\omega^2\sin{\varphi}[/itex]
Where phi is 45 degrees.

3. The attempt at a solution

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The equations for the x and y directions, respectively:
[itex]T_u\sin{\varphi}-T_d\sin{\varphi}=ml\omega^2\sin{\varphi}[/itex]
[itex]T_u\cos{\varphi}-mg-T_d\cos{\varphi}=ml\omega^2\sin{\varphi}[/itex]

Solving for the tension in the upper cord, I get this:
[itex]T_u=-mg[/itex]

Assuming that the tangent of 45 is 1. Am I on the right track or am I making a mistake right now? If it has to do with reference frames, I wouldn’t know what to do; I know that:
[itex]F_{apparent}=F_{true}+F_{fictitious}[/itex]
I don’t know how to apply it though. Should I continue solving for the tension in the lower rope, or do I need to correct something?

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