Maximum angular velocity from a swing carousel

The problem is to find out the maximum angular velocity that the swing carousel can spin at before the chain holding the passenger breaks. The maximum tension that the chain can withstand is known (T). The radius from the centre of the ride to the chain when stationary is known (r). The maximum mass that a passenger can be is known (m). The length of the chain is known (l). Given this information, derive a formula that will tell us the maximum angular velocity before the chain breaks.

2. Relevant equations

At first, I thought this was easy where T = mgcosθ + mω2rsinθ . But then I realised there were several problems with this firstly that we don’t know theta and so can’t work out ω when θ is also unknown. But also that using r is incorrect as the true radius that the passenger is revolving round is larger due to the centripetal force making them swing out.

I derived this formula for the necesarry condintions for the passenger being in equilibrium mgsinθ = mω2cosθ(r + lcosθ) . However it doesn’t help me much at this stage. The problem is I know that θ and ω are dependant in that if I determine an angular velocity, I will get a corresponding value for θ when the system is in equilibrium.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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