The forces acting on any seat are: the seat weight (mg), which is vertical, and the chain tension T, which acts along the chain and is slanted by θ, so it has a vertical component Ty = Tcosθ and a horizontal component Tx = Tsinθ.
If we project Newton's second law along y and x, we get:
Ty = mg. (along y)
Tx = m v^2/d. (along x)
since Tx must provide the necessary centripetal force to enable the rotation. Now, d is the radius of the circle the seat rotates along, and is equal to the radius R of the platform plus the projection of the chain L along x, i.e.
d = R + Lsinθ
Back to Newton's equation:
Tcosθ = mg
Tsinθ = mv^2/(R + Lsinθ)
If we divide the second equation by the first equation, we have tanθ = v^2/[g(R + Lsinθ)], so that:
v^2 = tanθ *g * (R + Lsinθ)
which is the squared of the seat velocity
