The physics
A car turns only if a force pulls it toward the curve's centre. It needs a pull of mv²/r.
On a flat road, only friction can give that pull. Friction gives at most μmg. So the top speed has μmg = mv²/r.
flat road: vmax = √(μ g r)
Tilt the road toward the centre. Now the normal force N leans inward too. Its sideways part pulls the car round.
At one speed, N alone does the whole job. Then the car needs no friction at all.
no friction: tan θ = v² / (r g)
Go faster, and the car starts to slide up. Friction then acts down the slope. Go slower, and friction acts up the slope. At each limit, friction is μN.
vmax² = r g (sin θ + μ cos θ) / (cos θ − μ sin θ) vmin² = r g (sin θ − μ cos θ) / (cos θ + μ sin θ)
A swing ride works the same way. The chain's pull T holds the chair up and pulls it round. The chair moves round a circle of radius L sin θ.
T cos θ = m g T = m ω² L ω² = g / (L cos θ)