The physics
Round a vertical circle, the speed changes. Gravity slows the body on the way up. It speeds it up on the way down.
On a smooth track, no energy is lost. So the speed at any height comes from the drop.
½ v² = g (h − y)
At the top, the track pushes down and gravity pulls down. Together they give mv²/R. The slowest safe pass has zero push there, so v² = gR.
top: v² = g R bottom of a string circle: v² = 5 g R loop track: h = 2.5 R
A string can only pull. At the bottom, it must pull up against gravity too. So its pull is largest there.
bottom: T = m g + m v² / R top: T = m v² / R − m g
A ring can push a bead either way. Its push is N = mg(2 − 3 cos θ), with + toward the centre.
On a board tilted at α, only g sin α acts along the board. Use it in place of g.