Treat the carriage as translating along the circular track, neglecting wheel rotational kinetic energy and its finite size. At the top, gravity and the inward normal force supply the centripetal acceleration:

m g + N = \frac{m v _{top}^{2}}{r} .

For unilateral contact,

N \geq 0

, so the limiting speed is

v_{top}^{2} = g r

. Conservation of energy from rest at height

h

then gives

m g h = 2 m g r + \frac{1}{2} m g r, h_{m i n} = \frac{5}{2} r .

(2)

This top contact condition is the most restrictive: at angle

ϕ

measured from the bottom,

N / (m g) = 2 h / r - 2 + 3 cos ϕ

, whose minimum occurs at the top. A mechanically captive carriage able to sustain an outward rail reaction has a different constraint; the result here uses the usual requirement of positive normal contact.