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 Incoming links: Toppling speed on level rails

Past exam of the physics 1 paper of the University of Oxford / 2016 / 3 / Solution 2026-10-25

In the frame following the turn, the outward inertial force

m v^{2} / R

acts at height

h

and tends to unload the inner rail. The two upward normal forces satisfy

N_{out} + N_{in} = m g, \frac{d}{2} (N_{out} - N_{in}) = \frac{m v ^{2} h}{R} .

(1)

The second equation is moment balance about the midpoint between the rails. Thus

N_{in} = \frac{m g}{2} - \frac{m v ^{2} h}{R d}, N_{out} = \frac{m g}{2} + \frac{m v ^{2} h}{R d} .

(2)

Full contact requires a nonnegative inner normal force. Its limiting value is zero, giving the toppling speed on level rails

v_{0} = \frac{g R d}{2 h} .

(3)

For strictly positive contact force on both rails,

v < v_{0}

. This ideal result assumes level rails, negligible suspension motion and enough lateral contact force to follow the curve; lateral sliding could impose a different limit.

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