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Zero applied end force and the force balance ${\mathbf{n}}^{\prime }=0$ imply $\mathbf{n}(S)=0$ everywhere. The moment balance then reduces to ${\mathbf{m}}^{\prime }=0$.

Choose an untwisted straight configuration: all three directors are constant, ${\mathbf{r}}^{\prime }={\mathbf{d}}_3$, and $\mathbf{u}=0$. The constitutive equation giveswhich is constant in space and therefore satisfies the moment balance. The required end bending moment is consequentlyIts sign opposes the positive intrinsic curvature of an elastic filament. If the initial position and frame from part (ii) are retained, this solution is simply $\mathbf{r}(S)=S{\mathbf{e}}_3$ with ${\mathbf{d}}_i={\mathbf{e}}_i$; the clamp supplies the opposite reaction moment at the other end.

Local force balance and moment balance on a slender rod areIn a planar rod, $\mathbf{M}$ and $\mathbf{m}$ point perpendicular to the plane.

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 satisfyThe second equation is moment balance about the midpoint between the rails. ThusFull contact requires a nonnegative inner normal force. Its limiting value is zero, giving the toppling speed on level railsFor 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.

A vehicle with centre of mass midway between level rails separated by d, at height h, follows a turn of radius R. Vertical force balance and cross-sectional moment balance give inner rail reaction $mg/2-m{v}^{2}h/(Rd)$. Its first zero occurs at the displayed speed. This is a toppling criterion under maintained lateral contact, not a guarantee against sliding or suspension instability.