= Solution
{c}
Zero applied end <force> and the <force balance> $\mathbf n'=\mathbf0$ imply $\mathbf n(S)=\mathbf0$ everywhere. The <moment balance> then reduces to $\mathbf m'=\mathbf0$.
Choose an untwisted straight configuration: all three directors are constant, $\mathbf r'=\mathbf d_3$, and $\mathbf u=\mathbf0$. The <constitutive equation> gives
$$
\mathbf m=-EI_2\hat u\mathbf d_2,
$$
which is constant in space and therefore satisfies the <moment balance>. The required end <bending moment> is consequently
$$
\boxed{m_2=-EI_2\hat u.}
$$
Its 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.
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