Local <force balance> and <moment balance> on a slender rod are
$$
\boxed{
\partial_s\mathbf F+\mathbf f=0,
\qquad
\partial_s\mathbf M+\mathbf t\times\mathbf F+\mathbf m=0
}.
$$
In a planar rod, $\mathbf M$ and $\mathbf m$ point perpendicular to the plane.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 3 / c / i / Solution

Moment balance equates the rate of change of angular momentum to applied torques. A static slender rod obeys $\partial_s\mathbf M+\mathbf t\times\mathbf F+\mathbf m=0$.


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