= Solution
An <Euler strut> is a slender initially straight elastic column under axial compression. For a uniform <bending stiffness> $EI$ and a small transverse <displacement> $y(x)$, the second-order <potential energy> under a conservative compressive load $P$ is
$$
\Pi^{(2)}=\frac12\int_0^L\left[EI(y'')^2-P(y')^2\right]dx.
$$
The bending term is positive; shortening releases load potential. Stationarity yields $EIy^{(4)}+Py''=0$. Pinned ends impose $y(0)=y(L)=0$ and zero end <bending moments>, $y''(0)=y''(L)=0$. Substitution of $y=A\sin(n\pi x/L)$ gives
$$
P_n=\frac{n^2\pi^2EI}{L^2},\qquad
\boxed{P_{\rm cr}=\frac{\pi^2EI}{L^2}.}
$$
The straight configuration loses stability first in the longest admissible bending mode. The <second moment of area> $I=\int z^2dA$ is geometric, not a <mass> <moment of inertia>. Boundary conditions matter: a fixed-free column has effective <length> $2L$, while fixed-fixed ends have effective <length> $L/2$, changing the threshold by factors $1/4$ and $4$. Slenderness, elastic material response, alignment and the weak bending direction must be checked. Imperfections produce bending before the ideal threshold, and a short column can yield or crush first. Linear buckling theory does not predict an unlimited deflection or a complete postbuckling curve.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2010/paper-1-euler-strut.png]
{title=Euler strut instability}
{description=Ideal pinned modes and energy curvature.}
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