= Solution
{c}
If $gf''(a)<\omega^2$, put
$$
\lambda^2=\frac{\omega^2-gf''(a)}{1+f'(a)^2}>0.
$$
The <linearized equation> becomes $\ddot\epsilon-\lambda^2\epsilon=0$, so
$$
\epsilon(t)=C_1e^{\lambda t}+C_2e^{-\lambda t}.
$$
Arbitrarily small initial perturbations can have $C_1\ne0$ and grow exponentially. Hence the strict <unstable equilibrium> condition is
$$
\boxed{f''(a)<\frac{\omega^2}{g}.}
$$
For the particular wire, this occurs precisely at $a=2j\pi b$, $j=1,2,\ldots$; the alternating radii $a=(2j-1)\pi b$ satisfy the <stable equilibrium> condition.
Back to article page
