Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S7 / 2 / d / Solution

Put $X={\omega }^2$ and $S={\omega }_1^2+{\omega }_2^2+4{\Omega }^2$. Its equation is $X^2-SX+{\omega }_1^2{\omega }_2^2=0$. If both effective stiffnesses are positive, the sum and product of roots are positive and $S^2-4{\omega }_1^2{\omega }_2^2\ge 0$, so both roots are positive. They can coincide in a degenerate zero-rotation isotropic case; “two” need not mean distinct.

If ${\omega }_1^2<0<{\omega }_2^2$, the product is negative, so one root is negative. Its imaginary frequency gives a growing exponential displacement: centrifugal softening has overwhelmed one elastic restoring direction, making the central configuration unstable. Coriolis coupling mixes the two coordinates but does not remove this negative-product instability.