= Solution
As $R\to0$, the short joining fibre is very stiff and the lower <normal mode> has $\phi_1\simeq\phi_2$. The spheres therefore behave as one torsional load with combined <moment of inertia> $2I$, suspended by the upper fibre of <torsion constant> $K_t/\ell_1$. Its <angular frequency> satisfies
$$
\boxed{\omega_-^2\to\frac{K_t/\ell_1}{2I}=\frac{\omega_0^2}{2}.}
$$
The high-frequency opposite-twist mode costs increasingly large elastic <energy> and moves to arbitrarily high frequency in the massless-fibre model.
As $R\to\infty$, the joining fibre is very compliant. In the higher <normal mode> the lower sphere scarcely moves: the amplitude ratio tends to $A_2/A_1\simeq-1/R\to0$. The upper sphere approaches the original single <torsion pendulum>, giving
$$
\boxed{\omega_+^2\to\omega_0^2.}
$$
The other mode is then a slow twist of the almost freely suspended lower sphere. These interpretations assume the formal limits do not invalidate the thin massless-fibre approximation.
Back to article page
