= Solution
Let $I$ be each sphere's <moment of inertia> about the fibre, and let $K_t$ be the common fibre <filament torsional stiffness>, so a length $\ell$ exerts restoring <torque> $-K_t\phi/\ell$. The single-sphere <torsion pendulum> obeys
$$
I\ddot\phi=-\frac{K_t}{\ell_1}\phi,\qquad
\omega_0^2=\frac{K_t}{I\ell_1}.
$$
For the two <coupled torsional pendulums>, the upper fibre has twist $\phi_1$ and the joining fibre has relative twist $\phi_2-\phi_1$. The joining fibre exerts equal and opposite <torques> on the two spheres:
$$
I\ddot\phi_1=-\frac{K_t}{\ell_1}\phi_1+\frac{K_t}{\ell_2}(\phi_2-\phi_1),\qquad
I\ddot\phi_2=-\frac{K_t}{\ell_2}(\phi_2-\phi_1).
$$
Using the measured $\omega_0$ gives
$$
\boxed{\ddot\phi_1=-\omega_0^2\left[\phi_1-\frac{\ell_1}{\ell_2}(\phi_2-\phi_1)\right],\qquad
\ddot\phi_2=-\omega_0^2\frac{\ell_1}{\ell_2}(\phi_2-\phi_1).}
$$
The fibre's own rotational inertia and damping are neglected, so its twists supply elastic <torques> without additional dynamic coordinates.
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