<p>We consider a linear initial-boundary-value problem generated by the problem of small motions of a spatial pendulum with a cavity partially filled with a homogeneous incompressible liquid, in the case where the moment of friction forces in the spherical hinge is proportional to the angular velocity. We propose an operator interpretation of the problem and prove a theorem on the strong solvability of the Cauchy problem on a finite time interval. For the corresponding spectral problem, the discreteness of the spectrum and its localization in a strip are proved, power asymptotics of eigenvalues are found, and the summability of the system of eigenvectors is established by the Abel–Lidsky method.</p>