Light propagating in a nondispersive medium is accompanied by a mass density wave (MDW) of atoms set in motion by the optical force of the field itself [Phys. Rev. A 95, 063850 (2017)2469-992610.1103/PhysRevA.95.063850]. This recent result is in strong contrast with the approximation of fixed atoms, which assumes that atoms are fixed to their equilibrium positions when light propagates in a medium and which is deeply rooted in the conventional electrodynamics of continuous media. In many photonic materials, the atoms carry the majority of the total momentum of light and their motion also gives rise to net transfer of medium mass with a light pulse. In this work we use optoelastic continuum dynamics combining the optical force field, elasticity theory, and Newtonian mechanics to analyze the angular momentum carried by the MDW. Our calculations are based on classical physics, but by dividing the numerically calculated angular momenta of Laguerre-Gaussian (LG) pulses with the photon number, we can also study the single-quantum values. We show that accounting for the MDW in the analysis of the angular momentum gives for the field's share of the total angular momentum of light a quantized value that is generally a fraction of. In contrast, the total angular momentum of the mass-polariton (MP) quasiparticle, which is a coupled state of the field and the MDW, and also the elementary quantum of light in a medium, is an integer multiple of. Thus, the angular momentum of the MP has coupled field and medium components, which cannot be separately experimentally measured. This discovery is related to the previous observation that a bare photon including only the field part cannot propagate in a medium. The same coupling is found for orbital and spin angular momentum components. The physical picture of the angular momentum of light emerging from our theory is fundamentally more general than earlier theoretical models, in which the total angular momentum of light is assumed to be carried by the electromagnetic field only or by an electronic polariton state, which also involves dipolar electronic oscillations. These models cannot describe the MDW shift of atoms associated with light. We simulate the MDW of LG pulses in silicon and present a schematic experimental setup for measuring the contribution of the atomic MDW to the total angular momentum of light.