  In this work, we study the spin density matrix element $\rho_{00}$ of the
$\phi$ in the decay $J/\psi \rightarrow \eta \pi \phi$. In previous studies, a
band around 1.4 GeV on the $\pi^0\phi$ distribution in Dalitz plot was reported
by the BESIII Collaboration. This structure may be caused by the production of
a resonance or the triangle singularity mechanism. We find that the predictions
of the spin density matrix elements of the final $\phi$ based on these
mechanisms show distinct features. Thus the measurement of the spin density
matrix elements of the $\phi$ in this reaction may offer an alternative way to
study the triangle singularity and to clarify the reaction mechanisms, i.e.
resonance production or kinematic effects. This work also shows the potential
of spin observables in studying kinematic singularities.
