In application as hyperthermia and nanowarming, power dissipation arises when the time-dependent magnetization $M(t)$ of an out-of-equilibrium system of nanoparticles lags behind the applied field $H(t)$. The key parameter governing this process is the relaxation time $\tau$ of the system, which induces a phase shift $\phi_n$ between $H(t)$ and every nth harmonic component of $M(t)$. In this work, we present an expression for $M(t)$ in terms of $\tau$ and the equilibrium magnetization, valid for any magnetic system exhibiting odd equilibrium response. From this calculation, we obtain a method for determining the effective $\tau$ of a MNPs sample directly from the experimental measurement of $M(t)$. Additionally, we demonstrate that the power dissipation (SAR: Specific Absorption Rate) of any magnetic sample under a sinusoidal field can be obtained from the first harmonic component of $M(t)$. As an illustrative application, we explore the variation of $\tau$ for magnetic MNPs in aqueous suspension during the melting process of the matrix. In this case, the change in $\tau$ can be understood as a result of the reorientation of the MNPs in the direction of the applied field as the matrix becomes liquid.
