  We measure the energy relaxation rate of single- and few-layer molybdenum
disulphide ($\mathrm{MoS_2}$) nanomechanical resonators by detecting the
resonator ring-down. Recent experiments on these devices show a remarkably low
quality (Q)-factor when taking spectrum measurements at room temperature. The
origin of the low spectral Q-factor is an open question, and it has been
proposed that besides dissipative processes, frequency fluctuations contribute
significantly to the resonance line-width. The spectral measurements performed
thus far however, do not allow one to distinguish these two processes. Here, we
use time-domain measurements to quantify the dissipation. We compare the
Q-factor obtained from the ring-down measurements to those obtained from the
thermal noise spectrum and from the frequency response of the driven device. In
few-layer and single-layer $\mathrm{MoS_2}$ resonators the two are in close
agreement, which demonstrates that the spectral line-width in $\mathrm{MoS_2}$
membranes at room temperature is limited by dissipation, and that excess
spectral broadening plays a negligible role.
