  Recent progress shows that a surface-acoustic-wave (SAW) cavity can not only
induce quantum acoustic dynamics but also can form optomechanical-like systems.
Its operating frequencies in the microwave band make it resistant to the
thermal noise of surrounding environments, while its radiation-pressure
couplings make it susceptible to weak forces. Based on these advantages, we
propose a gyroscope comprising coupled microwave-SAW cavities. In this paper,
we systematically consider the three indices including range, signal-to-noise
ratio, and sensitivity, which are the most important to gyroscopes but only
partially considered in existing works. Additionally, we establish the
fundamental limits of sensitivity when the quantum input is in the vacuum state
and the squeezed vacuum state. We find that squeezing improves sensitivity and
can surpass the standard quantum limit. However, this improvement can only
reach up to $\sqrt{2}/2$ even as the squeezed parameter approaches infinity,
which is rarely noted in recent works. Finally, we also offer analytical
constraints for cooperativity and squeezed parameters. These constraints can be
utilized to design gyroscopes based on coupled cavities in experiments.
