The onset of instability induced by impulsive spin-down of the rigid-body flow placed in the gap between two coaxial cylinders is analyzed by using the energy method. In the present stability analysis the growth rate of the kinetic energy of the base state and also that of disturbances are taken into consideration. In the present system the primary flow is a transient, laminar one. But for the Reynolds number equal or larger than a certain one, i.e. Re ≥ ReG secondary motion sets in, starting at a certain time. For Re ≥ ReG the dimensionless critical time to mark the onset of vortex instabilities, τc, is here presented as a function of the Reynolds number Re and the radius ratio η. For the wide gap case of small η, the transient instability is possible in the range of ReG ≤ Re ≤ ReS. It is found that the predicted τc-value is much smaller than experimental detection time of first observable secondary motion. It seems evident that small disturbances initiated at τc require some growth period until they are detected experimentally.