Why not just build wind turbines that will keep spinning (and generating) for days after the wind dies down?

Prof: Use wind turbines as flywheels to smooth output

Thus, according to the engineers, any power grid intending to use large amounts of wind normally needs to use expensive power storage units and/or backup generation to smooth out short-term spikes and troughs in output. But they believe that a useful amount of this can be avoided, by letting the turbine spin faster during gusts - so storing energy in the same way as a rotating flywheel - and only extracting as much power as the grid would like to have. Then, should a lull follow, the energy stored in the fast-spinning windmill can be extracted to bridge the gap.

It seems unlikely that this technique would be able to cope with the days-long calms which some critics of wind power have pointed out as the main weakness of the idea. The potential of wind turbines to fail catastrophically when spun too fast might also be a factor of concern. Nonetheless, the idea of dual-purposing the actual generating turbine itself as a storage flywheel is certainly an elegant one.