Any real attempt to use the momentum of the spinning disk to generate power would cause the disk to come to a halt very quickly. In essence, this is the principle behing regenerative braking of electric vehicles. Otherwise, you would have a perpetual motion machine.
However, there certainly were computers that used the momentum of a disk that was spinning to be able to write to it.
There were various generations and configuations of the DEC PDP-11, both in terms of the CPU and the disk drives. They had one common feature, that disk writes (and for that matter tape deck writes) were cached. The disk controller held an amount of RAM, writes to the drive by the CPU went into this RAM buffer. The buffer was written to the physical drive when it reached a defined threshold.
The configuration of these drives and the threshold at which writing occured could be changed by the System Manager, or other privileged account. In the process control uses, for which I used to use these computers, increasing the write frequency reduced the risk of data loss due to power failure at the expense of disk speed.
The drives also contained their own power supplies, with large capacitors. If the power failed, an interrupt would cause the disk controller to write the contents of the RAM buffer to the drive before the drive span down and the memory was lost.
There is good information about the configuration of various generations of PDP-11 disks here.
This was not a guaranteed fail-safe mechanism. Sometimes data was lost. However, the system did successfully prevent data loss most of the time.
The later MicroVAX computers also had this system and would very rarely lose data on power loss - in my experience.
(Of course, the VAX-FT could survive all sorts of failures - but that was a different animal.)
The system wasn't intended to hibernate the computer, in the way that a modern laptop can hibernate. It just prevented the loss or corruption of open files, including the page file and swap file.