Power and/or heat are the first two things that come to mind.
Problem: The computer comes on at 9:00 AM, a power interruption occurs any time after that and the computer shuts down.
Solution: Plug the computer into a UPS (Uninterruptable Power Supply).
Problem: The computer comes on at 9:00 AM, overheats sometime after that and shuts down.
Solution: Thoroughly clean the computer and fans, replace the thermal grease on the CPU heat sink, and while you're in there, might as well reseat all cable and card connectors several times to clean the contacts.
Problem: The computer comes on at 9:00 AM, the power supply is flaky/failing/lazy/overloaded and the computer shuts down.
Solution: Replace the power supply.
If you think the problem is the operating system (OS), boot and run the computer from a Live CD/USB for a while.
A Live CD/DVD or USB flash drive contains a complete bootable computer operating system (OS). When you boot a computer from it, the OS loads into, and then runs from memory instead of a hard drive. This allows you to run an OS without installing it or making any changes to a computer's current configuration.
To create one, download an ISO file containing the OS you want to use, (such as Ubuntu, Slax or Zorin), then use something like ISO Recorder to burn it to a CD/DVD or USB key.
Now boot your computer from this disc/key and your computer will be running that OS instead of the one on its hard drive.
When you're done, simply reboot your PC from the hard drive.
If the problem persists even when running a different OS, then the issue is probably hardware related.