'09 iMac now randomly stopping with pinwheel for 1/2-1 min

Question

My trusty '09 iMac has started randomly stopping, with pinwheel, for 1/2-1 min. then continuing.  Happens from sign-on to internet to opening apps, anytime. Averages every 2-3 min.  Other computer, MacAir on network, has no problem.  Timemachine backs up to Airport Express tower.  Turning off/on has no effect.  OS X 10.9.1.

Answer 1

When you next have the problem, note the exact time: hour, minute, second.
If you have more than one user account, these instructions must be carried out as an administrator.
Launch the Console application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad. Click Utilities, then Console in the icon grid.
Make sure the title of the Console window is All Messages. If it isn't, select All Messages from the SYSTEM LOG QUERIES menu on the left. If you don't see that menu, select
View ▹ Show Log List
from the menu bar.
Each message in the log begins with the date and time when it was entered. Scroll back to the time you noted above. Select the messages entered from then until the end of the episode, or until they start to repeat, whichever comes first. Copy the messages to the Clipboard by pressing the key combination command-C. Paste into a reply to this message (command-V).
The log contains a vast amount of information, almost all of it useless for solving any particular problem. When posting a log extract, be selective. In most cases, a few dozen lines are more than enough. It is never necessary or helpful to post more than about 100 lines. "The more, the better" is not the rule here.
Please do not indiscriminately dump thousands of lines from the log into this discussion.
Important: Some private information, such as your name, may appear in the log. Anonymize before posting.