No problem Pat, just needed to know what we are dealing with.
There are some things that will fix this that require no trips to Apple and the tools are already built-in to the iMac.
First, if you have a USB type keyboard of any type, including Windows, it would help to use it. If you do not then wireless will work but timing is critical.
Second is repairing permissions, this should have been done before doing any update but it is something that Apple does not tell you.
The above is a PDF that explains and shows how to do it, save a copy or print it out for future use.
Next is a pram reset, this is where a USB keyboard is handy.
To do a PRAM reset, you need to restart your machine and hold down these 4 keys all together at the same time.
‘Option’, ‘Command (Cloverleaf or Apple)’, ‘R’ and 'P' all together at the same time.
Keep them down until you hear 4 startup chimes, this is the sound that your computer makes when it starts up, it is like a boonngg. The screen should also cycle from white to black and then back to white along with the chimes.
The sound should also get louder.
If you have a wireless keyboard then the procedure is a little bit different.
Use the very same keys but wait until the chime and immediately press and hold those keys. I recommend getting your fingers ready because timing is critical.
It can take many trys to get it, any USB keyboard is preferable.
If using a Windows keyboard do the following.
Substitute the Control key for the Command key and Alt for Option, all others stay the same.
This resets all of the system buses on your computer, USB, FireWire, display, ethernet and your keyboard/mouse/trackpad.
This will correct many faults but not necessarily yours.
Try that, I think that the beachball will go away. Let me know please.